Finite Group Databasebeta

← #29737570

Subgroups — #29737570

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455685C211yesmaximal, characteristic, complemented
2#21455674C211yesmaximal, characteristic, complemented
3#21454345C211yesmaximal, characteristic
4#21454345C211yesmaximal, characteristic
5#21447340C211yesmaximal, characteristic, complemented
6#21447175C211yesmaximal, characteristic, complemented
7#21447175C211yesmaximal, characteristic, complemented
8#21446975C211yesmaximal, characteristic, complemented
9#21446092C211yesmaximal, characteristic, complemented
10#21446092C211yesmaximal, characteristic, complemented
11#21446054C211yesmaximal, characteristic, complemented
12#21446054C211yesmaximal, characteristic, complemented
13#21445881C211yesmaximal, characteristic, complemented
14#21445876C211yesmaximal, characteristic, complemented
15C61×#1792C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#16840393C2211yescharacteristic, complemented
19#16840245C2211yescharacteristic
20#16840245C2211yescharacteristic
21#16840071—21no—
22#16840067—21no—
23#16840064—21no—
24#16840060—21no—
25#16840055—21no—
26#16840050—21no—
27#16840045—21no—
28#16840040—21no—
29C2×#14519712C2211yescharacteristic, complemented
30C2×#14519712C2211yescharacteristic, complemented
31#16839804C2211yescharacteristic, complemented
32#16839804C2211yescharacteristic, complemented
33C2×#14519710C2211yescharacteristic, complemented
34C2×#14519708C2211yescharacteristic, complemented
35#16839790C2211yescharacteristic, complemented
36#16839790C2211yescharacteristic, complemented
37#16839767—21no—
38#16839767—21no—
39#16839767C2211yescharacteristic
40#16839767C2211yescharacteristic
41#16839763—21no—
42#16839763—21no—
43#16839763C2211yescharacteristic
44#16839763C2211yescharacteristic
45#16839758—21no—
46#16839758—21no—
47#16839758C2211yescharacteristic
48#16839758C2211yescharacteristic
49C2×#14519698—21no—
50C2×#14519698—21no—
51C2×#14519698C2211yescharacteristic
52C2×#14519698C2211yescharacteristic
53#16839628C2211yescharacteristic, complemented
54#16839531C2211yescharacteristic, complemented
55#16839530C2211yescharacteristic, complemented
56#16839530C2211yescharacteristic, complemented
57#16839522C2211yescharacteristic, complemented
58#16839522C2211yescharacteristic, complemented
59#16839517C2211yescharacteristic, complemented
60C2×H(5,𝔽2)×C61C2211yescharacteristic
61C2×#104×C61C2211yescharacteristic
62C2×#104×C61C2211yescharacteristic
63C61×#417C2211yescharacteristic
64C61×#417C2211yescharacteristic
65C2×#100×C61C2211yescharacteristic
66C61×#390C2211yescharacteristic
67#14519784—22no—
68#14519784—22no—
69#14519784—41no—
70#14519784—41no—
71C2×#11758194—21no—
72C2×#11758194—21no—
73C2×#11758194C2311yescharacteristic
74C2×#11758194C2311yescharacteristic
75C22×#92418—21no—
76C22×#92418—21no—
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79#14519776—21no—
80#14519776—21no—
81#14519768—21no—
82#14519768—21no—
83#14519756—41no—
84Q8×Dic61—21no—
85Q8×Dic61—21no—
86D4×Dic61—21no—
87D4×Dic61—21no—
88D4×Dic61—41no—
89D4×Dic61—41no—
90#14519736—21no—
91#14519736—21no—
92#14519736C2311yescharacteristic
93#14519736C2311yescharacteristic
94C2×C4⋊Dic61—21no—
95C2×C4⋊Dic61—21no—
96C2×C4⋊Dic61C2311yescharacteristic
97C2×C4⋊Dic61C2311yescharacteristic
98#14519712—41no—
99#14519712—41no—
100#14519711—21no—
101#14519711—21no—
102#14519710—21no—
103#14519710—21no—
104#14519709—21no—
105#14519709—21no—
106#14519708—21no—
107#14519708—21no—
108#14519706—21no—
109#14519706—21no—
110#14519706—21no—
111#14519706—21no—
112#14519701—21no—
113#14519701—21no—
114#14519701D411yescharacteristic
