Finite Group Databasebeta

← #29743457

Subgroups — #29743457

Order 15616.

714 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455679C211yesmaximal, characteristic
2#21455662C211yesmaximal, characteristic, complemented
3#21454665C211yesmaximal, characteristic
4#21454414C211yesmaximal, characteristic
5#21447361C211yesmaximal, characteristic, complemented
6#21447329C211yesmaximal, characteristic, complemented
7#21447181C211yesmaximal, characteristic, complemented
8#21447163C211yesmaximal, characteristic, complemented
9#21446600C211yesmaximal, characteristic, complemented
10#21446419C211yesmaximal, characteristic, complemented
11#21446334C211yesmaximal, characteristic
12#21446186C211yesmaximal, characteristic
13#21446047C211yesmaximal, characteristic
14#21446035C211yesmaximal, characteristic, complemented
15C61×#1866C211yesmaximal, characteristic
16C2×#14519807C2211yescharacteristic
17#16840390C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840239C2211yescharacteristic
21#16840070—21no—
22#16840070—21no—
23#16840067—21no—
24#16840064—21no—
25#16840058—21no—
26#16840050—21no—
27#16840048—21no—
28#16840045—21no—
29#16839866—21no—
30#16839859—21no—
31#16839854—21no—
32#16839853—21no—
33#16839805C2211yescharacteristic
34#16839805C2211yescharacteristic
35#16839804—21no—
36#16839804—21no—
37#16839804C2211yescharacteristic
38#16839804C2211yescharacteristic
39#16839804C2211yescharacteristic
40#16839800—21no—
41#16839800C2211yescharacteristic
42C2×#14519709—21no—
43C2×#14519709C2211yescharacteristic
44C2×#14519709C2211yescharacteristic
45#16839788C2211yescharacteristic
46#16839772C2211yescharacteristic
47#16839763C2211yescharacteristic
48C2×#14519702C2211yescharacteristic
49C2×#14519702C2211yescharacteristic
50C2×#14519695C2211yescharacteristic
51#16839717C2211yescharacteristic
52#16839668C2211yescharacteristic
53#16839632C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839527C2211yescharacteristic
58#16839522C2211yescharacteristic
59#16839522C2211yescharacteristic
60C2×#131×C61C2211yescharacteristic
61C2×#108×C61C2211yescharacteristic
62C61×#424C2211yescharacteristic
63C2×#104×C61C2211yescharacteristic
64C61×#416C2211yescharacteristic
65C61×#339C2211yescharacteristic
66C61×#336C2211yescharacteristic
67C22×Dic122C2311yescharacteristic
68#14519807—22no—
69#14519807Q814yescomplemented
70C2×#11758202—21no—
71C2×#11758200—21no—
72C2×#11758194C2311yescharacteristic
73C2×#11758194C2311yescharacteristic
74#14519777—21no—
75#14519776—21no—
76#14519769—21no—
77#14519768—21no—
78#14519754—41no—
79Q8×Dic61—21no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122—21no—
83#14519742—21no—
84#14519742—21no—
85D4×Dic61—21no—
86D4×Dic61—21no—
87D4×Dic61—41no—
88D4×Dic61—41no—
89#14519736—21no—
90#14519736—21no—
91#14519736C2311yescharacteristic
92#14519736C2311yescharacteristic
93#14519733—41no—
94C2×C4⋊Dic61—21no—
95C2×C4⋊Dic61—21no—
96C2×C4⋊Dic61—41no—
97C2×C4⋊Dic61C2311yescharacteristic
98C2×C4⋊Dic61C2311yescharacteristic
99C2×C4⋊Dic61C2311yescharacteristic
100#14519711—21no—
101#14519711—21no—
102#14519711—21no—
103#14519711D411yescharacteristic
104#14519711D411yescharacteristic
105#14519710—21no—
106#14519710—21no—
107#14519709—21no—
108#14519709—21no—
109#14519709—21no—
110#14519709—21no—
111#14519709—21no—
