Finite Group Databasebeta

← #29737592

Subgroups — #29737592

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 696 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#21454352C211yesmaximal, characteristic
4#21447360C211yesmaximal, characteristic, complemented
5#21447186C211yesmaximal, characteristic, complemented
6#21447184C211yesmaximal, characteristic, complemented
7#21446978C211yesmaximal, 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#21445778C211yesmaximal, characteristic
15C61×#2898C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17#16840390C2211yescharacteristic
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
28C2×#14519712C2211yescharacteristic
29#16839805C2211yescharacteristic
30#16839802C2211yescharacteristic
31C2×#14519709C2211yescharacteristic
32C2×#14519708C2211yescharacteristic
33#16839789C2211yescharacteristic
34#16839788C2211yescharacteristic
35C2×#14519704C2211yescharacteristic
36C2×#14519704C2211yescharacteristic
37#16839771—21no—
38#16839771C2211yescharacteristic
39#16839763—21no—
40#16839763C2211yescharacteristic
41#16839763C2211yescharacteristic
42#16839763C2211yescharacteristic
43#16839758—21no—
44#16839758C2211yescharacteristic
45C2×#14519701—21no—
46C2×#14519701C2211yescharacteristic
47C2×#14519696C2211yescharacteristic
48#16839669C2211yescharacteristic
49#16839631C2211yescharacteristic
50#16839531C2211yescharacteristic
51#16839530C2211yescharacteristic
52#16839530C2211yescharacteristic
53#16839523C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×2−1+4×C61C2211yescharacteristic
57C2×#131×C61C2211yescharacteristic
58C61×#520C2211yescharacteristic
59C61×#419—21no—
60C61×#419C2211yescharacteristic
61C61×#417—21no—
62C61×#417C2211yescharacteristic
63C61×#416—21no—
64C61×#416C2211yescharacteristic
65C2×#103×C61—21no—
66C2×#103×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×C4×Dic61C2311yescharacteristic
90#14519713—41no—
91#14519712—41no—
92#14519710—21no—
93#14519710—21no—
94#14519709—21no—
95#14519709—21no—
96#14519708—21no—
97#14519708—21no—
98#14519708—21no—
99#14519708—21no—
100C4⋊#92418—21no—
101C4⋊#92418—21no—
102#14519706—21no—
103#14519706—21no—
104#14519704—22no—
105#14519704—22no—
106#14519701—21no—
107#14519701—21no—
108#14519701—21no—
109#14519701—21no—
110#14519701—21no—
111#14519701—21no—
112#14519701—21no—
113#14519701D411yescharacteristic
114#14519701D411yescharacteristic
115#14519698—21no—
116#14519698—21no—
117#14519698—21no—
118#14519698D411yescharacteristic
119#14519698D411yescharacteristic
120#14519696—41no—
121#14519684—41no—
1222−1+4×C61—22no—
1232−1+4×C61—22no—
1242−1+4×C61—41no—
1252−1+4×C61Q814yescomplemented
126C2×#41×C61—21no—
127C2×#41×C61—21no—
128C2×#41×C61—41no—
129C2×#41×C61C2311yescharacteristic