115#14519701D411yescharacteristic
116#14519701D411yescharacteristic
117#14519701D411yescharacteristic
118#14519698—21no—
119#14519698—21no—
120#14519698—21no—
121#14519698—21no—
122#14519698—21no—
123#14519698—21no—
124#14519698—21no—
125#14519698—21no—
126#14519698—21no—
127#14519698—21no—
128#14519698D411yescharacteristic
129#14519698D411yescharacteristic
130#14519698D411yescharacteristic
131#14519698D411yescharacteristic
132#14519684—41no—
133H(5,𝔽2)×C61—22no—
134H(5,𝔽2)×C61—22no—
135H(5,𝔽2)×C61—41no—
136H(5,𝔽2)×C61Q814yescomplemented
137C2×#41×C61—21no—
138C2×#41×C61—21no—
139C2×#41×C61C2311yescharacteristic
140C2×#41×C61C2311yescharacteristic
141C22×D4×C61—21no—
142C22×D4×C61—21no—
143C22×D4×C61—41no—
144C22×D4×C61C2311yescharacteristic
145C2×M16×C61C2311yescharacteristic
146C2×M16×C61C2311yescharacteristic
147C2×C4⋊C4×C61C2311yescharacteristic
148C2×C22⋊C4×C61—41no—
149C2×C42×C61C2311yescharacteristic
150#104×C61—22no—
151#104×C61—22no—
152#103×C61—21no—
153#103×C61—21no—
154#102×C61—21no—
155#102×C61—21no—
156#100×C61—41no—
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—21no—
195C22×C4×C61—41no—
196C22×C4×C61C2411yescharacteristic
197C22×Dic61—21no—
198C22×Dic61—21no—
199C22×Dic61—41no—
200C22×Dic61—41no—
201C2×Dic122—41no—
202M16×C61—21no—
203M16×C61—21no—
204M16×C61—21no—
205M16×C61—21no—
206C2×C8×C61—21no—
207C2×C8×C61—21no—
208C4⋊C4×C61—21no—
209C4⋊C4×C61—21no—
210C22⋊C4×C61—82no—
211C42×C61—21no—
212C42×C61—21no—
213C22⋊Dic61—21no—
214C22⋊Dic61—21no—
215C22⋊Dic61—41no—
216C22⋊Dic61—41no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61—21no—
221C4⋊Dic61—41no—
222C4⋊Dic61—41no—
223C4⋊Dic61—41no—
224C4⋊Dic61C2×D411yescharacteristic
225C4⋊Dic61C2×D411yescharacteristic
226C4⋊Dic61C2×D411yescharacteristic
227C4⋊Dic61C2×D411yescharacteristic
228#11758196—41no—
229#11758196—41no—
230C4×Dic61—21no—
231C4×Dic61—21no—
232C4×Dic61—41no—
233C4×Dic61—41no—
234#11758194—21no—
235#11758194—21no—
236#11758194—21no—
237#11758194—21no—
238#11758194—42no—
239#11758194—42no—
240C2×#92418—21no—
241C2×#92418—21no—
242C2×#92418—21no—
243C2×#92418—21no—
244C2×#92418—21no—
245C2×#92418—21no—
246C2×#92418—42no—
247C2×#92418—42no—
248C2×#92418C2×D411yescharacteristic
249C2×#92418C2×D411yescharacteristic
250C2×#92418C2×D411yescharacteristic
251C2×#92418C2×D411yescharacteristic
252C23×C61—21no—
253C23×C61—21no—
254C23×C61—22no—
255C23×C61—44no—
256C23×C61—41no—
257C23×C61—41no—
258C23×C61—81no—
259C23×C61—81no—
260C23×C61—161no—
261C23×C61—161no—
262C23×C61H(5,𝔽2)11yescharacteristic
263Q8×C61—21no—
264Q8×C61—21no—
265Q8×C61C22⋊Q812yes—
266Q8×C61C22⋊Q812yes—
267D4×C61—22no—
268D4×C61—22no—
269D4×C61—22no—
270D4×C61—22no—
271D4×C61—22no—
272D4×C61—22no—
273D4×C61—21no—
274D4×C61—21no—
275D4×C61—44no—
276D4×C61—44no—
277D4×C61—41no—
278D4×C61—41no—
279D4×C61C22⋊Q812yes—
280D4×C61C22⋊Q812yes—
281C2×C4×C61—21no—
282C2×C4×C61—21no—
283C2×C4×C61—22no—
284C2×C4×C61—22no—
285C2×C4×C61—22no—
286C2×C4×C61—22no—
287C2×C4×C61—44no—
288C2×C4×C61—41no—
289C2×C4×C61—41no—
290C2×C4×C61—41no—
291C2×C4×C61—41no—
292C2×C4×C61—81no—
293C2×C4×C61C22×D411yescharacteristic
294C2×C4×C61C22×D411yescharacteristic
295C2×C4×C61C22×Q811yescharacteristic
296C2×C4×C61C2×#4111yescharacteristic
297C2×C4×C61C2×#4111yescharacteristic
298C2×C4×C61H(5,𝔽2)11yescharacteristic
299C2×Dic61—21no—
300C2×Dic61—21no—
301C2×Dic61—21no—
302C2×Dic61—21no—
303C2×Dic61—41no—
304C2×Dic61—41no—
305C2×Dic61—81no—
306C2×Dic61—81no—
307C2×Dic61—161no—
308C2×Dic61—161no—
309Dic122—161no—
310C8×C61—41no—
311C8×C61—41no—
312#92418—21no—
313#92418—21no—
314#92418—21no—
315#92418—21no—
316#92418—41no—
317#92418—41no—
