Finite Group Databasebeta

← #30611697

Subgroups — #30611697

Order 15616.

900 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462665C211yesmaximal, characteristic, complemented
2#21462656C211yesmaximal, characteristic, complemented
3#21461310C211yesmaximal, characteristic, complemented
4#21461306C212yesmaximal, complemented
5#21461297C212yesmaximal, complemented
6#21461287C212yesmaximal, complemented
7#21461280C211yesmaximal, characteristic, complemented
8#21461280C211yesmaximal, characteristic, complemented
9#21461276C211yesmaximal, characteristic, complemented
10#21461272C212yesmaximal, complemented
11#21459718C211yesmaximal, characteristic
12#16840813—21no—
13Q8×#92420—21no—
14#16840797—21no—
15#16840797—22no—
16#16840793—22no—
17D4×#92420—21no—
18#16840783C2211yescharacteristic
19#16840783C2211yescharacteristic
20#16840783C2212yescomplemented
21#16840782C2212yescomplemented
22C4×#11758217C2211yescharacteristic
23C4×#11758217C2211yescharacteristic
24C4×#11758217C2212yescomplemented
25C2×#14519832C2212yescomplemented
26#16840619C2211yescharacteristic, complemented
27#16840619C2212yescomplemented
28#16840619C2211yescharacteristic, complemented
29#16840619C2211yescharacteristic, complemented
30#16840619C2211yescharacteristic, complemented
31#16840619C2212yescomplemented
32#16840618C2212yes—
33#16840618C2212yes—
34#16840618C2212yes—
35#16840618C2212yes—
36#16840494—21no—
37#16840494—21no—
38#16840486—21no—
39#16840486—21no—
40#16840435—22no—
41#16840435—22no—
42#16840385C412yes—
43#16840385C412yes—
44#16840326C411yescharacteristic
45#16840315C2211yescharacteristic
46#16840298C2212yes—
47#16840298C2212yes—
48C2×#14519766C2211yescharacteristic
49#16840240C411yescharacteristic
50#16840236C411yescharacteristic
51#41×Dic61C2211yescharacteristic
52C61×#536C411yescharacteristic
53C2×#11758217—21no—
54C2×#11758217—21no—
55C2×#11758217—22no—
56C22×#92420—22no—
57#14519844—22no—
58C4⋊#92420—21no—
59C4⋊#92420—21no—
60C4⋊#92420—22no—
61C22⋊#92420—22no—
62C22⋊#92420—21no—
63C22⋊#92420—21no—
64C22⋊#92420—22no—
65#14519832—22no—
66#14519832C2311yescharacteristic
67#14519832C2312yes—
68#14519832C2311yescharacteristic
69#14519832C2311yescharacteristic
70#14519832C2311yescharacteristic
71#14519832C2312yes—
72C4×#92420—21no—
73C4×#92420—21no—
74C4×#92420—22no—
75C22×D244C2×C411yescharacteristic
76#14519776C2×C411yescharacteristic
77#14519776C2×C411yescharacteristic
78#14519773—21no—
79#14519772—22no—
80#14519772—22no—
81#14519772—22no—
82#14519772—22no—
83#14519768C2311yescharacteristic
84#14519768C2311yescharacteristic
85#14519766—21no—
86#14519766—21no—
87#14519766—21no—
88#14519766C2×C412yes—
89#14519766C2312yes—
90#14519763—22no—
91#14519763—22no—
92#14519761—22no—
93#14519761—22no—
94#14519759—22no—
95#14519759—22no—
96#14519757—22no—
97#14519757—22no—
98#14519754—21no—
99#14519754—21no—
100#14519751C2×C412yes—
101#14519751C2×C412yes—
102#14519745—22no—
103Q8×Dic61—21no—
104Q8×Dic61—21no—
105Q8×Dic61C2×C412yes—
106Q8×Dic61C2×C412yes—
107Q8×Dic61D411yescharacteristic
108Q8×Dic61D411yescharacteristic
109C4×Dic122—21no—
110C4×Dic122—21no—
