Finite Group Databasebeta

← #30622272

Subgroups — #30622272

Order 15616.

844 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462580C211yesmaximal, characteristic, complemented
2#21462577C212yesmaximal
3#21462474C211yesmaximal, characteristic
4#21461965C211yesmaximal, characteristic, complemented
5#21461937C211yesmaximal, characteristic, complemented
6#21461904C212yesmaximal, complemented
7#21461818C212yesmaximal
8#21461775C212yesmaximal, complemented
9#21461624C211yesmaximal, characteristic, complemented
10#21461593C211yesmaximal, characteristic
11#473×D61C211yesmaximal, characteristic
12#16840831C2212yes—
13#16840830C2212yes—
14#16840802—22no—
15#16840802—22no—
16#16840801—22no—
17Q16×#6923—21no—
18Q16×#6923—21no—
19Q16×#6923—22no—
20SD16×#6923—22no—
21#16840773—22no—
22#16840773C2212yes—
23#16840773C2212yes—
24#16840772—21no—
25#16840772C2211yescharacteristic
26C2×#14519846—21no—
27C2×#14519846C2211yescharacteristic
28D61⋊#104C2211yescharacteristic
29#16840755C2212yes—
30#16840752C2211yescharacteristic
31C2×#14519840C2212yes—
32#16840728C2212yes—
33#16840725C2211yescharacteristic
34C2×C8×#6923C2211yescharacteristic
35(C2×#6923)⋊Q8C2212yes—
36#16840641C2212yes—
37#16840637C2211yescharacteristic
38#16840634C2211yescharacteristic
39#16840634C2212yes—
40#137×D61C2212yes—
41C2×Q16×D61C2211yescharacteristic
42#131×D61C2211yescharacteristic
43#16840038C411yescharacteristic, complemented
44#16840037C411yescharacteristic, complemented
45#16840036C412yescomplemented
46#16840035C412yescomplemented
47#16840034C411yescharacteristic, complemented
48#108×D61C2211yescharacteristic
49#101×D61C2212yes—
50C61×#473C411yescharacteristic, complemented
51D61⋊M16C2311yescharacteristic
52C2×D61⋊C8C2311yescharacteristic
53Q8×#6923—22no—
54Q8×#6923—22no—
55Q8×#6923—22no—
56Q8×#6923—22no—
57Q8×#6923—22no—
58D4×#6923—22no—
59D4×#6923—42no—
60#14519856—22no—
61#14519856—42no—
62#14519856C2311yescharacteristic
63#14519856C2312yes—
64C2×C4⋊#6923—22no—
65C2×C4⋊#6923C2312yes—
66C2×C4×#6923C2311yescharacteristic
67#14519846—22no—
68#14519846—22no—
69#14519846—21no—
70#14519846—21no—
71#14519846D411yescharacteristic
72#14519846D411yescharacteristic
73#14519845—22no—
74#14519845—22no—
75#14519845D412yes—
76#14519842—21no—
77#14519841—21no—
78#14519840—22no—
79#14519840—22no—
80#14519840—22no—
81#14519840—22no—
82#14519840D411yescharacteristic
83#14519840D411yescharacteristic
84#14519839D411yescharacteristic
85#14519839D411yescharacteristic
86#14519831—22no—
87#14519831—21no—
88C8×#6923—22no—
89C8×#6923—21no—
90C8×#6923—21no—
91C8×#6923—21no—
92(C2×#6923)⋊C4—42no—
93#41×D61C2311yescharacteristic
94C2×Q8×D61C2312yes—
95#14519810C2×C412yes—
96#14519809—22no—
97#14519808C2×C412yes—
98#14519807C2×C412yes—
99#14519806—22no—
100C2×Dic244C2×C411yescharacteristic, complemented
101C2×#11758202C2×C412yescomplemented
102Q16×D61—22no—
103Q16×D61—21no—
104Q16×D61—21no—
105Q16×D61—21no—
106Q16×D61—21no—
107SD16×D61—22no—
108#14519785—21no—
109#14519785C2×C411yescharacteristic
110#14519784C2×C411yescharacteristic
111#14519783—21no—
112#14519783C2×C411yescharacteristic
113M16×D61—21no—
114M16×D61C2312yes—
115M16×D61C2311yescharacteristic
116C2×C8×D61—21no—
117C2×C8×D61C2311yescharacteristic
118#14519713C2×C411yescharacteristic, complemented
119#14519712C2×C412yescomplemented
120#14519696C2×C412yescomplemented
