Finite Group Databasebeta

← #29737553

Subgroups — #29737553

Order 15616.

884 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455681C211yesmaximal, characteristic, complemented
2#21455679C211yesmaximal, characteristic, complemented
3#21454345C211yesmaximal, characteristic
4#21447350C211yesmaximal, characteristic, complemented
5#21447179C211yesmaximal, characteristic, complemented
6#21447177C211yesmaximal, characteristic, complemented
7#21446973C211yesmaximal, characteristic, complemented
8#21446096C211yesmaximal, characteristic, complemented
9#21446094C211yesmaximal, characteristic, complemented
10#21446045C211yesmaximal, characteristic, complemented
11#21446043C211yesmaximal, characteristic, complemented
12#21445878C211yesmaximal, characteristic, complemented
13#21445876C211yesmaximal, characteristic, complemented
14#21445773C211yesmaximal, characteristic
15C61×#2897C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17#16840390C2211yescharacteristic, complemented
18#16840245C2211yescharacteristic
19#16840069—21no—
20#16840067—21no—
21#16840064—21no—
22#16840062—21no—
23#16840052—21no—
24#16840050—21no—
25#16840047—21no—
26#16840045—21no—
27C2×#14519713C2211yescharacteristic, complemented
28C2×#14519712C2211yescharacteristic, complemented
29#16839805C2211yescharacteristic, complemented
30#16839802C2211yescharacteristic, complemented
31C2×#14519709C2211yescharacteristic, complemented
32C2×#14519708C2211yescharacteristic, complemented
33#16839789C2211yescharacteristic, complemented
34#16839788C2211yescharacteristic, complemented
35C2×#14519704C2211yescharacteristic
36C2×#14519704C2211yescharacteristic
37#16839767—21no—
38#16839767C2211yescharacteristic
39#16839763—21no—
40#16839763C2211yescharacteristic
41#16839763C2211yescharacteristic
42#16839763C2211yescharacteristic
43#16839758—21no—
44#16839758C2211yescharacteristic
45C2×#14519698—21no—
46C2×#14519698C2211yescharacteristic
47C2×#14519694C2211yescharacteristic
48#16839665C2211yescharacteristic
49#16839631C2211yescharacteristic, complemented
50#16839531C2211yescharacteristic, complemented
51#16839530C2211yescharacteristic, complemented
52#16839530C2211yescharacteristic, complemented
53#16839523C2211yescharacteristic, complemented
54#16839521C2211yescharacteristic, complemented
55#16839516C2211yescharacteristic, complemented
56C2×H(5,𝔽2)×C61C2211yescharacteristic
57C2×#131×C61C2211yescharacteristic
58C61×#520C2211yescharacteristic
59C61×#418—21no—
60C61×#418C2211yescharacteristic
61C61×#417—21no—
62C61×#417C2211yescharacteristic
63C61×#416—21no—
64C61×#416C2211yescharacteristic
65C2×#102×C61—21no—
66C2×#102×C61C2211yescharacteristic
67#14519784—22no—
68#14519784—41no—
69C2×#11758194—21no—
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73C22×#92418—21no—
