Finite Group Databasebeta

← #30622274

Subgroups — #30622274

Order 15616.

840 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462580C211yesmaximal, characteristic, complemented
2#21462577C212yesmaximal
3#21462473C211yesmaximal, characteristic
4#21461967C211yesmaximal, characteristic, complemented
5#21461936C211yesmaximal, characteristic, complemented
6#21461904C212yesmaximal, complemented
7#21461818C212yesmaximal
8#21461775C212yesmaximal, complemented
9#21461628C211yesmaximal, characteristic, complemented
10#21461597C211yesmaximal, characteristic
11#473×D61C211yesmaximal, characteristic
12#16840831C2212yes—
13#16840830C2212yes—
14#16840802—21no—
15#16840802—21no—
16#16840802—22no—
17#16840801—22no—
18Q16×#6923—22no—
19Q16×#6923—22no—
20SD16×#6923—22no—
21#16840773—22no—
22#16840773C2212yes—
23#16840773C2212yes—
24#16840772—21no—
25#16840772C2211yescharacteristic
26C2×#14519845—21no—
27C2×#14519845C2211yescharacteristic
28D61⋊#104C2211yescharacteristic
29#16840755C2212yes—
30#16840752C2211yescharacteristic
31C2×#14519840C2212yes—
32#16840728C2212yes—
33#16840725C2211yescharacteristic
34C2×#14519831C2211yescharacteristic
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#14519846D412yes—
70#14519845—22no—
71#14519845—22no—
72#14519845—21no—
73#14519845—21no—
74#14519845D411yescharacteristic
75#14519845D411yescharacteristic
76#14519842—21no—
77#14519841—21no—
78#14519840—22no—
79#14519840—22no—
80#14519840—22no—
81#14519840—22no—
82#14519840D412yes—
83#14519839D412yes—
84#14519831—22no—
85#14519831—21no—
86#14519831—21no—
87#14519831—21no—
88C8×#6923—22no—
89C8×#6923—21no—
90(C2×#6923)⋊C4—42no—
91#41×D61C2311yescharacteristic
92C2×Q8×D61C2312yes—
93#14519810C2×C412yes—
94#14519809—22no—
95#14519808C2×C412yes—
96#14519807C2×C412yes—
97#14519806—22no—
98C2×Dic244C2×C411yescharacteristic, complemented
99C2×#11758202C2×C412yescomplemented
100Q16×D61—22no—
101Q16×D61—21no—
102Q16×D61—21no—
103Q16×D61—21no—
104Q16×D61—21no—
105SD16×D61—22no—
106#14519785—21no—
107#14519785C2×C411yescharacteristic
108#14519784C2×C411yescharacteristic
109#14519783—21no—
110#14519783C2×C411yescharacteristic
111M16×D61—21no—
112M16×D61C2312yes—
113M16×D61C2311yescharacteristic
114C2×C8×D61—21no—
115C2×C8×D61C2311yescharacteristic
116#14519713C2×C411yescharacteristic, complemented
117#14519712C2×C412yescomplemented
118#14519696C2×C412yescomplemented
119#14519695C2×C412yescomplemented
120#14519695C2×C412yescomplemented
121#137×C61C2×C412yes—
122C2×Q16×C61C2×C411yescharacteristic, complemented
123#131×C61C2×C411yescharacteristic
124#108×C61C2×C411yescharacteristic, complemented
125#101×C61C2×C412yescomplemented
126C22×#6923—21no—
127C22×#6923—22no—
128C22×#6923—42no—
129#41×C61C22×C411yescharacteristic
130C2×Q8×C61C22×C412yes—
131#11758226#4111yescharacteristic
132Q8×D61—22no—
133Q8×D61—22no—
134Q8×D61—22no—
135Q8×D61—22no—
136Q8×D61C2×D411yescharacteristic
137#11758224—21no—
138#11758224#4111yescharacteristic
139D4×D61—21no—
140D4×D61C2×D411yescharacteristic