112#14519709—21no—
113#14519709—21no—
114#14519709D411yescharacteristic
115#14519709D411yescharacteristic
116#14519708—21no—
117#14519708—21no—
118#14519706—21no—
119#14519706—21no—
120#14519702—21no—
121#14519702—21no—
122#14519702—21no—
123#14519702—21no—
124#14519702—21no—
125#14519702—21no—
126#14519701—21no—
127#14519698—21no—
128#14519695—22no—
129#14519687—21no—
130C8×Dic61—21no—
131#14519684—41no—
132C2×#41×C61C2311yescharacteristic
133#131×C61—22no—
134#131×C61—22no—
135C2×M16×C61—21no—
136C2×M16×C61C2311yescharacteristic
137C2×M16×C61C2311yescharacteristic
138C22×C8×C61—21no—
139C22×C8×C61C2311yescharacteristic
140C22×C8×C61C2311yescharacteristic
141#117×C61C2311yescharacteristic
142C2×C42×C61C2311yescharacteristic
143#108×C61—22no—
144C4⋊C8×C61—21no—
145C4⋊C8×C61—21no—
146#104×C61—22no—
147#103×C61D411yescharacteristic
148#103×C61D411yescharacteristic
149#102×C61D411yescharacteristic
150#102×C61D411yescharacteristic
151#41×C61—21no—
152#41×C61—21no—
153#41×C61C2×Q812yes—
154#41×C61#4112yes—
155C2×Q8×C61C2×D411yescharacteristic
156C2×D4×C61—21no—
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C22×Dic61—41no—
166C22×Dic61—41no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—41no—
173C2×Dic122C2×Q812yes—
174M16×C61—24no—
175M16×C61—21no—
176M16×C61—21no—
177M16×C61—21no—
178M16×C61#4112yes—
179C2×C8×C61—24no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61C2×D411yescharacteristic
187C2×C8×C61C2×D411yescharacteristic
188C2×C8×C61C2×D411yescharacteristic
189C2×C8×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C22⋊C4×C61—21no—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61—21no—
196C22⋊Dic61—21no—
197C22⋊Dic61—21no—
198C22⋊Dic61—41no—
199C22⋊Dic61—41no—
200#11758202—24no—
201#11758200—24no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—41no—
209C4⋊Dic61—41no—
210C4⋊Dic61—41no—
211C4⋊Dic61—82no—
212C4⋊Dic61C2×D411yescharacteristic
213C4⋊Dic61C2×D411yescharacteristic
214C4⋊Dic61C2×D411yescharacteristic
215C4⋊Dic61C2×D411yescharacteristic
216#11758196—41no—
217#11758196—41no—
218#11758196—41no—
219#11758196—41no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—41no—
223C4×Dic61—41no—
224C4×Dic61—41no—
225C4×Dic61—41no—
226#11758194—21no—
227#11758194—21no—
228#11758194—21no—
229#11758194C2×Q812yes—
230C2×#92418—21no—
231C2×#92418—21no—
232C23×C61—21no—
233C23×C61H(5,𝔽2)11yescharacteristic
234Q8×C61—21no—
235Q8×C61C22⋊Q812yes—
236D4×C61—22no—
237D4×C61—22no—
238D4×C61—21no—
239D4×C61C22⋊Q812yes—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×Q811yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C612−1+411yescharacteristic
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—81no—
265C2×Dic61—81no—
266C2×Dic61—81no—
267C2×Dic61—161no—
268C2×Dic61—161no—
269Dic122—22no—
270Dic122—22no—
271Dic122—41no—
272Dic122—41no—
273Dic122—161no—
274C8×C61—21no—
275C8×C61—22no—
276C8×C61—21no—
277C8×C61—21no—
278C8×C61—21no—
279C8×C61—41no—
280#92418—22no—
281#92418—41no—
282#26401—611nomaximal, Sylow
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—22no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61#53911yescharacteristic
289C22×C61D4211yescharacteristic
290C22×C61#55411yescharacteristic
291C4×C61—22no—
292C4×C61—42no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61C2×C22⋊Q811yescharacteristic