130C2×#41×C61C2311yescharacteristic
131C22×Q8×C61—21no—
132C22×Q8×C61—21no—
133C22×Q8×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×C61—21no—
149#103×C61—21no—
150#103×C61—21no—
151#103×C61—21no—
152#103×C61D411yescharacteristic
153#103×C61D411yescharacteristic
154#102×C61—21no—
155#102×C61D411yescharacteristic
156#102×C61D411yescharacteristic
157#41×C61—21no—
158#41×C61—21no—
159#41×C61—21no—
160#41×C61—21no—
161#41×C61—24no—
162#41×C61—24no—
163#41×C61—44no—
164#41×C61—42no—
165#41×C61—42no—
166#41×C61—44no—
167#41×C61C2×Q812yes—
168#41×C61C2×Q812yes—
169#41×C61#4112yes—
170#41×C61#4112yes—
171C2×Q8×C61—21no—
172C2×Q8×C61—24no—
173C2×Q8×C61—24no—
174C2×Q8×C61—21no—
175C2×Q8×C61—21no—
176C2×Q8×C61—21no—
177C2×Q8×C61—21no—
178C2×Q8×C61—41no—
179C2×Q8×C61—42no—
180C2×Q8×C61—42no—
181C2×Q8×C61C2×D411yescharacteristic
182C2×Q8×C61C2×D411yescharacteristic
183C2×Q8×C61C2×Q812yes—
184C2×D4×C61—21no—
185C2×D4×C61—21no—
186C2×D4×C61—41no—
187C2×D4×C61C2×D411yescharacteristic
188C2×D4×C61C2×D411yescharacteristic
189C22×C4×C61—21no—
190C22×C4×C61—21no—
191C22×C4×C61—21no—
192C22×C4×C61—21no—
193C22×C4×C61—41no—
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—
222C4⋊Dic61—21no—
223C4⋊Dic61—21no—
224C4⋊Dic61—21no—
225C4⋊Dic61—21no—
226C4⋊Dic61—41no—
227C4⋊Dic61—41no—
228C4⋊Dic61C2×D411yescharacteristic
229C4⋊Dic61C2×D411yescharacteristic
230#11758196—41no—
231#11758196—41no—
232#11758196—82no—
233C4×Dic61—21no—
234C4×Dic61—21no—
235C4×Dic61—21no—
236C4×Dic61—41no—
237#11758194—21no—
238#11758194—21no—
239#11758194—21no—
240#11758194—21no—
241#11758194—21no—
242#11758194—21no—
243#11758194—42no—
244C2×#92418—21no—
245C2×#92418—21no—
246C2×#92418—21no—
247C2×#92418—21no—
248C2×#92418—21no—
249C2×#92418—42no—
250C2×#92418C2×D411yescharacteristic
251C2×#92418C2×D411yescharacteristic
252C23×C61—21no—
253C23×C61—21no—
254C23×C61H(5,𝔽2)11yescharacteristic
255Q8×C61—21no—
256Q8×C61—21no—
257Q8×C61—22no—
258Q8×C61—22no—
259Q8×C61—44no—
260Q8×C61—41no—
261Q8×C61—41no—
262Q8×C61C22⋊Q812yes—
263Q8×C61C22⋊Q812yes—
264D4×C61—22no—
265D4×C61—22no—
266D4×C61—22no—
267D4×C61—22no—
268D4×C61—21no—
269D4×C61—21no—
270D4×C61—44no—
271D4×C61C22⋊Q812yes—
272D4×C61C22⋊Q812yes—
273C2×C4×C61—21no—
274C2×C4×C61—21no—
275C2×C4×C61—22no—
276C2×C4×C61—22no—
277C2×C4×C61—22no—
278C2×C4×C61—22no—
279C2×C4×C61—22no—
280C2×C4×C61—21no—
281C2×C4×C61—21no—
282C2×C4×C61—44no—
283C2×C4×C61—44no—
284C2×C4×C61—41no—
285C2×C4×C61—41no—
286C2×C4×C61—41no—
287C2×C4×C61—81no—
288C2×C4×C61—161no—
289C2×C4×C61C22×D411yescharacteristic
290C2×C4×C61C22×D411yescharacteristic
291C2×C4×C61C22×Q811yescharacteristic
292C2×C4×C61C2×#4111yescharacteristic
293C2×C4×C61C2×#4111yescharacteristic
294C2×C4×C61H(5,𝔽2)11yescharacteristic
