Finite Group Databasebeta

← #29743465

Subgroups — #29743465

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#21455685C211yesmaximal, characteristic
2#21455657C211yesmaximal, characteristic, complemented
3#21454663C211yesmaximal, characteristic
4#21454414C211yesmaximal, characteristic
5#21447360C211yesmaximal, characteristic, complemented
6#21447331C211yesmaximal, characteristic, complemented
7#21447182C211yesmaximal, characteristic, complemented
8#21447165C211yesmaximal, characteristic, complemented
9#21446598C211yesmaximal, characteristic, complemented
10#21446417C211yesmaximal, characteristic, complemented
11#21446190C211yesmaximal, characteristic
12#21446094C211yesmaximal, characteristic
13#21446043C211yesmaximal, characteristic
14#21446036C211yesmaximal, characteristic, complemented
15C61×#1908C211yesmaximal, characteristic
16#16840393C2211yescharacteristic
17#16840245C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840240C2211yescharacteristic
20#16840071—21no—
21#16840071—21no—
22#16840067—21no—
23#16840060—21no—
24#16840055—21no—
25#16840050—21no—
26#16840045—21no—
27#16840045—21no—
28#16839869—21no—
29#16839863—21no—
30Q16×Dic61—21no—
31SD16×Dic61—21no—
32C2×#14519712C2211yescharacteristic
33#16839804—21no—
34#16839804—21no—
35#16839804C2211yescharacteristic
36#16839804C2211yescharacteristic
37#16839804C2211yescharacteristic
38#16839804C2211yescharacteristic
39#16839802C2211yescharacteristic
40#16839800—21no—
41#16839800C2211yescharacteristic
42C2×#14519709—21no—
43C2×#14519709C2211yescharacteristic
44C2×#14519708C2211yescharacteristic
45#16839790C2211yescharacteristic
46#16839788C2211yescharacteristic
47C2×#14519704C2211yescharacteristic
48#16839771C2211yescharacteristic
49#16839758C2211yescharacteristic
50C2×#14519701C2211yescharacteristic
51#16839720C2211yescharacteristic
52#16839669C2211yescharacteristic
53#16839631C2211yescharacteristic
54#16839530C2211yescharacteristic
55#16839530C2211yescharacteristic
56#16839523C2211yescharacteristic
57#16839523C2211yescharacteristic
58#16839523C2211yescharacteristic
59#16839516C2211yescharacteristic
60C2×#137×C61C2211yescharacteristic
61C2×#131×C61C2211yescharacteristic
62C61×#417C2211yescharacteristic
63C2×#103×C61C2211yescharacteristic
64C2×#101×C61C2211yescharacteristic
65C61×#343C2211yescharacteristic
66C61×#341C2211yescharacteristic
67C2×#11758194C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519776—21no—
70#14519776—21no—
71#14519768—21no—
72#14519768—21no—
73#14519756—41no—
74Q8×Dic61—21no—
75Q8×Dic61—21no—
76Q8×Dic61—21no—
77Q8×Dic61—21no—
78Q8×Dic61—21no—
79Q8×Dic61—21no—
80D4×Dic61—21no—
81D4×Dic61—21no—
82D4×Dic61—41no—
83D4×Dic61—41no—
84#14519736—21no—
85#14519736—21no—
86#14519736—41no—
87#14519736C2311yescharacteristic
88#14519736C2311yescharacteristic
89#14519736C2311yescharacteristic
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×#11758196—41no—
95C2×C4×Dic61C2311yescharacteristic
96#14519712—22no—
97#14519711—21no—
98#14519711—21no—
99#14519711—21no—
100#14519711—21no—
101#14519711D411yescharacteristic
102#14519711D411yescharacteristic
103#14519710—21no—
104#14519709—21no—
105#14519709—21no—
106#14519709—21no—
107#14519709—21no—
108#14519709—21no—
109#14519709—21no—
110#14519709D411yescharacteristic
111#14519709D411yescharacteristic
112#14519708—21no—
113#14519708—21no—
114#14519708—21no—
115#14519706—21no—
116#14519706—21no—
117#14519706—21no—
118#14519706—21no—
119#14519704—22no—
120#14519701—21no—
121#14519701—21no—
122#14519701—21no—
123#14519701—21no—
124#14519701D411yescharacteristic
125#14519701D411yescharacteristic
126#14519698D411yescharacteristic
127#14519698D411yescharacteristic
128#14519687—21no—
129C8×Dic61—21no—
130#14519684—41no—
131C2×#41×C61C2311yescharacteristic
132C22×Q8×C61C2311yescharacteristic
133#137×C61—22no—
134#137×C61Q814yescomplemented
135C2×Q16×C61—21no—
136C2×SD16×C61—21no—
137#131×C61—22no—
138#131×C61—22no—
139C2×M16×C61—21no—
140C2×M16×C61C2311yescharacteristic