121#14519695C2×C412yescomplemented
122#14519695C2×C412yescomplemented
123#137×C61C2×C412yes—
124C2×Q16×C61C2×C411yescharacteristic, complemented
125#131×C61C2×C411yescharacteristic
126#108×C61C2×C411yescharacteristic, complemented
127#101×C61C2×C412yescomplemented
128C22×#6923—21no—
129C22×#6923—22no—
130C22×#6923—42no—
131#41×C61C22×C411yescharacteristic
132C2×Q8×C61C22×C412yes—
133#11758226#4111yescharacteristic
134Q8×D61—22no—
135Q8×D61—22no—
136Q8×D61—22no—
137Q8×D61—22no—
138Q8×D61C2×D411yescharacteristic
139#11758224—21no—
140#11758224#4111yescharacteristic
141D4×D61—21no—
142D4×D61C2×D411yescharacteristic
143#11758222—21no—
144#11758222C22×C411yescharacteristic
145C2×C4×D61—21no—
146C2×C4×D61—22no—
147C2×C4×D61C2411yescharacteristic
148C2×Dic122—22no—
149C2×Dic122C22×C412yes—
150C22⋊#6923—21no—
151C22⋊#6923—22no—
152C22⋊#6923—42no—
153#11758217—21no—
154C2×#92420—21no—
155C4⋊#6923—22no—
156C4⋊#6923—22no—
157C4⋊#6923—42no—
158C4⋊#6923—42no—
159C4⋊#6923—42no—
160C4⋊#6923C2×D411yescharacteristic
161C4⋊#6923C2×D411yescharacteristic
162C4⋊#6923C2×D412yes—
163C4⋊#6923C2×D412yes—
164C4×#6923—22no—
165C4×#6923—21no—
166C4×#6923—21no—
167C4×#6923—21no—
168C4×#6923—42no—
169C4×#6923—42no—
170C4×#6923—42no—
171#11758213—21no—
172D61⋊C8—21no—
173D61⋊C8—21no—
174D61⋊C8C2×D411yescharacteristic
175D61⋊C8C2×D411yescharacteristic
176Q16×C61—22no—
177Q16×C61—21no—
178Q16×C61—21no—
179SD16×C61—22no—
180M16×C61—21no—
181M16×C61C22×C412yes—
182M16×C61C22×C411yescharacteristic
183C2×C8×C61—21no—
184C2×C8×C61C22×C411yescharacteristic
185#11758202—22no—
186#11758202—22no—
187#11758202—22no—
188#11758202—22no—
189#11758201—22no—
190#11758200—22no—
191#11758194—21no—
192#11758194C22×C412yes—
193#11758194C22×C411yescharacteristic
194C2×#92418—21no—
195C2×#92418C22×C411yescharacteristic
196Dic244—22no—
197Dic244—21no—
198Dic244—21no—
199#11758190—22no—
200#11758189—22no—
201#11758189—21no—
202#11758189—21no—
203#11758189—21no—
204C8×D61—22no—
205C8×D61—21no—
206C8×D61—21no—
207C8×D61—21no—
208C8×D61C2×D411yescharacteristic
209C8×D61C2×D411yescharacteristic
210C8×D61#4111yescharacteristic
211C8×D61#4111yescharacteristic
212C22×D61—21no—
213C22×D61H(5,𝔽2)11yescharacteristic
214C2×#6923—22no—
215C2×#6923—21no—
216C2×#6923—21no—
217C2×#6923—22no—
218C2×#6923—22no—
219C2×#6923—41no—
220C2×#6923—41no—
221C2×#6923—42no—
222C2×#6923—82no—
223C2×#6923—162no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61C4×D411yescharacteristic
227D4×C61—21no—
228D4×C61C4×D411yescharacteristic
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61C23×C411yescharacteristic
232C22⋊D61—21no—
233C22⋊D61—21no—
234C22⋊D61—21no—
235C2×Dic61—21no—
236C2×Dic61—22no—
237C2×Dic612−1+411yescharacteristic
238D244—21no—
239D244C4×D411yescharacteristic
240C4×D61—21no—
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—42no—
244C4×D61—42no—
245C4×D61C22×D411yescharacteristic
246C4×D61C22×D411yescharacteristic
247C4×D61C2×#4111yescharacteristic
248C4×D61C2×#4111yescharacteristic
249Dic122—21no—
250Dic122—22no—
251Dic122—22no—
252Dic122—42no—
253Dic122—42no—
254Dic122C4×D411yescharacteristic
255#92420—21no—
256#92420—21no—
257#92420—41no—
258C8×C61—21no—
259C8×C61—22no—
260C8×C61C4×D411yescharacteristic
261C8×C61C4×D411yescharacteristic
262#92418—21no—
263#92418—22no—
264#92418C4×D411yescharacteristic
265#92418C4×D411yescharacteristic