295C2×Dic61—22no—
296C2×Dic61—21no—
297C2×Dic61—21no—
298C2×Dic61—41no—
299C2×Dic61—41no—
300C2×Dic61—41no—
301C2×Dic61—41no—
302C2×Dic61—81no—
303C2×Dic61—81no—
304C2×Dic61—161no—
305Dic122—161no—
306C8×C61—21no—
307C8×C61—21no—
308C8×C61—41no—
309#92418—21no—
310#92418—21no—
311#92418—41no—
312#92418—41no—
313#92418—41no—
314#25986—611nomaximal, Sylow
315C22×C61—21no—
316C22×C61—21no—
317C22×C61—22no—
318C22×C61—22no—
319C22×C61—41no—
320C22×C61—41no—
321C22×C61—41no—
322C22×C61—41no—
323C22×C61#54311yescharacteristic
324C22×C61D4211yescharacteristic
325C22×C61#55111yescharacteristic
326C4×C61—22no—
327C4×C61—22no—
328C4×C61—42no—
329C4×C61—41no—
330C4×C61—41no—
331C4×C61—81no—
332C4×C61C2×C22⋊Q811yescharacteristic
333C4×C61C2×C22⋊Q811yescharacteristic
334C4×C61#54611yescharacteristic
335C4×C61#54611yescharacteristic
336Dic61—42no—
337Dic61—41no—
338Dic61—41no—
339Dic61—81no—
340Dic61—81no—
341#3084—611no—
342#3084—611no—
343#2898—611no—
344#2898D6111yescharacteristic
345#2096—611no—
346#2064—611no—
347#2064—611no—
348#2028—611no—
349#1866—611no—
350#1866—611no—
351#1859—611no—
352#1859—611no—
353#1815—611no—
354#1815—611no—
355#1793—611no—
356C2×C61—21no—
357C2×C61—21no—
358C2×C61—42no—
359C2×C61—42no—
360C2×C61#243711yescharacteristic
361C2×C61#328911yescharacteristic
362C2×C61#329111yescharacteristic
363C2×2−1+4C2×D6111yescharacteristic
364C2×#131—611no—
365C2×#131C2×D6111yescharacteristic
366#544—611no—
367#520—611no—
368#520C2×D6111yescharacteristic
369#479—1221no—
370#479—1221no—
371#478—1221no—
372#478—1221no—
373#476—1221no—
374#476—1221no—
375#475—1221no—
376#475—1221no—
377C2×#108—611no—
378C2×#108—611no—
379#428—611no—
380#428—611no—
381C2×#106—611no—
382C2×#106—611no—
383#424—611no—
384#424—611no—
385C2×#104—611no—
386C2×#104—611no—
387#419—21no—
388#419—611no—
389#419—1221no—
390#419C2×D6111yescharacteristic
391#417—21no—
392#417—611no—
393#417—611no—
394#417—611no—
395#417—1221no—
396#417C2×D6111yescharacteristic
397#416—21no—
398#416—611no—
399#416—1221no—
400#416C2×D6111yescharacteristic
401C2×#103—21no—
402C2×#103—611no—
403C2×#103—1221no—
404C2×#103C2×D6111yescharacteristic
405C2×#101—611no—
406#391—611no—
407#384—611no—
408#343—611no—
409#343—611no—
410#343—611no—
411#339—611no—
412#339—611no—
413#336—611no—
414C61#2598611yescharacteristic, complemented, Sylow
4152−1+4—22no—
4162−1+4—22no—
4172−1+4—41no—
4182−1+4Dic12214yescomplemented
419C2×#41—21no—
420C2×#41—21no—
421C2×#41—41no—
422C2×#41C22×D6111yescharacteristic
423C2×#41C22×D6111yescharacteristic
424C22×Q8—21no—
425C22×Q8—21no—
426C22×Q8C22×D6111yescharacteristic
427#131—22no—
428#131—41no—
429#131—1222no—
430#131—2441no—
431C2×M16—21no—
432C2×M16—611no—
433C2×M16—611no—
434C2×M16—611no—