141#11758222—21no—
142#11758222C22×C411yescharacteristic
143C2×C4×D61—21no—
144C2×C4×D61—22no—
145C2×C4×D61C2411yescharacteristic
146C2×Dic122—22no—
147C2×Dic122C22×C412yes—
148C22⋊#6923—21no—
149C22⋊#6923—22no—
150C22⋊#6923—42no—
151#11758217—21no—
152C2×#92420—21no—
153C4⋊#6923—22no—
154C4⋊#6923—22no—
155C4⋊#6923—42no—
156C4⋊#6923—42no—
157C4⋊#6923—42no—
158C4⋊#6923C2×D412yes—
159C4⋊#6923C2×D412yes—
160C4⋊#6923C2×D412yes—
161C4×#6923—22no—
162C4×#6923—21no—
163C4×#6923—21no—
164C4×#6923—21no—
165C4×#6923—42no—
166C4×#6923—42no—
167C4×#6923—42no—
168#11758213—21no—
169D61⋊C8—21no—
170D61⋊C8—21no—
171D61⋊C8C2×D412yes—
172Q16×C61—22no—
173Q16×C61—21no—
174Q16×C61—21no—
175SD16×C61—22no—
176M16×C61—21no—
177M16×C61C22×C412yes—
178M16×C61C22×C411yescharacteristic
179C2×C8×C61—21no—
180C2×C8×C61C22×C411yescharacteristic
181#11758202—22no—
182#11758202—22no—
183#11758202—22no—
184#11758202—22no—
185#11758201—22no—
186#11758200—22no—
187#11758194—21no—
188#11758194C22×C412yes—
189#11758194C22×C411yescharacteristic
190C2×#92418—21no—
191C2×#92418C22×C411yescharacteristic
192Dic244—22no—
193Dic244—21no—
194Dic244—21no—
195#11758190—22no—
196#11758189—22no—
197#11758189—21no—
198#11758189—21no—
199#11758189—21no—
200C8×D61—22no—
201C8×D61—21no—
202C8×D61—21no—
203C8×D61—21no—
204C8×D61C2×D411yescharacteristic
205C8×D61C2×D411yescharacteristic
206C8×D61#4111yescharacteristic
207C8×D61#4111yescharacteristic
208C22×D61—21no—
209C22×D61H(5,𝔽2)11yescharacteristic
210C2×#6923—22no—
211C2×#6923—22no—
212C2×#6923—22no—
213C2×#6923—42no—
214C2×#6923—42no—
215C2×#6923—41no—
216C2×#6923—82no—
217C2×#6923—162no—
218Q8×C61—22no—
219Q8×C61—22no—
220Q8×C61C4×D411yescharacteristic
221D4×C61—21no—
222D4×C61C4×D411yescharacteristic
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61C23×C411yescharacteristic
226C22⋊D61—21no—
227C22⋊D61—21no—
228C22⋊D61—21no—
229C2×Dic61—21no—
230C2×Dic61—22no—
231C2×Dic612−1+411yescharacteristic
232D244—21no—
233D244C4×D411yescharacteristic
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—42no—
238C4×D61—42no—
239C4×D61C22×D411yescharacteristic
240C4×D61C22×D411yescharacteristic
241C4×D61C2×#4111yescharacteristic
242C4×D61C2×#4111yescharacteristic
243Dic122—21no—
244Dic122—22no—
245Dic122—22no—
246Dic122—42no—
247Dic122—42no—
248Dic122C4×D411yescharacteristic
249#92420—22no—
250#92420—41no—
251C8×C61—21no—
252C8×C61—22no—
253C8×C61C4×D411yescharacteristic
254C8×C61C4×D411yescharacteristic
255#92418—21no—
256#92418—22no—
257#92418C4×D411yescharacteristic
258#92418C4×D411yescharacteristic
259#23047—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61#52011yescharacteristic
262C2×D61—22no—
263C2×D61—21no—
264C2×D61—42no—
265C2×D61D4211yescharacteristic
266C2×D61#55311yescharacteristic
267#6923—42no—
268#6923—42no—
269#6923—42no—
270#6923—81no—
271#6923—82no—
272C4×C61—21no—
273C4×C61—42no—
274C4×C61C2×C4×D411yescharacteristic
275C4×C61C2×C4×D411yescharacteristic
276Dic61—21no—
277Dic61—42no—