266#23045—611nomaximal, Sylow
267C22×C61—21no—
268C22×C61#52011yescharacteristic
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—41no—
273C2×D61—41no—
274C2×D61D4211yescharacteristic
275C2×D61#55311yescharacteristic
276#6923—42no—
277#6923—41no—
278#6923—42no—
279#6923—41no—
280#6923—42no—
281#6923—82no—
282C4×C61—21no—
283C4×C61—42no—
284C4×C61C2×C4×D411yescharacteristic
285C4×C61C2×C4×D411yescharacteristic
286Dic61—21no—
287Dic61—42no—
288Dic61#54711yescharacteristic
289Dic61#54811yescharacteristic
290C2×#473—611no—
291#2950—611no—
292#2947—612no—
293#2913—611no—
294#1969—611no—
295#1952—611no—
296#1930—612no—
297#1882—612no—
298#1857—612no—
299#1772—611no—
300#1764—611no—
301C2×C61—21no—
302C2×C61—41no—
303C2×C61#236111yescharacteristic
304D61—21no—
305D61—41no—
306D61#329011yescharacteristic
307D61#329311yescharacteristic
308C2×#137—612no—
309C22×Q16—611no—
310C2×#131—611no—
311#520—612no—
312#519—612no—
313#473—612no—
314#473—611no—
315#473—611no—
316#473—611no—
317#473—612no—
318#473#692311yescharacteristic, complemented
319#441—1222no—
320#441—1222no—
321#440—1222no—
322C4×Q16—1222no—
323C4×Q16—1221no—
324C4×Q16—1221no—
325C4×SD16—1222no—
326C2×#108—611no—
327#428—612no—
328#428—612no—
329#428—1222no—
330#427—611no—
331#427—1221no—
332C2×#107—611no—
333C2×#107—1221no—
334C2×#104—611no—
335#419—612no—
336#416—611no—
337C2×#103—612no—
338C2×#101—612no—
339#405—611no—
340#405—612no—
341C2×C4×C8—611no—
342#391—612no—
343#343—612no—
344#341—611no—
345#339—611no—
346#339—612no—
347C61#2304511yescharacteristic, complemented, Sylow
348C2×#41—611no—
349C22×Q8—612no—
350#137—612no—
351#137—612no—
352#137—612no—
353#137—1222no—
354#137—1222no—
355#137C2×#692312yes—
356C2×Q16—612no—
357C2×Q16—611no—
358C2×Q16—1221no—
359C2×Q16—1221no—
360C2×Q16—1221no—
361C2×Q16—1221no—
362C2×Q16—1222no—
363C2×Q16C2×#692311yescharacteristic, complemented
364C2×SD16—1222no—
365#131—611no—
366#131—611no—
367#131—611no—
368#131—1221no—
369#131—1221no—
370#131C2×#692311yescharacteristic
371C2×M16—611no—
372C2×M16—612no—
373C2×M16—611no—
374C2×M16—1221no—
375C22×C8—611no—
376C22×C8—611no—
377C22×C8—1221no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×Q8—1222no—
381C4×Q8—1222no—
382C4×Q8—1222no—
383C4×D4—1222no—
384C4×D4—2442no—
385#117—611no—
386#117—612no—
387#117—1222no—
388#117—2442no—
389C2×C4⋊C4—612no—
390C2×C4⋊C4—1222no—
391C2×C42—611no—
392#108—612no—
393#108—611no—
394#108C2×#692311yescharacteristic, complemented
395#107—611no—
396#107—611no—
397#107—1222no—
398#107—1222no—
399#107—1221no—
400#107—1221no—
401#106—612no—
402#106—1222no—
403#106—1222no—
404#104—1221no—
405#104—1221no—
406#103—611no—
407#103—611no—
408#103—1222no—
409#103—1222no—
410#103—1222no—
411#103—1222no—
412#102—611no—
413#102—611no—
414#101—612no—
415#101—612no—
416#101—612no—
417#101C2×#692312yescomplemented
418#97—1222no—
419#97—1221no—
420C4×C8—1221no—
421C4×C8—1221no—
422C4×C8—1222no—
423C4×C8—1221no—
424#95—2442no—
425#41—611no—
426#41—611no—
427#41—611no—
428#41—1221no—
429#41—1221no—
430#41C22×#692311yescharacteristic
431C2×Q8—611no—
432C2×Q8—612no—
433C2×Q8—1222no—
434C2×Q8—1222no—
435C2×Q8—1222no—
436C2×Q8—1222no—
437C2×Q8—1222no—
438C2×Q8C22×#692312yes—
439C2×D4—611no—
440C2×D4—1221no—
441C22×C4—611no—
442C22×C4—1221no—
443C22×C4—1222no—
444C22×C4—1221no—
445C22×C4—1222no—
446C22×C4—2442no—
447Q16—22no—
448Q16—21no—