318#25984—611nomaximal, Sylow
319C22×C61—21no—
320C22×C61—21no—
321C22×C61—22no—
322C22×C61—22no—
323C22×C61—41no—
324C22×C61—41no—
325C22×C61—41no—
326C22×C61—41no—
327C22×C61—42no—
328C22×C61—82no—
329C22×C61—81no—
330C22×C61—81no—
331C22×C61—161no—
332C22×C61—164no—
333C22×C61#54311yescharacteristic
334C22×C61D4211yescharacteristic
335C22×C61#55111yescharacteristic
336C4×C61—22no—
337C4×C61—22no—
338C4×C61—81no—
339C4×C61—81no—
340C4×C61C2×C22⋊Q811yescharacteristic
341C4×C61C2×C22⋊Q811yescharacteristic
342C4×C61#54611yescharacteristic
343C4×C61#54611yescharacteristic
344Dic61—41no—
345Dic61—41no—
346Dic61—41no—
347Dic61—41no—
348Dic61—81no—
349Dic61—81no—
350#3084—611no—
351#3083—611no—
352#2897—611no—
353#2897—611no—
354#2092—611no—
355#2063—611no—
356#2063—611no—
357#2027—611no—
358#1866—611no—
359#1866—611no—
360#1860—611no—
361#1860—611no—
362#1816—611no—
363#1815—611no—
364#1792D6111yescharacteristic
365C2×C61—21no—
366C2×C61—21no—
367C2×C61—42no—
368C2×C61—42no—
369C2×C61—82no—
370C2×C61#243711yescharacteristic
371C2×C61#328811yescharacteristic
372C2×C61#329011yescharacteristic
373C2×H(5,𝔽2)C2×D6111yescharacteristic
374C2×#131—611no—
375C2×#131—611no—
376#544—611no—
377#520—611no—
378#520—611no—
379#479—1221no—
380#479—1221no—
381#478—1221no—
382#478—1221no—
383#477—1221no—
384#476—1221no—
385#475—1221no—
386#474—1221no—
387C2×#108—611no—
388C2×#108—611no—
389#428—611no—
390#428—611no—
391C2×#107—611no—
392C2×#106—611no—
393#424—611no—
394#424—611no—
395C2×#104C2×D6111yescharacteristic
396C2×#104C2×D6111yescharacteristic
397#418—611no—
398#418—611no—
399#418—1221no—
400#418—1221no—
401#417—611no—
402#417—611no—
403#417—1221no—
404#417—1221no—
405#417C2×D6111yescharacteristic
406#417C2×D6111yescharacteristic
407#416—611no—
408#416—611no—
409#416—1221no—
410#416—1221no—
411C2×#102—611no—
412C2×#102—611no—
413C2×#102—1221no—
414C2×#102—1221no—
415C2×#100C2×D6111yescharacteristic
416#390C2×D6111yescharacteristic
417#382—611no—
418#343—611no—
419#343—611no—
420#343—611no—
421#339—611no—
422#339—611no—
423#336—611no—
424C61#2598411yescharacteristic, complemented, Sylow
425H(5,𝔽2)—22no—
426H(5,𝔽2)—22no—
427H(5,𝔽2)—41no—
428H(5,𝔽2)Dic12214yescomplemented
429C2×#41—21no—
430C2×#41—21no—
431C2×#41C22×D6111yescharacteristic
432C2×#41C22×D6111yescharacteristic
433C22×D4—21no—
434C22×D4—21no—
435C22×D4—41no—
436C22×D4C22×D6111yescharacteristic
437#131—1222no—
438#131—1222no—
439#131—2441no—
440#131—2441no—
441C2×M16—611no—
442C2×M16—611no—
443C2×M16—1221no—
444C2×M16—1221no—
445C2×M16C22×D6111yescharacteristic
446C2×M16C22×D6111yescharacteristic
447C22×C8—611no—
448C22×C8—611no—
449C22×C8—1221no—
450C22×C8—1221no—
451C4⋊Q8—1221no—
452C4⋊Q8—1221no—
453#125—1221no—
454#125—1221no—
455C22⋊Q8—2441no—
456C4×Q8—1221no—
457C4×Q8—1221no—
458C4×D4—1221no—
459C4×D4—1221no—
460C4×D4—2441no—
461C4×D4—2441no—
462#117—611no—
463#117—611no—
464#117—1221no—
465#117—1221no—
466C2×C4⋊C4—611no—
467C2×C4⋊C4—611no—
468C2×C4⋊C4—1221no—
469C2×C4⋊C4—1221no—
470C2×C4⋊C4C22×D6111yescharacteristic
471C2×C22⋊C4—41no—
472C2×C42C22×D6111yescharacteristic
473#108—2441no—
474#108—2441no—
475#107—1221no—
476#107—1221no—
477#107—1221no—
478#107—1221no—
479#106—1221no—
480#106—1221no—
481#106—1221no—
482#106—1221no—
483C4⋊C8—1221no—
484C4⋊C8—1221no—
485C4⋊C8—1221no—
486C4⋊C8—1221no—
487#104—22no—
488#104—22no—
489#103—21no—
490#103—21no—
491#103—611no—
492#103—611no—
493#103—611no—
494#103—611no—
495#103—1221no—
496#103—1221no—
497#102—21no—
498#102—21no—
499#102—611no—
500#102—611no—
501#102—611no—
502#102—611no—
503#102—1221no—
504#102—1221no—
505#102—1221no—
506#102—1221no—
507#102—1221no—