74C22×#92418C2311yescharacteristic
75#14519777—21no—
76#14519776—21no—
77#14519769—21no—
78#14519768—21no—
79#14519754—41no—
80Q8×Dic61—21no—
81D4×Dic61—21no—
82D4×Dic61—41no—
83#14519736—21no—
84#14519736C2311yescharacteristic
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—41no—
89C2×C22⋊Dic61—41no—
90C2×C4×Dic61C2311yescharacteristic
91#14519713—41no—
92#14519712—41no—
93#14519710—21no—
94#14519710—21no—
95#14519709—21no—
96#14519709—21no—
97#14519708—21no—
98#14519708—21no—
99#14519708—21no—
100#14519708—21no—
101C4⋊#92418—21no—
102C4⋊#92418—21no—
103#14519706—21no—
104#14519706—21no—
105#14519704—22no—
106#14519704—22no—
107#14519701—21no—
108#14519701—21no—
109#14519701—21no—
110#14519701D411yescharacteristic
111#14519701D411yescharacteristic
112#14519698—21no—
113#14519698—21no—
114#14519698—21no—
115#14519698—21no—
116#14519698—21no—
117#14519698—21no—
118#14519698—21no—
119#14519698D411yescharacteristic
120#14519698D411yescharacteristic
121#14519694—41no—
122H(5,𝔽2)×C61—22no—
123H(5,𝔽2)×C61—22no—
124H(5,𝔽2)×C61—41no—
125H(5,𝔽2)×C61Q814yescomplemented
126C2×#41×C61—21no—
127C2×#41×C61—21no—
128C2×#41×C61C2311yescharacteristic
129C2×#41×C61C2311yescharacteristic
130C22×D4×C61—21no—
131C22×D4×C61—21no—
132C22×D4×C61—41no—
133C22×D4×C61C2311yescharacteristic
134#131×C61—22no—
135#131×C61—41no—
136C2×M16×C61—21no—
137C2×M16×C61C2311yescharacteristic
138C22×C8×C61—21no—
139C22×C8×C61C2311yescharacteristic
140C4×Q8×C61—21no—
141C4×D4×C61—21no—
142C4×D4×C61—41no—
143#117×C61—21no—
144#117×C61C2311yescharacteristic
145C2×C4⋊C4×C61—21no—
146C2×C4⋊C4×C61C2311yescharacteristic
147#103×C61—21no—
148#103×C61D411yescharacteristic
149#103×C61D411yescharacteristic
150#102×C61—21no—
151#102×C61—21no—
152#102×C61—21no—
153#102×C61—21no—
154#102×C61—21no—
155#102×C61D411yescharacteristic
156#102×C61D411yescharacteristic
157C24×C61—21no—
158C24×C61—41no—
159#41×C61—21no—
160#41×C61—21no—
161#41×C61—21no—
162#41×C61—21no—
163#41×C61—24no—
164#41×C61—24no—
165#41×C61—42no—
166#41×C61—42no—
167#41×C61C2×Q812yes—
168#41×C61C2×Q812yes—
169#41×C61#4112yes—
170#41×C61#4112yes—
171C2×Q8×C61C2×D411yescharacteristic
172C2×Q8×C61C2×D411yescharacteristic
173C2×D4×C61—21no—
174C2×D4×C61—21no—
175C2×D4×C61—24no—
176C2×D4×C61—24no—
177C2×D4×C61—21no—
178C2×D4×C61—21no—
179C2×D4×C61—21no—
180C2×D4×C61—21no—
181C2×D4×C61—21no—
182C2×D4×C61—42no—
183C2×D4×C61—42no—
184C2×D4×C61—41no—
185C2×D4×C61—41no—
186C2×D4×C61—44no—
187C2×D4×C61—44no—
188C2×D4×C61C2×D411yescharacteristic
189C2×D4×C61C2×D411yescharacteristic
190C2×D4×C61C2×Q812yes—
191C22×C4×C61—21no—
192C22×C4×C61—21no—
193C22×C4×C61—21no—
194C22×C4×C61—41no—
195C22×C4×C61—41no—
196C22×C4×C61C2411yescharacteristic
197C22×Dic61—21no—
198C22×Dic61—21no—