111C4×D244—21no—
112C4×D244—21no—
113D4×Dic61D411yescharacteristic
114D4×Dic61D411yescharacteristic
115D4×Dic61D412yes—
116D4×Dic61D412yes—
117#14519739C2×C412yes—
118#14519739C2×C412yes—
119#14519737—22no—
120#14519737—22no—
121#14519736—22no—
122#14519736C2×C411yescharacteristic
123#14519736C2×C411yescharacteristic
124#14519736C2×C412yes—
125#14519735—22no—
126#14519735—22no—
127C2×#11758198—21no—
128C2×#11758198—21no—
129C22⋊C4×D61—22no—
130C2×C4×Dic61C2312yes—
131C2×C4×Dic61C2311yescharacteristic
132C2×#41×C61C2×C411yescharacteristic
133C22×Q8×C61C2×C411yescharacteristic
134C4⋊Q8×C61—21no—
135#124×C61—21no—
136#123×C61—22no—
137C22⋊Q8×C61C2×C412yes—
138C22⋊Q8×C61C2×C412yes—
139#117×C61C2×C411yescharacteristic
140C23×D61—21no—
141#41×C61—42no—
142C2×Q8×C61—21no—
143C2×Q8×C61—42no—
144C2×Q8×C61C22⋊C411yescharacteristic
145C2×Q8×C61C22×C412yes—
146C2×D4×C61C22⋊C411yescharacteristic
147C2×D4×C61C22⋊C412yes—
148C22×C4×C61—21no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C22×C412yes—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C22×Dic61C2×D412yes—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D412yes—
157C2×D244—21no—
158C2×D244—42no—
159C2×D244C22×C412yes—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164#11758217—42no—
165#11758217—42no—
166#11758217—44no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—22no—
171C2×#92420—21no—
172C2×#92420—22no—
173C2×#92420—44no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C22×C411yescharacteristic
177C4⋊C4×C61C22×C411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C42×C61—21no—
182C22⋊Dic61—21no—
183C22⋊Dic61C22⋊C412yes—
184C22⋊Dic61C22⋊C412yes—
185C22⋊Dic61C22⋊C412yes—
186#11758198—22no—
187#11758198—22no—
188#11758198—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61C22⋊C411yescharacteristic
199C4⋊Dic61C22⋊C411yescharacteristic
200C4⋊Dic61C22⋊C412yes—
201C4⋊Dic61C22⋊C412yes—
202C4⋊Dic61C22×C411yescharacteristic
203C4⋊Dic61C22×C411yescharacteristic
204#11758196—21no—
205#11758196—21no—
206#11758196—21no—
207#11758196—21no—
208C4×Dic61—21no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61C22×C412yes—
212C4×Dic61C22×C411yescharacteristic
213C4×Dic61C2×D412yes—
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D412yes—
217C4×Dic61C2411yescharacteristic
218C23×C61C2×C22⋊C411yescharacteristic
219C23×C61C2×C22⋊C412yes—
220C22×D61—22no—
221C22×D61—42no—
222C22×D61—81no—
223C22×D61—81no—
224Q8×C61—22no—
225Q8×C61—42no—
226Q8×C61—42no—
227D4×C61—22no—
228D4×C61—42no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C22⋊D61—82no—
239C22⋊D61—82no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—42no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—42no—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C22×D411yescharacteristic
251C2×Dic61C22×D411yescharacteristic
252D244—42no—
253D244—42no—
254C4×D61—82no—
255C4×D61—82no—
256Dic122—81no—
257Dic122—81no—
258#92420—22no—
259#92420—22no—
260#92420—42no—
261#92420—41no—
262#92420—42no—
263#92420—41no—
264#17691—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61C22×C22⋊C411yescharacteristic