141C2×M16×C61C2311yescharacteristic
142C2×M16×C61C2311yescharacteristic
143C22×C8×C61—21no—
144C22×C8×C61C2311yescharacteristic
145C2×C4⋊C4×C61C2311yescharacteristic
146#103×C61—21no—
147#103×C61—21no—
148#103×C61—21no—
149#102×C61—21no—
150#101×C61—22no—
151#41×C61—21no—
152#41×C61—21no—
153#41×C61C2×Q812yes—
154#41×C61#4112yes—
155C2×Q8×C61—21no—
156C2×Q8×C61—21no—
157C2×Q8×C61—21no—
158C2×Q8×C61—21no—
159C2×Q8×C61—21no—
160C2×Q8×C61C2×D411yescharacteristic
161C2×Q8×C61C2×Q812yes—
162C2×D4×C61—21no—
163C2×D4×C61C2×D411yescharacteristic
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61C2411yescharacteristic
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61—21no—
171C22×Dic61—41no—
172C22×Dic61—41no—
173C2×Dic122—41no—
174Q16×C61—24no—
175SD16×C61—24no—
176M16×C61—24no—
177M16×C61—21no—
178M16×C61—21no—
179M16×C61—21no—
180M16×C61—21no—
181M16×C61C2×Q812yes—
182M16×C61#4112yes—
183C2×C8×C61—24no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61C2×D411yescharacteristic
190C2×C8×C61C2×D411yescharacteristic
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C22⋊Dic61—21no—
194C22⋊Dic61—21no—
195C22⋊Dic61—21no—
196C22⋊Dic61—41no—
197C22⋊Dic61—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—41no—
203C4⋊Dic61—41no—
204C4⋊Dic61—41no—
205C4⋊Dic61—41no—
206C4⋊Dic61—41no—
207C4⋊Dic61C2×D411yescharacteristic
208C4⋊Dic61C2×D411yescharacteristic
209C4⋊Dic61C2×D411yescharacteristic
210C4⋊Dic61C2×D411yescharacteristic
211C4⋊Dic61C2×D411yescharacteristic
212C4⋊Dic61C2×D411yescharacteristic
213#11758196—41no—
214#11758196—41no—
215#11758196—82no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—41no—
222C4×Dic61—41no—
223C4×Dic61—41no—
224C4×Dic61—41no—
225#11758194—21no—
226#11758194—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418C2×D411yescharacteristic
231C2×#92418C2×D411yescharacteristic
232C23×C61—21no—
233C23×C61H(5,𝔽2)11yescharacteristic
234Q8×C61—21no—
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—41no—
238Q8×C61—41no—
239Q8×C61C22⋊Q812yes—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—21no—
243D4×C61C22⋊Q812yes—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—81no—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×Q811yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—81no—
270C2×Dic61—81no—
271C2×Dic61—161no—
272C2×Dic61—161no—
273Dic122—161no—
274C8×C61—22no—
275C8×C61—21no—
276C8×C61—22no—
277C8×C61—21no—
278C8×C61—41no—
279#92418—21no—
280#92418—21no—
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—81no—
294C4×C61C2×C22⋊Q811yescharacteristic
295C4×C61#52511yescharacteristic
296C4×C61#54711yescharacteristic
297C4×C61D4×Q811yescharacteristic
298Dic61—42no—
299Dic61—41no—
300Dic61—41no—
301Dic61—41no—
302Dic61—41no—
303Dic61—41no—
304Dic61—41no—
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#1908D6111yescharacteristic
318#1882—611no—
319#1866—611no—
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×#137C2×D6111yescharacteristic
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×#108—611no—
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#424—611no—
361C2×#104—611no—
362#419—611no—
363#417C2×D6111yescharacteristic
364#416—611no—
365C2×#103—611no—
366C2×#103C2×D6111yescharacteristic
367C2×#101C2×D6111yescharacteristic
368#405—611no—
369#391—611no—
370#384—611no—
371#343—611no—
372#343—611no—
373#343C2×D6111yescharacteristic
374#341C2×D6111yescharacteristic
375#339—611no—
376#339—611no—
377#339—611no—
378#336—611no—
379C61#2640111yescharacteristic, complemented, Sylow
380C2×#41C22×D6111yescharacteristic
381C22×Q8C22×D6111yescharacteristic
382#137—22no—
383#137Dic12214yescomplemented
384C2×Q16—21no—
385C2×SD16—21no—
386#131—22no—
387#131—22no—
388C2×M16—21no—
389C2×M16—611no—
390C2×M16C22×D6111yescharacteristic
391C2×M16C22×D6111yescharacteristic
392C2×M16C22×D6111yescharacteristic
393C22×C8—21no—
394C22×C8—611no—
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—611no—
414#117—1221no—
415#117—1221no—