435C2×M16—1221no—
436C2×M16C22×D6111yescharacteristic
437C22×C8—21no—
438C22×C8—611no—
439C22×C8—1221no—
440C22×C8C22×D6111yescharacteristic
441C4⋊Q8—1221no—
442C4⋊Q8—1221no—
443#125—1221no—
444#125—1221no—
445C22⋊Q8—2441no—
446C4×Q8—21no—
447C4×Q8—1221no—
448C4×D4—21no—
449C4×D4—41no—
450C4×D4—1221no—
451C4×D4—2441no—
452#117—21no—
453#117—611no—
454#117—1221no—
455#117C22×D6111yescharacteristic
456C2×C4⋊C4—21no—
457C2×C4⋊C4—611no—
458C2×C4⋊C4—611no—
459C2×C4⋊C4—1221no—
460C2×C4⋊C4—2441no—
461C2×C4⋊C4C22×D6111yescharacteristic
462C2×C42—611no—
463#108—2441no—
464#108—2441no—
465#107—1221no—
466#107—1221no—
467#106—1221no—
468#106—1221no—
469#106—1221no—
470#106—1221no—
471#106—1221no—
472#106—1221no—
473C4⋊C8—1221no—
474C4⋊C8—1221no—
475C4⋊C8—1221no—
476C4⋊C8—1221no—
477#104—1222no—
478#104—1222no—
479#103—21no—
480#103—21no—
481#103—21no—
482#103—21no—
483#103—21no—
484#103—611no—
485#103—611no—
486#103—1221no—
487#103—1221no—
488#103—1221no—
489#103—1221no—
490#103—1221no—
491#103—1221no—
492#103—1221no—
493#103C22⋊D6111yescharacteristic
494#103C22⋊D6111yescharacteristic
495#102—21no—
496#102—611no—
497#102—611no—
498#102—1221no—
499#102—1221no—
500#102—1221no—
501#102C22⋊D6111yescharacteristic
502#102C22⋊D6111yescharacteristic
503#101—2441no—
504#95—2441no—
505#41—21no—
506#41—21no—
507#41—21no—
508#41—21no—
509#41—24no—
510#41—24no—
511#41—44no—
512#41—42no—
513#41—42no—
514#41—44no—
515#41C2×Dic12212yes—
516#41C2×Dic12212yes—
517#41#1175822212yes—
518#41#1175822412yes—
519C2×Q8—21no—
520C2×Q8—24no—
521C2×Q8—24no—
522C2×Q8—21no—
523C2×Q8—21no—
524C2×Q8—21no—
525C2×Q8—21no—
526C2×Q8—41no—
527C2×Q8—42no—
528C2×Q8—42no—
529C2×Q8—2441no—
530C2×Q8C2×Dic12212yes—
531C2×Q8D4×D6111yescharacteristic
532C2×Q8C2×C22⋊D6111yescharacteristic
533C2×D4—21no—
534C2×D4—21no—
535C2×D4—41no—
536C2×D4D4×D6111yescharacteristic
537C2×D4C2×C22⋊D6111yescharacteristic
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—21no—
541C22×C4—21no—
542C22×C4—41no—
543C22×C4—41no—
544C22×C4—41no—
545C22×C4—1221no—
546C22×C4—1221no—
547C22×C4—1221no—
548C22×C4—2441no—
549C22×C4C23×D6111yescharacteristic
550M16—21no—
551M16—21no—
552M16—42no—
553M16—1221no—
554M16—1221no—
555M16—1221no—
556M16—1221no—
557M16—1221no—
558M16—1221no—
559M16—2442no—
560C2×C8—21no—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—42no—
564C2×C8—611no—
565C2×C8—611no—
566C2×C8—1221no—
567C2×C8—1221no—
568C2×C8—1221no—
569C2×C8—1221no—
570C2×C8—1221no—
571C2×C8—2442no—
572C2×C8C2×C22⋊D6111yescharacteristic
573C2×C8C2×C22⋊D6111yescharacteristic
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—41no—
577C4⋊C4—611no—
578C4⋊C4—611no—
579C4⋊C4—1221no—
580C4⋊C4—1221no—
581C4⋊C4—1221no—
582C4⋊C4—1221no—
583C4⋊C4—4882no—