508#102—1221no—
509#102—1221no—
510#102—1221no—
511#102—1221no—
512#102—1221no—
513#100—41no—
514#95—2441no—
515C24—21no—
516C24—41no—
517#41—21no—
518#41—21no—
519#41—21no—
520#41—21no—
521#41—24no—
522#41—24no—
523#41—42no—
524#41—42no—
525#41C2×Dic12212yes—
526#41C2×Dic12212yes—
527#41#1175822412yes—
528#41#1175822412yes—
529C2×Q8—2441no—
530C2×Q8D4×D6111yescharacteristic
531C2×Q8D4×D6111yescharacteristic
532C2×D4—21no—
533C2×D4—21no—
534C2×D4—24no—
535C2×D4—24no—
536C2×D4—21no—
537C2×D4—21no—
538C2×D4—21no—
539C2×D4—21no—
540C2×D4—21no—
541C2×D4—42no—
542C2×D4—42no—
543C2×D4—41no—
544C2×D4—41no—
545C2×D4—44no—
546C2×D4—44no—
547C2×D4C2×Dic12212yes—
548C2×D4D4×D6111yescharacteristic
549C2×D4D4×D6111yescharacteristic
550C22×C4—21no—
551C22×C4—21no—
552C22×C4—21no—
553C22×C4—21no—
554C22×C4—41no—
555C22×C4—1221no—
556C22×C4—1221no—
557C22×C4—2441no—
558C22×C4—2441no—
559C22×C4C23×D6111yescharacteristic
560M16—21no—
561M16—21no—
562M16—21no—
563M16—21no—
564M16—1221no—
565M16—1221no—
566M16—1221no—
567M16—1221no—
568M16—2442no—
569M16—2442no—
570C2×C8—21no—
571C2×C8—21no—
572C2×C8—611no—
573C2×C8—611no—
574C2×C8—611no—
575C2×C8—611no—
576C2×C8—1221no—
577C2×C8—1221no—
578C2×C8—1221no—
579C2×C8—1221no—
580C2×C8—1221no—
581C2×C8—1221no—
582C2×C8—2442no—
583C2×C8—2442no—
584C4⋊C4—21no—
585C4⋊C4—21no—
586C4⋊C4—611no—
587C4⋊C4—611no—
588C4⋊C4—611no—
589C4⋊C4—611no—
590C4⋊C4—1221no—
591C4⋊C4—1221no—
592C4⋊C4—1221no—
593C4⋊C4—1221no—
594C4⋊C4—2441no—
595C4⋊C4—2441no—
596C4⋊C4—2441no—
597C4⋊C4—2441no—
598C4⋊C4—2441no—
599C22⋊C4—82no—
600C22⋊C4—1221no—
601C22⋊C4—1221no—
602C22⋊C4—2441no—
603C22⋊C4—2441no—
604C42—21no—
605C42—21no—
606C42—1221no—
607C42—1221no—
608C42—2441no—
609C42—2441no—
610C23—21no—
611C23—21no—
612C23—22no—
613C23—44no—
614C23—41no—
615C23—41no—
616C23—81no—
617C23—81no—
618C23—161no—
619C23—161no—
620C23#1451982211yescharacteristic
621Q8—21no—
622Q8—21no—
623Q8—9761no—
624Q8#1451975412yes—
625Q8#1451975412yes—
626D4—22no—
627D4—22no—
628D4—22no—
629D4—22no—
630D4—22no—
631D4—22no—
632D4—21no—
633D4—21no—
634D4—44no—
635D4—44no—
636D4—41no—
637D4—41no—
638D4#1451975412yes—
639D4#1451975412yes—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—81no—
652C2×C4—1221no—
653C2×C4—1221no—
654C2×C4—1221no—
655C2×C4—1221no—
656C2×C4—9761no—
657C2×C4—9761no—
658C2×C4—4881no—
659C2×C4—4881no—
660C2×C4—2441no—
661C2×C4—2441no—
662C2×C4C2×D4×D6111yescharacteristic
663C2×C4C2×D4×D6111yescharacteristic
664C2×C4C22×Dic12211yescharacteristic
665C2×C4C2×#1175822411yescharacteristic
666C2×C4C2×#1175822411yescharacteristic
667C2×C4#1451982211yescharacteristic
668C8—41no—
669C8—41no—
670C8—1221no—
671C8—1221no—
672C8—1221no—
673C8—1221no—
674C8—2441no—
675C8—2441no—
676C22—21no—
677C22—21no—
678C22—22no—
679C22—22no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—42no—
685C22—82no—
686C22—81no—
687C22—81no—
688C22—161no—
689C22—164no—
690C22#1684038811yescharacteristic
691C22#1684039411yescharacteristic
692C22#1684044311yescharacteristic
693C4—22no—
694C4—22no—
695C4—81no—
696C4—81no—
697C4—4881no—
698C4—4881no—
699C4—2441no—
700C4—2441no—
701C4—2441no—
702C4—2441no—
703C4C2×#1451975411yescharacteristic
704C4C2×#1451975411yescharacteristic
705C4#1684039911yescharacteristic
706C4#1684039911yescharacteristic
707C2—21no—
708C2—21no—
709C2—42no—
710C2—42no—
711C2—82no—
712C2#2144948511yescharacteristic
713C2#2145758011yescharacteristic
714C2#2145760311yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1792C2
socleC22×C61D42
prime radicalC2×H(5,𝔽2)×C61C22
hypercenter#1792D61
hypocenterC61#25984
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1792