296C4×C61#52511yescharacteristic
297C4×C61#54711yescharacteristic
298C4×C61D4×Q811yescharacteristic
299Dic61—42no—
300Dic61—41no—
301Dic61—41no—
302Dic61—41no—
303Dic61—41no—
304Dic61—81no—
305Dic61—81no—
306Dic61—81no—
307#3084—611no—
308#3082—611no—
309#2947—611no—
310#2912—611no—
311#2096—611no—
312#2090—611no—
313#2064—611no—
314#2060—611no—
315#1951—611no—
316#1922—611no—
317#1908—611no—
318#1882—611no—
319#1866D6111yescharacteristic
320#1859—611no—
321#1857—611no—
322C2×C61—21no—
323C2×C61—21no—
324C2×C61—42no—
325C2×C61#244911yescharacteristic
326C2×C61#328911yescharacteristic
327C2×C61#329111yescharacteristic
328C2×#137—611no—
329C2×#131C2×D6111yescharacteristic
330#544—611no—
331#520—611no—
332#520—611no—
333#519—611no—
334#479—1221no—
335#479—1221no—
336#479—1221no—
337#478—1221no—
338#477—1221no—
339#476—1221no—
340#475—1221no—
341#475—1221no—
342#441—1221no—
343#440—1221no—
344C4×Q16—1221no—
345C4×SD16—1221no—
346C2×#108C2×D6111yescharacteristic
347#428—611no—
348#428—611no—
349#428—611no—
350#428—611no—
351#428—611no—
352#428—1221no—
353#428—1221no—
354#427—611no—
355#427—1221no—
356C2×#106—611no—
357C2×#106—611no—
358C2×#106—1221no—
359#424—611no—
360#424C2×D6111yescharacteristic
361C2×#104C2×D6111yescharacteristic
362#419—611no—
363#417—611no—
364#416C2×D6111yescharacteristic
365C2×#103—611no—
366C2×#103—611no—
367C2×#101—611no—
368#405—611no—
369#391—611no—
370#384—611no—
371#343—611no—
372#343—611no—
373#343—611no—
374#341—611no—
375#339—611no—
376#339—611no—
377#339C2×D6111yescharacteristic
378#336C2×D6111yescharacteristic
379C61#2640111yescharacteristic, complemented, Sylow
380C2×#41C22×D6111yescharacteristic
381C22×Q8—611no—
382#137—614no—
383#137—1222no—
384C2×Q16—1221no—
385C2×SD16—1221no—
386#131—22no—
387#131—22no—
388C2×M16—21no—
389C2×M16—611no—
390C2×M16—611no—
391C2×M16C22×D6111yescharacteristic
392C2×M16C22×D6111yescharacteristic
393C22×C8—21no—
394C22×C8C22×D6111yescharacteristic
395C22×C8C22×D6111yescharacteristic
396C4⋊Q8—1221no—
397C4⋊Q8—1221no—
398#125—1221no—
399#125—1221no—
400C22⋊Q8—2441no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×Q8—1221no—
407C4×D4—1221no—
408C4×D4—1221no—
409C4×D4—2441no—
410C4×D4—2441no—
411#117—611no—
412#117—611no—
413#117—1221no—
414#117—1221no—
415#117—2441no—
416#117C22×D6111yescharacteristic
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—2441no—
423C2×C42C22×D6111yescharacteristic
424#108—22no—
425#107—611no—
426#107—611no—
427#107—1221no—
428#107—1221no—
429#107—1221no—
430#107—1221no—
431#107—1221no—
432#106—611no—
433#106—611no—
434#106—1221no—
435#106—1221no—
436#106—1221no—
437#106—1221no—
438#106—1221no—
439#106—1221no—
440#106—1221no—
441#106—1221no—
442#106—1221no—
443C4⋊C8—21no—
444C4⋊C8—21no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447#104—22no—
448#103—1221no—
449#103—1221no—
450#103—1221no—
451#103—1221no—
452#103—1221no—
453#103—1221no—
454#103—1221no—
455#103D24411yescharacteristic
456#103D24411yescharacteristic
457#102—1221no—
458#102D24411yescharacteristic
459#102D24411yescharacteristic
460#101—1222no—
461#97—1221no—
462C4×C8—1221no—
463#95—2441no—
464#41—21no—
465#41—21no—
466#41#1175822412yes—
467#41Q8×D6112yes—
468C2×Q8—612no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—2441no—