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—162no—
276C4×C61—21no—
277C4×C61—22no—
278C4×C61—42no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61#41312yes—
282Dic61—21no—
283Dic61—21no—
284Dic61—21no—
285Dic61—21no—
286Dic61—22no—
287Dic61—22no—
288Dic61—41no—
289Dic61—41no—
290#3510—611no—
291#2971—611no—
292#2970—611no—
293#1537—611no—
294#1535—612no—
295#1532—612no—
296#1529—612no—
297#1527—611no—
298#1527—611no—
299#1526—611no—
300#1524—612no—
301C2×C61—41no—
302C2×C61—42no—
303C2×C61C2×#41311yescharacteristic
304C2×C61#290111yescharacteristic
305C2×C61#290111yescharacteristic
306D61—82no—
307#559—1221no—
308#559—1221no—
309#556—1221no—
310#556—1221no—
311#551—1222no—
312#551—1222no—
313#542—612no—
314#542—612no—
315#536#692311yescharacteristic
316#535—611no—
317#533—611no—
318#529—612no—
319#529—612no—
320C2×#124—611no—
321#519—611no—
322#518—611no—
323C4×#41—611no—
324#446—1221no—
325C8×Q8—1221no—
326#436—1222no—
327#436—1221no—
328#435—1222no—
329C8×D4—1221no—
330#432—611no—
331#432—611no—
332#432—612no—
333#432—612no—
334C4×M16—611no—
335C4×M16—611no—
336C4×M16—612no—
337C2×#97—612no—
338#330—612no—
339#330—612no—
340#330—611no—
341#330—611no—
342#330—611no—
343#330—611no—
344#329—612no—
345#329—612no—
346#329—612no—
347#329—612no—
348C61#1769111yescharacteristic, complemented, Sylow
349C2×#41C2×#692311yescharacteristic
350C22×Q8C2×#692311yescharacteristic
351C22×D4—611no—
352C2×M16—1221no—
353C2×M16—1221no—
354C2×M16—1222no—
355C22×C8—1222no—
356C4⋊Q8—21no—
357C4⋊Q8—611no—
358C4⋊Q8—611no—
359#127—1221no—
360#126—1222no—
361#126—1222no—
362#126—1222no—
363#126—1222no—
364#125—611no—
365#125—611no—
366#124—21no—
367#124—612no—
368#124—612no—
369#124—1221no—
370#124—1221no—
371#124—1222no—
372#124—1222no—
373#124—1221no—
374#123—22no—
375#123—1222no—
376#123—1222no—
377#123—1222no—
378#123—1222no—
379C22⋊Q8—1221no—
380C22⋊Q8—1221no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8C2×#692312yes—
384C22⋊Q8C2×#692312yes—
385#121—612no—
386#121—612no—
387#120—1222no—
388C4×Q8—611no—
389C4×Q8—611no—
390C4×Q8—612no—
391C4×Q8—612no—
392C4×Q8—1221no—
393C4×Q8—1221no—
394C4×Q8—1221no—
395C4×Q8—1221no—
396C4×D4—612no—
397C4×D4—612no—
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—612no—
401C4×D4—612no—
402C4×D4—1221no—
403C4×D4—1221no—
404C4×D4—1222no—
405C4×D4—1222no—
406#117—611no—
407#117—612no—
408#117—611no—
409#117—1222no—
410#117—1222no—
411#117—1222no—
412#117C2×#692311yescharacteristic
413C2×C22⋊C4—1222no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C42—611no—
417C2×C42—612no—
418C4⋊C8—1221no—
419C4⋊C8—1222no—
420C4⋊C8—1221no—
421C4⋊C8—1222no—
422C22⋊C8—1222no—
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1222no—
426#97—611no—
427#97—612no—
428#97—611no—
429#97—611no—
430#97—611no—
431#97—612no—
432#97—1222no—
433C4×C8—1221no—
434C4×C8—1222no—
435C4×C8—1221no—
436C24—1221no—
437#41—42no—
438C2×Q8—21no—
439C2×Q8—42no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8C22⋊#692311yescharacteristic
443C2×Q8C22×#692312yes—
444C2×D4—612no—
445C2×D4—1222no—