449Q16—21no—
450Q16—1222no—
451Q16—1222no—
452Q16—1222no—
453Q16—1222no—
454Q16—1222no—
455Q16—1221no—
456Q16—1221no—
457SD16—22no—
458SD16—1222no—
459SD16—1222no—
460SD16—1222no—
461M16—21no—
462M16—611no—
463M16—612no—
464M16—1221no—
465M16—1221no—
466M16—1222no—
467M16—1221no—
468M16—1221no—
469M16—1221no—
470M16—1221no—
471M16C22×#692311yescharacteristic
472M16C22×#692312yes—
473C2×C8—21no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—611no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—1221no—
482C2×C8—1222no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8C22×#692311yescharacteristic
490C4⋊C4—611no—
491C4⋊C4—611no—
492C4⋊C4—612no—
493C4⋊C4—612no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—2442no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C22⋊C4—1221no—
500C22⋊C4—1222no—
501C22⋊C4—2442no—
502C42—1222no—
503C42—1221no—
504C42—1221no—
505C42—1221no—
506C42—2442no—
507C42—2442no—
508C42—2442no—
509C23—611no—
510C23—1221no—
511Q8—22no—
512Q8—22no—
513Q8—611no—
514Q8—1221no—
515Q8—1222no—
516Q8—1222no—
517Q8—2442no—
518Q8—2442no—
519Q8D4×#692311yescharacteristic
520D4—21no—
521D4—611no—
522D4—1221no—
523D4—1221no—
524D4—1221no—
525D4—1221no—
526D4D4×#692311yescharacteristic
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—611no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1222no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—9762no—
545C2×C4—4882no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4C23×#692311yescharacteristic
552C8—21no—
553C8—22no—
554C8—611no—
555C8—611no—
556C8—1221no—
557C8—1221no—
558C8—1221no—
559C8—1222no—
560C8—2441no—
561C8D4×#692311yescharacteristic
562C8D4×#692311yescharacteristic
563C22—21no—
564C22—611no—
565C22—611no—
566C22—1221no—
567C22—1221no—
568C22—1221no—
569C22—2441no—
570C22—2441no—
571C22#1684083111yescharacteristic
572C4—21no—
573C4—42no—
574C4—611no—
575C4—611no—
576C4—1221no—
577C4—4882no—
578C4—2442no—
579C4—2441no—
580C4—2442no—
581C4—2441no—
582C4—2442no—
583C4—2442no—
584C4C2×D4×#692311yescharacteristic
585C4C2×D4×#692311yescharacteristic
586C2—21no—
587C2—41no—
588C2—611no—
589C2—611no—
590C2—1221no—
591C2—2441no—
592C2#2146234411yescharacteristic
subgroupG/H
centerC2#21462344
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#473C4
socleC2×C61#2361
prime radical#41×D61C23
cosocleC2×C4×D61C24
hypercenter#473#6923
hypocenterC61#23045
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #21462580C2semidirect
4. #21461965C2semidirect
5. #21461937C2semidirect
6. #21461904C2semidirect
8. #21461775C2semidirect
9. #21461624C2semidirect
43. #16840038C4semidirect
44. #16840037C4semidirect
45. #16840036C4semidirect
46. #16840035C4semidirect
47. #16840034C4semidirect
50. C61×#473C4semidirect
100. C2×Dic244C2×C4semidirect
101. C2×#11758202C2×C4semidirect
118. #14519713C2×C4semidirect
119. #14519712C2×C4semidirect
120. #14519696C2×C4semidirect
121. #14519695C2×C4semidirect
122. #14519695C2×C4semidirect
124. C2×Q16×C61C2×C4semidirect
126. #108×C61C2×C4semidirect
127. #101×C61C2×C4semidirect
318. #473#6923semidirect
347. C61#23045semidirect
363. C2×Q16C2×#6923semidirect
394. #108C2×#6923semidirect
417. #101C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214625802
#214625772
#214624742
#214619652
#214619372
#214619042
#214618182
#214617752
#214616242
#214615932
#473×D612
#2304561