475C2×Q8C2×D24411yescharacteristic
476C2×D4—21no—
477C2×D4C2×D24411yescharacteristic
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—2441no—
485C22×C4—2441no—
486C22×C4C23×D6111yescharacteristic
487Q16—1224no—
488SD16—1224no—
489M16—24no—
490M16—21no—
491M16—21no—
492M16—21no—
493M16—612no—
494M16—1221no—
495M16—1221no—
496M16—1221no—
497M16#1175822412yes—
498C2×C8—24no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8C2×D24411yescharacteristic
508C2×C8C2×D24411yescharacteristic
509C2×C8D4×D6111yescharacteristic
510C2×C8D4×D6111yescharacteristic
511C4⋊C4—611no—
512C4⋊C4—611no—
513C4⋊C4—611no—
514C4⋊C4—611no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—4882no—
522C4⋊C4—2441no—
523C4⋊C4—2441no—
524C4⋊C4—2441no—
525C4⋊C4—2441no—
526C4⋊C4—2441no—
527C4⋊C4—2441no—
528C4⋊C4—2441no—
529C4⋊C4C2×D24411yescharacteristic
530C4⋊C4C2×D24411yescharacteristic
531C22⋊C4—21no—
532C22⋊C4—1221no—
533C22⋊C4—1221no—
534C22⋊C4—2441no—
535C22⋊C4—2441no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—1221no—
540C42—1221no—
541C42—2441no—
542C42—2441no—
543C42—2441no—
544C42—2441no—
545C23—21no—
546C23#1451982311yescharacteristic
547Q8—21no—
548Q8—1222no—
549Q8—1222no—
550Q8—9761no—
551Q8—2441no—
552Q8—2441no—
553Q8#1451975712yes—
554D4—22no—
555D4—22no—
556D4—21no—
557D4#1451975712yes—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—1222no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—9761no—
572C2×C4—9761no—
573C2×C4—4881no—
574C2×C4—4881no—
575C2×C4—4881no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C22×D24411yescharacteristic
583C2×C4C2×Q8×D6111yescharacteristic
584C2×C4C2×#1175822411yescharacteristic
585C2×C4C2×#1175822411yescharacteristic
586C2×C4#1451982411yescharacteristic
587C8—21no—
588C8—22no—
589C8—21no—
590C8—21no—
591C8—21no—
592C8—41no—
593C8—1222no—
594C8—2441no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22#1684035611yescharacteristic
601C22D4×D24411yescharacteristic
602C22#1684047011yescharacteristic
603C4—22no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—4881no—
608C4—4881no—
609C4—4881no—
610C4—2442no—
611C4—2441no—
612C4—2441no—
613C4—2441no—
614C4—2441no—
615C4C2×#1451975711yescharacteristic
616C4#1684026811yescharacteristic
617C4#1684041611yescharacteristic
618C4#1684042511yescharacteristic
619C2—21no—
620C2—21no—
621C2—42no—
622C2#2144962611yescharacteristic
623C2#2145759511yescharacteristic
624C2#2145761311yescharacteristic
subgroupG/H
centerC22D4×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1866C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#1866D61
hypocenterC61#26401
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1866

Complements

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

NH ≅ G/Ntype
2. #21455662C2semidirect
5. #21447361C2semidirect
6. #21447329C2semidirect
7. #21447181C2semidirect
8. #21447163C2semidirect
9. #21446600C2semidirect
10. #21446419C2semidirect
14. #21446035C2semidirect
69. #14519807Q8semidirect
379. C61#26401semidirect

Maximal subgroups

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

subgroupindex
#214556792
#214556622
#214546652
#214544142
#214473612
#214473292
#214471812
#214471632
#214466002
#214464192
#214463342
#214461862
#214460472
#214460352
C61×#18662
#2640161