199C22×Dic61—21no—
200C22×Dic61—41no—
201C2×Dic122—41no—
202M16×C61—21no—
203M16×C61—21no—
204M16×C61—42no—
205C2×C8×C61—21no—
206C2×C8×C61—21no—
207C2×C8×C61—21no—
208C2×C8×C61—42no—
209C2×C8×C61C2×D411yescharacteristic
210C2×C8×C61C2×D411yescharacteristic
211C4⋊C4×C61—21no—
212C4⋊C4×C61—21no—
213C4⋊C4×C61—41no—
214C4⋊C4×C61C2×D411yescharacteristic
215C4⋊C4×C61C2×D411yescharacteristic
216C22⋊C4×C61—21no—
217C22⋊C4×C61—41no—
218C42×C61—21no—
219C42×C61—41no—
220C22⋊Dic61—21no—
221C22⋊Dic61—41no—
222C22⋊Dic61—82no—
223C4⋊Dic61—21no—
224C4⋊Dic61—21no—
225C4⋊Dic61—21no—
226C4⋊Dic61—21no—
227C4⋊Dic61—41no—
228C4⋊Dic61—41no—
229C4⋊Dic61C2×D411yescharacteristic
230C4⋊Dic61C2×D411yescharacteristic
231#11758196—41no—
232#11758196—41no—
233#11758196—82no—
234C4×Dic61—21no—
235C4×Dic61—21no—
236C4×Dic61—21no—
237C4×Dic61—41no—
238#11758194—21no—
239#11758194—21no—
240#11758194—21no—
241#11758194—21no—
242#11758194—21no—
243#11758194—21no—
244#11758194—42no—
245C2×#92418—21no—
246C2×#92418—21no—
247C2×#92418—21no—
248C2×#92418—21no—
249C2×#92418—21no—
250C2×#92418—42no—
251C2×#92418C2×D411yescharacteristic
252C2×#92418C2×D411yescharacteristic
253C23×C61—21no—
254C23×C61—21no—
255C23×C61—22no—
256C23×C61—44no—
257C23×C61—41no—
258C23×C61—41no—
259C23×C61—81no—
260C23×C61—81no—
261C23×C61—161no—
262C23×C61—161no—
263C23×C61H(5,𝔽2)11yescharacteristic
264Q8×C61—21no—
265Q8×C61—21no—
266Q8×C61C22⋊Q812yes—
267Q8×C61C22⋊Q812yes—
268D4×C61—22no—
269D4×C61—22no—
270D4×C61—22no—
271D4×C61—22no—
272D4×C61—22no—
273D4×C61—22no—
274D4×C61—21no—
275D4×C61—21no—
276D4×C61—44no—
277D4×C61—44no—
278D4×C61—41no—
279D4×C61—41no—
280D4×C61C22⋊Q812yes—
281D4×C61C22⋊Q812yes—
282C2×C4×C61—21no—
283C2×C4×C61—21no—
284C2×C4×C61—22no—
285C2×C4×C61—22no—
286C2×C4×C61—22no—
287C2×C4×C61—22no—
288C2×C4×C61—21no—
289C2×C4×C61—21no—
290C2×C4×C61—44no—
291C2×C4×C61—41no—
292C2×C4×C61—81no—
293C2×C4×C61—161no—
294C2×C4×C61C22×D411yescharacteristic
295C2×C4×C61C22×D411yescharacteristic
296C2×C4×C61C22×Q811yescharacteristic
297C2×C4×C61C2×#4111yescharacteristic
298C2×C4×C61C2×#4111yescharacteristic
299C2×C4×C61H(5,𝔽2)11yescharacteristic
300C2×Dic61—22no—
301C2×Dic61—21no—
302C2×Dic61—21no—
303C2×Dic61—41no—
304C2×Dic61—41no—
305C2×Dic61—41no—
306C2×Dic61—41no—
307C2×Dic61—81no—
308C2×Dic61—81no—
309C2×Dic61—161no—
310Dic122—161no—
311C8×C61—21no—
312C8×C61—21no—
313C8×C61—41no—
314#92418—21no—
315#92418—21no—
316#92418—41no—
317#92418—41no—
318#92418—41no—
319#25983—611nomaximal, Sylow
320C22×C61—21no—
321C22×C61—21no—
322C22×C61—22no—
323C22×C61—22no—
324C22×C61—41no—
325C22×C61—41no—
326C22×C61—41no—
327C22×C61—41no—
328C22×C61—42no—
329C22×C61—82no—
330C22×C61—81no—
331C22×C61—81no—
332C22×C61—161no—
333C22×C61—164no—
334C22×C61#54311yescharacteristic