584C4⋊C4—2441no—
585C4⋊C4—2441no—
586C4⋊C4—2441no—
587C4⋊C4—2441no—
588C4⋊C4C2×C22⋊D6111yescharacteristic
589C4⋊C4C2×C22⋊D6111yescharacteristic
590C22⋊C4—21no—
591C22⋊C4—41no—
592C22⋊C4—1221no—
593C22⋊C4—2441no—
594C42—21no—
595C42—41no—
596C42—1221no—
597C42—1221no—
598C42—1221no—
599C42—2441no—
600C23—21no—
601C23—21no—
602C23#1451982211yescharacteristic
603Q8—21no—
604Q8—21no—
605Q8—22no—
606Q8—22no—
607Q8—44no—
608Q8—41no—
609Q8—41no—
610Q8—9761no—
611Q8#1451975412yes—
612Q8#1451975612yes—
613D4—22no—
614D4—22no—
615D4—22no—
616D4—22no—
617D4—21no—
618D4—21no—
619D4—44no—
620D4#1451975412yes—
621D4#1451975612yes—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—81no—
637C2×C4—161no—
638C2×C4—1222no—
639C2×C4—1221no—
640C2×C4—1221no—
641C2×C4—9761no—
642C2×C4—4881no—
643C2×C4—4881no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4C2×D4×D6111yescharacteristic
649C2×C4C22×C22⋊D6111yescharacteristic
650C2×C4C22×Dic12211yescharacteristic
651C2×C4C2×#1175822411yescharacteristic
652C2×C4C2×#1175822211yescharacteristic
653C2×C4#1451982211yescharacteristic
654C8—21no—
655C8—21no—
656C8—41no—
657C8—1221no—
658C8—1221no—
659C8—2441no—
660C8—2441no—
661C8—2441no—
662C22—21no—
663C22—21no—
664C22—22no—
665C22—22no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22#1684038811yescharacteristic
671C22D4×C22⋊D6111yescharacteristic
672C22#1684043811yescharacteristic
673C4—22no—
674C4—22no—
675C4—42no—
676C4—41no—
677C4—41no—
678C4—81no—
679C4—4881no—
680C4—4881no—
681C4—2442no—
682C4—2441no—
683C4—2441no—
684C4C2×#1451975411yescharacteristic
685C4C2×#1451975611yescharacteristic
686C4#1684040211yescharacteristic
687C4#1684040611yescharacteristic
688C2—21no—
689C2—21no—
690C2—42no—
691C2—42no—
692C2#2144948011yescharacteristic
693C2#2145759011yescharacteristic
694C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2898C2
socleC22×C61D42
prime radicalC2×2−1+4×C61C22
hypercenter#2898D61
hypocenterC61#25986
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2898

Complements

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

NH ≅ G/Ntype
1. #21455681C2semidirect
2. #21455679C2semidirect
4. #21447360C2semidirect
5. #21447186C2semidirect
6. #21447184C2semidirect
7. #21446978C2semidirect
8. #21446096C2semidirect
9. #21446094C2semidirect
10. #21446045C2semidirect
11. #21446043C2semidirect
12. #21445878C2semidirect
13. #21445876C2semidirect
125. 2−1+4×C61Q8semidirect
414. C61#25986semidirect
418. 2−1+4Dic122semidirect

Maximal subgroups

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

subgroupindex
#214556812
#214556792
#214543522
#214473602
#214471862
#214471842
#214469782
#214460962
#214460942
#214460452
#214460432
#214458782
#214458762
#214457782
C61×#28982
#2598661