Complements

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

NH ≅ G/Ntype
1. #21455685C2semidirect
2. #21455674C2semidirect
5. #21447340C2semidirect
6. #21447175C2semidirect
7. #21447175C2semidirect
8. #21446975C2semidirect
9. #21446092C2semidirect
10. #21446092C2semidirect
11. #21446054C2semidirect
12. #21446054C2semidirect
13. #21445881C2semidirect
14. #21445876C2semidirect
18. #16840393C22semidirect
29. C2×#14519712C22semidirect
30. C2×#14519712C22semidirect
31. #16839804C22semidirect
32. #16839804C22semidirect
33. C2×#14519710C22semidirect
34. C2×#14519708C22semidirect
35. #16839790C22semidirect
36. #16839790C22semidirect
53. #16839628C22semidirect
54. #16839531C22semidirect
55. #16839530C22semidirect
56. #16839530C22semidirect
57. #16839522C22semidirect
58. #16839522C22semidirect
59. #16839517C22semidirect
136. H(5,𝔽2)×C61Q8semidirect
424. C61#25984semidirect
428. H(5,𝔽2)Dic122semidirect

Maximal subgroups

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

subgroupindex
#214556852
#214556742
#214543452
#214473402
#214471752
#214469752
#214460922
#214460542
#214458812
#214458762
C61×#17922
#2598461