335C22×C61D4211yescharacteristic
336C22×C61#55111yescharacteristic
337C4×C61—22no—
338C4×C61—22no—
339C4×C61—41no—
340C4×C61—41no—
341C4×C61—81no—
342C4×C61C2×C22⋊Q811yescharacteristic
343C4×C61C2×C22⋊Q811yescharacteristic
344C4×C61#54611yescharacteristic
345C4×C61#54611yescharacteristic
346Dic61—42no—
347Dic61—41no—
348Dic61—41no—
349Dic61—81no—
350Dic61—81no—
351#3084—611no—
352#3084—611no—
353#2897—611no—
354#2897D6111yescharacteristic
355#2094—611no—
356#2063—611no—
357#2063—611no—
358#2027—611no—
359#1866—611no—
360#1866—611no—
361#1859—611no—
362#1859—611no—
363#1815—611no—
364#1815—611no—
365#1792—611no—
366C2×C61—21no—
367C2×C61—21no—
368C2×C61—42no—
369C2×C61—42no—
370C2×C61—82no—
371C2×C61#243711yescharacteristic
372C2×C61#328811yescharacteristic
373C2×C61#329111yescharacteristic
374C2×H(5,𝔽2)C2×D6111yescharacteristic
375C2×#131—611no—
376C2×#131C2×D6111yescharacteristic
377#544—611no—
378#520—611no—
379#520C2×D6111yescharacteristic
380#479—1221no—
381#479—1221no—
382#478—1221no—
383#478—1221no—
384#476—1221no—
385#476—1221no—
386#475—1221no—
387#475—1221no—
388C2×#108—611no—
389C2×#108—611no—
390#428—611no—
391#428—611no—
392C2×#106—611no—
393C2×#106—611no—
394#424—611no—
395#424—611no—
396C2×#104—611no—
397C2×#104—611no—
398#418—21no—
399#418—611no—
400#418—1221no—
401#418C2×D6111yescharacteristic
402#417—21no—
403#417—611no—
404#417—611no—
405#417—611no—
406#417—1221no—
407#417C2×D6111yescharacteristic
408#416—21no—
409#416—611no—
410#416—1221no—
411#416C2×D6111yescharacteristic
412C2×#102—21no—
413C2×#102—611no—
414C2×#102—1221no—
415C2×#102C2×D6111yescharacteristic
416C2×#100—611no—
417#390—611no—
418#384—611no—
419#343—611no—
420#343—611no—
421#343—611no—
422#339—611no—
423#339—611no—
424#336—611no—
425C61#2598311yescharacteristic, complemented, Sylow
426H(5,𝔽2)—22no—
427H(5,𝔽2)—22no—
428H(5,𝔽2)—41no—
429H(5,𝔽2)Dic12214yescomplemented
430C2×#41—21no—
431C2×#41—21no—
432C2×#41C22×D6111yescharacteristic
433C2×#41C22×D6111yescharacteristic
434C22×D4—21no—
435C22×D4—21no—
436C22×D4—41no—
437C22×D4C22×D6111yescharacteristic
438#131—22no—
439#131—41no—
440#131—1222no—
441#131—2441no—
442C2×M16—21no—
443C2×M16—611no—
444C2×M16—611no—
445C2×M16—611no—
446C2×M16—1221no—
447C2×M16C22×D6111yescharacteristic
448C22×C8—21no—
449C22×C8—611no—
450C22×C8—1221no—
451C22×C8C22×D6111yescharacteristic
452C4⋊Q8—1221no—
453C4⋊Q8—1221no—
454#125—1221no—
455#125—1221no—
456C22⋊Q8—2441no—
457C4×Q8—21no—
458C4×Q8—1221no—
459C4×D4—21no—
460C4×D4—41no—
461C4×D4—1221no—
462C4×D4—2441no—
463#117—21no—
464#117—611no—
465#117—1221no—
466#117C22×D6111yescharacteristic
467C2×C4⋊C4—21no—
468C2×C4⋊C4—611no—
469C2×C4⋊C4—611no—
470C2×C4⋊C4—1221no—
471C2×C4⋊C4—2441no—
472C2×C4⋊C4C22×D6111yescharacteristic