446C2×D4—1222no—
447C2×D4—1221no—
448C2×D4—2442no—
449C2×D4C22⋊#692311yescharacteristic
450C2×D4C22⋊#692312yes—
451C22×C4—21no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4C22×#692311yescharacteristic
459C22×C4C22×#692312yes—
460M16—2442no—
461M16—2444no—
462M16—2442no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1222no—
467C2×C8—1221no—
468C2×C8—1222no—
469C2×C8—2444no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—611no—
473C4⋊C4—611no—
474C4⋊C4—612no—
475C4⋊C4—612no—
476C4⋊C4—611no—
477C4⋊C4—611no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4C22×#692311yescharacteristic
487C4⋊C4C22×#692311yescharacteristic
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—612no—
492C22⋊C4—612no—
493C22⋊C4—612no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—1221no—
501C22⋊C4—1222no—
502C22⋊C4—1222no—
503C42—21no—
504C42—612no—
505C42—611no—
506C42—611no—
507C42—612no—
508C42—611no—
509C42—612no—
510C42—611no—
511C42—1222no—
512C42—1222no—
513C42—1221no—
514C23—1222no—
515C23—4881no—
516C23—4881no—
517C23—2442no—
518C23C2×C22⋊#692312yes—
519C23C2×C22⋊#692311yescharacteristic
520Q8—22no—
521Q8—42no—
522Q8—42no—
523Q8—4881no—
524Q8—4881no—
525D4—22no—
526D4—42no—
527D4—4882no—
528D4—4882no—
529D4—2442no—
530D4—2442no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—4882no—
546C2×C4—4882no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4C2×C22⋊#692312yes—
552C2×C4C2×C22⋊#692311yescharacteristic
553C2×C4C23×#692311yescharacteristic
554C8—1222no—
555C8—1222no—
556C8—2442no—
557C8—2441no—
558C8—2441no—
559C8—2442no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—21no—
565C22—21no—
566C22—9762no—
567C22—2442no—
568C22—2442no—
569C22—2442no—
570C22C22×C22⋊#692311yescharacteristic
571C4—21no—
572C4—22no—
573C4—42no—
574C4—41no—
575C4—41no—
576C4—1221no—
577C4—1221no—
578C4—1222no—
579C4—1222no—
580C4—1221no—
581C4—1221no—
582C4—2441no—
583C4—2441no—
584C4#1684074912yes—
585C2—41no—
586C2—42no—
587C2—4882no—
588C2C2×#1684074911yescharacteristic
589C2#2146245011yescharacteristic
590C2#2146245011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#536C4
socleC22×C61C22×C22⋊C4
prime radical#16840326C4
cosocleC4×Dic61C24
hypercenter#536#6923
hypocenterC61#17691
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #21462665C2semidirect
2. #21462656C2semidirect
3. #21461310C2semidirect
4. #21461306C2semidirect
5. #21461297C2semidirect
6. #21461287C2semidirect
7. #21461280C2semidirect
8. #21461280C2semidirect
9. #21461276C2semidirect
10. #21461272C2semidirect
20. #16840783C22semidirect
21. #16840782C22semidirect
24. C4×#11758217C22semidirect
25. C2×#14519832C22semidirect
26. #16840619C22semidirect
27. #16840619C22semidirect
28. #16840619C22semidirect
29. #16840619C22semidirect
30. #16840619C22semidirect
31. #16840619C22semidirect
348. C61#17691semidirect

Maximal subgroups

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

subgroupindex
#214626652
#214626562
#214613102
#214613062
#214612972
#214612872
#214612802
#214612762
#214612722
#214597182
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