278Dic61#54711yescharacteristic
279Dic61#54811yescharacteristic
280C2×#473—611no—
281#2950—611no—
282#2947—612no—
283#2912—611no—
284#1971—611no—
285#1951—611no—
286#1930—612no—
287#1882—612no—
288#1857—612no—
289#1773—611no—
290#1765—611no—
291C2×C61—21no—
292C2×C61—41no—
293C2×C61#236111yescharacteristic
294D61—21no—
295D61—41no—
296D61#328912yes—
297C2×#137—612no—
298C22×Q16—611no—
299C2×#131—611no—
300#520—612no—
301#519—612no—
302#473—612no—
303#473—611no—
304#473—611no—
305#473—611no—
306#473—612no—
307#473#692311yescharacteristic, complemented
308#441—1222no—
309#441—1221no—
310#441—1221no—
311#440—1222no—
312C4×Q16—1222no—
313C4×Q16—1222no—
314C4×SD16—1222no—
315C2×#108—611no—
316#428—612no—
317#428—612no—
318#428—1222no—
319#427—611no—
320#427—1221no—
321C2×#106—611no—
322C2×#106—1221no—
323C2×#104—611no—
324#419—612no—
325#416—611no—
326C2×#103—612no—
327C2×#101—612no—
328#405—611no—
329#405—612no—
330C2×#97—611no—
331#391—612no—
332#343—612no—
333#341—611no—
334#339—611no—
335#339—612no—
336C61#2304711yescharacteristic, complemented, Sylow
337C2×#41—611no—
338C22×Q8—612no—
339#137—612no—
340#137—612no—
341#137—612no—
342#137—1222no—
343#137—1222no—
344#137C2×#692312yes—
345C2×Q16—612no—
346C2×Q16—611no—
347C2×Q16—1221no—
348C2×Q16—1221no—
349C2×Q16—1221no—
350C2×Q16—1221no—
351C2×Q16—1222no—
352C2×Q16C2×#692311yescharacteristic, complemented
353C2×SD16—1222no—
354#131—611no—
355#131—611no—
356#131—611no—
357#131—1221no—
358#131—1221no—
359#131C2×#692311yescharacteristic
360C2×M16—611no—
361C2×M16—612no—
362C2×M16—611no—
363C2×M16—1221no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—1221no—
367C4×Q8—1222no—
368C4×Q8—1222no—
369C4×Q8—1222no—
370C4×Q8—1222no—
371C4×Q8—1222no—
372C4×D4—1222no—
373C4×D4—2442no—
374#117—611no—
375#117—612no—
376#117—1222no—
377#117—2442no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—1222no—
380C2×C42—611no—
381#108—612no—
382#108—611no—
383#108C2×#692311yescharacteristic, complemented
384#107—612no—
385#107—1222no—
386#107—1222no—
387#106—611no—
388#106—611no—
389#106—1222no—
390#106—1222no—
391#106—1221no—
392#106—1221no—
393#104—1221no—
394#104—1221no—
395#103—612no—
396#103—1222no—
397#103—1222no—
398#103—1222no—
399#103—1222no—
400#102—612no—
401#101—612no—
402#101—612no—
403#101—612no—
404#101C2×#692312yescomplemented
405#97—1221no—
406#97—1221no—
407#97—1222no—
408#97—1221no—
409C4×C8—1222no—
410C4×C8—1221no—
411#95—2442no—
412#41—611no—
413#41—611no—
414#41—611no—
415#41—1221no—
416#41—1221no—
417#41C22×#692311yescharacteristic
418C2×Q8—611no—
419C2×Q8—612no—
420C2×Q8—1222no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×Q8—1222no—
424C2×Q8—1222no—
425C2×Q8C22×#692312yes—
426C2×D4—611no—
427C2×D4—1221no—
428C22×C4—611no—
429C22×C4—1221no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1221no—
433C22×C4—2442no—
434Q16—22no—
435Q16—21no—
436Q16—21no—
437Q16—1222no—
438Q16—1222no—
439Q16—1222no—
440Q16—1222no—
441Q16—1222no—
442Q16—1221no—
443Q16—1221no—
444SD16—22no—