473C2×C22⋊C4—2441no—
474C2×C42—611no—
475#108—2441no—
476#108—2441no—
477#107—1221no—
478#107—1221no—
479#106—1221no—
480#106—1221no—
481#106—1221no—
482#106—1221no—
483#106—1221no—
484#106—1221no—
485C4⋊C8—1221no—
486C4⋊C8—1221no—
487C4⋊C8—1221no—
488C4⋊C8—1221no—
489#104—1222no—
490#104—1222no—
491#103—21no—
492#103—611no—
493#103—611no—
494#103—1221no—
495#103—1221no—
496#103—1221no—
497#103C22⋊D6111yescharacteristic
498#103C22⋊D6111yescharacteristic
499#102—21no—
500#102—21no—
501#102—21no—
502#102—21no—
503#102—21no—
504#102—611no—
505#102—611no—
506#102—1221no—
507#102—1221no—
508#102—1221no—
509#102—1221no—
510#102—1221no—
511#102—1221no—
512#102—1221no—
513#102C22⋊D6111yescharacteristic
514#102C22⋊D6111yescharacteristic
515#100—2441no—
516C24—21no—
517C24—41no—
518#41—21no—
519#41—21no—
520#41—21no—
521#41—21no—
522#41—24no—
523#41—24no—
524#41—42no—
525#41—42no—
526#41C2×Dic12212yes—
527#41C2×Dic12212yes—
528#41#1175822212yes—
529#41#1175822412yes—
530C2×Q8—2441no—
531C2×Q8D4×D6111yescharacteristic
532C2×Q8C2×C22⋊D6111yescharacteristic
533C2×D4—21no—
534C2×D4—21no—
535C2×D4—24no—
536C2×D4—24no—
537C2×D4—21no—
538C2×D4—21no—
539C2×D4—21no—
540C2×D4—21no—
541C2×D4—21no—
542C2×D4—42no—
543C2×D4—42no—
544C2×D4—41no—
545C2×D4—41no—
546C2×D4—44no—
547C2×D4—44no—
548C2×D4C2×Dic12212yes—
549C2×D4D4×D6111yescharacteristic
550C2×D4C2×C22⋊D6111yescharacteristic
551C22×C4—21no—
552C22×C4—21no—
553C22×C4—21no—
554C22×C4—41no—
555C22×C4—41no—
556C22×C4—1221no—
557C22×C4—1221no—
558C22×C4—1221no—
559C22×C4—2441no—
560C22×C4C23×D6111yescharacteristic
561M16—21no—
562M16—21no—
563M16—42no—
564M16—1221no—
565M16—1221no—
566M16—1221no—
567M16—1221no—
568M16—1221no—
569M16—1221no—
570M16—2442no—
571C2×C8—21no—
572C2×C8—21no—
573C2×C8—21no—
574C2×C8—42no—
575C2×C8—611no—
576C2×C8—611no—
577C2×C8—1221no—
578C2×C8—1221no—
579C2×C8—1221no—
580C2×C8—1221no—
581C2×C8—1221no—
582C2×C8—2442no—
583C2×C8C2×C22⋊D6111yescharacteristic
584C2×C8C2×C22⋊D6111yescharacteristic
585C4⋊C4—21no—
586C4⋊C4—21no—
587C4⋊C4—41no—
588C4⋊C4—611no—
589C4⋊C4—611no—
590C4⋊C4—1221no—
591C4⋊C4—1221no—
592C4⋊C4—1221no—
593C4⋊C4—1221no—
594C4⋊C4—4882no—
595C4⋊C4—2441no—
596C4⋊C4—2441no—
597C4⋊C4—2441no—
598C4⋊C4—2441no—
599C4⋊C4C2×C22⋊D6111yescharacteristic
600C4⋊C4C2×C22⋊D6111yescharacteristic
601C22⋊C4—21no—
602C22⋊C4—41no—
603C22⋊C4—1221no—
604C22⋊C4—4882no—
605C22⋊C4—2441no—
606C42—21no—
607C42—41no—
608C42—1221no—
609C42—1221no—
610C42—1221no—
611C42—2441no—
612C23—21no—
613C23—21no—
614C23—22no—
615C23—44no—
616C23—41no—
617C23—41no—
618C23—81no—
619C23—81no—
620C23—161no—
621C23—161no—
622C23#1451982211yescharacteristic
623Q8—21no—
624Q8—21no—
625Q8—9761no—
626Q8#1451975412yes—
627Q8#1451975612yes—
628D4—22no—
629D4—22no—
630D4—22no—