416#117—2441no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—2441no—
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C42—611no—
424#108—1222no—
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—1221no—
444C4⋊C8—1221no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447#104—1222no—
448#103—21no—
449#103—21no—
450#103—21no—
451#103—611no—
452#103—611no—
453#103—1221no—
454#103—1221no—
455#103—1221no—
456#103—1221no—
457#102—21no—
458#102—611no—
459#102—611no—
460#101—22no—
461#97—1221no—
462C4×C8—1221no—
463#95—2441no—
464#41—21no—
465#41—21no—
466#41C2×Dic12212yes—
467#41#1175822412yes—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—2441no—
474C2×Q8C2×Dic12212yes—
475C2×Q8D4×D6111yescharacteristic
476C2×D4—21no—
477C2×D4D4×D6111yescharacteristic
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—2441no—
485C22×C4—2441no—
486C22×C4C23×D6111yescharacteristic
487Q16—24no—
488SD16—24no—
489M16—24no—
490M16—21no—
491M16—21no—
492M16—21no—
493M16—21no—
494M16—1221no—
495M16—1221no—
496M16C2×Dic12212yes—
497M16#1175822412yes—
498C2×C8—24no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—611no—
505C2×C8—611no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8D4×D6111yescharacteristic
510C2×C8D4×D6111yescharacteristic
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—611no—
514C4⋊C4—611no—
515C4⋊C4—611no—
516C4⋊C4—611no—
517C4⋊C4—611no—
518C4⋊C4—611no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—4882no—
524C4⋊C4—2441no—
525C4⋊C4—2441no—
526C4⋊C4—2441no—
527C4⋊C4—2441no—
528C4⋊C4—2441no—
529C4⋊C4—2441no—
530C4⋊C4—2441no—
531C22⋊C4—1221no—
532C22⋊C4—1221no—
533C22⋊C4—1221no—
534C22⋊C4—2441no—
535C22⋊C4—2441no—
536C42—1221no—
537C42—1221no—
538C42—1221no—
539C42—1221no—
540C42—1221no—
541C42—2441no—
542C42—2441no—
543C42—2441no—
544C42—2441no—
545C23—21no—
546C23#1451982311yescharacteristic
547Q8—21no—
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552Q8—9761no—
553Q8#1451975412yes—
554D4—22no—
555D4—22no—
556D4—21no—
557D4#1451975412yes—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—81no—
566C2×C4—1222no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—9761no—
574C2×C4—9761no—
575C2×C4—4881no—
576C2×C4—4881no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C22×Dic12211yescharacteristic
584C2×C4C2×#1175822411yescharacteristic
585C2×C4C2×#1175822411yescharacteristic
586C2×C4#1451982411yescharacteristic
587C8—22no—
588C8—21no—
589C8—22no—
590C8—21no—
591C8—41no—
592C8—1221no—
593C8—1221no—
594C8—2441no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22#1684035611yescharacteristic
601C22#1684039611yescharacteristic
602C22#1684046911yescharacteristic
603C4—22no—
604C4—42no—
605C4—81no—
606C4—4881no—
607C4—4881no—
608C4—2442no—
609C4—2441no—
610C4—2441no—
611C4—2441no—
612C4—2441no—
613C4—2441no—
614C4—2441no—
615C4C2×#1451975411yescharacteristic
616C4#1684026811yescharacteristic
617C4#1684041211yescharacteristic
618C4D4×Dic12211yescharacteristic
619C2—21no—
620C2—21no—
621C2—42no—
622C2#2144963411yescharacteristic
623C2#2145759311yescharacteristic
624C2#2145761111yescharacteristic
subgroupG/H
centerC22#16840396
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1908C2
socleC22×C61D42
prime radicalC2×#41×C61C23
hypercenter#1908D61
hypocenterC61#26401
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1908

Complements

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

NH ≅ G/Ntype
2. #21455657C2semidirect
5. #21447360C2semidirect
6. #21447331C2semidirect
7. #21447182C2semidirect
8. #21447165C2semidirect
9. #21446598C2semidirect
10. #21446417C2semidirect
14. #21446036C2semidirect
134. #137×C61Q8semidirect
379. C61#26401semidirect
383. #137Dic122semidirect

Maximal subgroups

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

subgroupindex
#214556852
#214556572
#214546632
#214544142
#214473602
#214473312
#214471822
#214471652
#214465982
#214464172
#214461902
#214460942
#214460432
#214460362
C61×#19082
#2640161