445SD16—1222no—
446SD16—1222no—
447SD16—1222no—
448M16—21no—
449M16—611no—
450M16—612no—
451M16—1221no—
452M16—1221no—
453M16—1222no—
454M16—1221no—
455M16—1221no—
456M16—1221no—
457M16—1221no—
458M16C22×#692311yescharacteristic
459M16C22×#692312yes—
460C2×C8—21no—
461C2×C8—611no—
462C2×C8—611no—
463C2×C8—611no—
464C2×C8—611no—
465C2×C8—611no—
466C2×C8—612no—
467C2×C8—1221no—
468C2×C8—1222no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8C22×#692311yescharacteristic
476C4⋊C4—612no—
477C4⋊C4—612no—
478C4⋊C4—612no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—2442no—
482C4⋊C4—2442no—
483C4⋊C4—2442no—
484C22⋊C4—1221no—
485C22⋊C4—1222no—
486C22⋊C4—2442no—
487C42—1222no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C42—2442no—
492C42—2442no—
493C42—2442no—
494C23—611no—
495C23—1221no—
496Q8—22no—
497Q8—22no—
498Q8—611no—
499Q8—1221no—
500Q8—1222no—
501Q8—1222no—
502Q8—2442no—
503Q8—2442no—
504Q8D4×#692311yescharacteristic
505D4—21no—
506D4—611no—
507D4—1221no—
508D4—1221no—
509D4—1221no—
510D4—1221no—
511D4D4×#692311yescharacteristic
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—611no—
515C2×C4—611no—
516C2×C4—611no—
517C2×C4—611no—
518C2×C4—611no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—9762no—
528C2×C4—4882no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2441no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4C23×#692311yescharacteristic
535C8—21no—
536C8—22no—
537C8—611no—
538C8—611no—
539C8—1222no—
540C8—1221no—
541C8—1222no—
542C8—2441no—
543C8D4×#692311yescharacteristic
544C8D4×#692311yescharacteristic
545C22—21no—
546C22—611no—
547C22—611no—
548C22—1222no—
549C22—1221no—
550C22—2442no—
551C22#1684083111yescharacteristic
552C4—21no—
553C4—42no—
554C4—611no—
555C4—611no—
556C4—1221no—
557C4—4882no—
558C4—4881no—
559C4—2442no—
560C4—2442no—
561C4—2442no—
562C4—2442no—
563C4C2×D4×#692311yescharacteristic
564C4C2×D4×#692311yescharacteristic
565C2—21no—
566C2—41no—
567C2—612no—
568C2—1221no—
569C2—2441no—
570C2#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#23047
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. #21461967C2semidirect
5. #21461936C2semidirect
6. #21461904C2semidirect
8. #21461775C2semidirect
9. #21461628C2semidirect
43. #16840038C4semidirect
44. #16840037C4semidirect
45. #16840036C4semidirect
46. #16840035C4semidirect
47. #16840034C4semidirect
50. C61×#473C4semidirect
98. C2×Dic244C2×C4semidirect
99. C2×#11758202C2×C4semidirect
116. #14519713C2×C4semidirect
117. #14519712C2×C4semidirect
118. #14519696C2×C4semidirect
119. #14519695C2×C4semidirect
120. #14519695C2×C4semidirect
122. C2×Q16×C61C2×C4semidirect
124. #108×C61C2×C4semidirect
125. #101×C61C2×C4semidirect
307. #473#6923semidirect
336. C61#23047semidirect
352. C2×Q16C2×#6923semidirect
383. #108C2×#6923semidirect
404. #101C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214625802
#214625772
#214624732
#214619672
#214619362
#214619042
#214618182
#214617752
#214616282
#214615972
#473×D612
#2304761