631D4—22no—
632D4—22no—
633D4—22no—
634D4—21no—
635D4—21no—
636D4—44no—
637D4—44no—
638D4—41no—
639D4—41no—
640D4#1451975412yes—
641D4#1451975612yes—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—44no—
651C2×C4—41no—
652C2×C4—81no—
653C2×C4—161no—
654C2×C4—1222no—
655C2×C4—1221no—
656C2×C4—1221no—
657C2×C4—9761no—
658C2×C4—4881no—
659C2×C4—4881no—
660C2×C4—2441no—
661C2×C4—2441no—
662C2×C4—2441no—
663C2×C4—2441no—
664C2×C4C2×D4×D6111yescharacteristic
665C2×C4C22×C22⋊D6111yescharacteristic
666C2×C4C22×Dic12211yescharacteristic
667C2×C4C2×#1175822411yescharacteristic
668C2×C4C2×#1175822211yescharacteristic
669C2×C4#1451982211yescharacteristic
670C8—21no—
671C8—21no—
672C8—41no—
673C8—1221no—
674C8—1221no—
675C8—2441no—
676C8—2441no—
677C8—2441no—
678C22—21no—
679C22—21no—
680C22—22no—
681C22—22no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—42no—
687C22—82no—
688C22—81no—
689C22—81no—
690C22—161no—
691C22—164no—
692C22#1684038811yescharacteristic
693C22D4×C22⋊D6111yescharacteristic
694C22#1684043811yescharacteristic
695C4—22no—
696C4—22no—
697C4—41no—
698C4—41no—
699C4—81no—
700C4—4881no—
701C4—4881no—
702C4—2442no—
703C4—2441no—
704C4—2441no—
705C4C2×#1451975411yescharacteristic
706C4C2×#1451975611yescharacteristic
707C4#1684040211yescharacteristic
708C4#1684040611yescharacteristic
709C2—21no—
710C2—21no—
711C2—42no—
712C2—42no—
713C2—82no—
714C2#2144948011yescharacteristic
715C2#2145758311yescharacteristic
716C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2897C2
socleC22×C61D42
prime radicalC2×H(5,𝔽2)×C61C22
hypercenter#2897D61
hypocenterC61#25983
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2897

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21455681C2semidirect
2. #21455679C2semidirect
4. #21447350C2semidirect
5. #21447179C2semidirect
6. #21447177C2semidirect
7. #21446973C2semidirect
8. #21446096C2semidirect
9. #21446094C2semidirect
10. #21446045C2semidirect
11. #21446043C2semidirect
12. #21445878C2semidirect
13. #21445876C2semidirect
17. #16840390C22semidirect
27. C2×#14519713C22semidirect
28. C2×#14519712C22semidirect
29. #16839805C22semidirect
30. #16839802C22semidirect
31. C2×#14519709C22semidirect
32. C2×#14519708C22semidirect
33. #16839789C22semidirect
34. #16839788C22semidirect
49. #16839631C22semidirect
50. #16839531C22semidirect
51. #16839530C22semidirect
52. #16839530C22semidirect
53. #16839523C22semidirect
54. #16839521C22semidirect
55. #16839516C22semidirect
125. H(5,𝔽2)×C61Q8semidirect
425. C61#25983semidirect
429. H(5,𝔽2)Dic122semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214556812
#214556792
#214543452
#214473502
#214471792
#214471772
#214469732
#214460962
#214460942
#214460452
#214460432
#214458782
#214458762
#214457732
C61×#28972
#2598361