Finite Group Databasebeta

← #30610642

Subgroups — #30610642

Order 15616.

920 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#21462738C211yesmaximal, characteristic, complemented
2#41×#92420C211yesmaximal, characteristic, complemented
3#21461181C212yesmaximal, complemented
4#21461174C212yesmaximal, complemented
5#21461170C212yesmaximal, complemented
6#21461167C212yesmaximal, complemented
7#21461123C211yesmaximal, characteristic, complemented
8#21461114C211yesmaximal, characteristic, complemented
9#21461110C211yesmaximal, characteristic, complemented
10C2×#16840612C211yesmaximal, characteristic, complemented
11#21459247C211yesmaximal, characteristic
12#16840813—21no—
13Q8×#92420—21no—
14#16840797—21no—
15#16840793—22no—
16D4×#92420—22no—
17D4×#92420—21no—
18#16840783C2212yes—
19#16840778C2211yescharacteristic
20#16840778C2211yescharacteristic
21#16840778C2212yes—
22#16840765C2212yes—
23C2×C4⋊#92420C2211yescharacteristic
24C2×C4×#92420C2211yescharacteristic
25C2×C4×#92420C2212yes—
26#16840612C2212yes—
27#16840612C2212yes—
28#16840612C2212yes—
29#16840612C2212yes—
30Q8⋊#92420C2211yescharacteristic
31Q8⋊#92420C2211yescharacteristic
32Q8⋊#92420C2212yes—
33D4⋊#92420C2211yescharacteristic
34D4⋊#92420C2211yescharacteristic
35D4⋊#92420C2212yes—
36#16840393C411yescharacteristic
37#16840379—21no—
38#16840379C412yes—
39#16840372—21no—
40#16840372C412yes—
41#16840357—22no—
42#16840333C411yescharacteristic
43#16840333C411yescharacteristic
44#16840314—21no—
45#16840314C2212yes—
46#16840293—22no—
47#16840293—21no—
48#16840293C2212yes—
49#41×Dic61C2211yescharacteristic
50#41×Dic61C2211yescharacteristic
51C2×#14519742C2211yescharacteristic
52C61×#537C411yescharacteristic
53C2×#11758217—22no—
54C22×#92420—21no—
55C22×#92420—22no—
56C22×#92420—21no—
57#14519844—22no—
58C4⋊#92420—21no—
59C4⋊#92420—21no—
60C4⋊#92420—22no—
61C4⋊#92420C2312yes—
62C4⋊#92420C2311yescharacteristic
63C4⋊#92420C2311yescharacteristic
64C22⋊#92420—21no—
65C22⋊#92420—22no—
66C22⋊#92420—21no—
67C22⋊#92420—22no—
68#14519832—22no—
69C4×#92420—21no—
70C4×#92420—22no—
71C4×#92420—21no—
72C4×#92420C2311yescharacteristic
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C22×Dic122C2×C411yescharacteristic
76#14519776—21no—
77#14519776C2×C412yes—
78#14519775—21no—
79#14519775C2×C412yes—
80#14519770—22no—
81#14519770—22no—
82#14519770—22no—
83#14519770—22no—
84#14519768—22no—
85#14519768—21no—
86#14519768C2×C412yes—
87#14519767—22no—
88#14519767—22no—
89#14519764—21no—
90#14519764—22no—
91#14519764C2×C412yes—
92#14519758—22no—
93#14519758—22no—
94#14519758—22no—
95#14519758—22no—
96#14519756—22no—
97#14519756—21no—
98#14519756—21no—
99#14519756C2×C412yes—
100#14519756C2×C412yes—
101#14519754—22no—
102#14519754—22no—
103Q8×Dic61C2×C411yescharacteristic
104Q8×Dic61D411yescharacteristic
105Q8×Dic61D411yescharacteristic
106Q8×Dic61C2311yescharacteristic
107#14519742—21no—
108#14519742—21no—
109#14519742—22no—
110#14519742C2×C412yes—
111#14519742C2312yes—
112D4×Dic61—21no—
113D4×Dic61—21no—
114D4×Dic61—22no—
115D4×Dic61C2×C411yescharacteristic
116D4×Dic61D412yes—
117D4×Dic61D411yescharacteristic
118D4×Dic61D412yes—
119D4×Dic61D411yescharacteristic
120D4×Dic61C2311yescharacteristic
121#14519736—21no—
122#14519736C2×C411yescharacteristic
123#14519736C2×C411yescharacteristic
124#14519736C2×C412yes—
125#14519733—22no—
126#14519733—22no—
127C2×#11758196—21no—
128C2×C4×Dic61—21no—
129C2×C4×Dic61—21no—
130C2×C4×Dic61C2311yescharacteristic
131C2×C4×Dic61C2312yes—
132C2×#41×C61C2×C411yescharacteristic
133C2×#41×C61C2×C411yescharacteristic
134C22⋊Q8×C61—21no—
135C22⋊Q8×C61C2×C412yes—
136#121×C61—22no—
137#121×C61—21no—
138#121×C61C2×C412yes—
139C2×C4⋊C4×C61C2×C411yescharacteristic
140#41×C61—24no—
141#41×C61—24no—
142C2×Q8×C61C22⋊C411yescharacteristic
143C2×Q8×C61C22×C411yescharacteristic
144C2×D4×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61C22⋊C412yes—
147C2×D4×C61C22⋊C411yescharacteristic
148C2×D4×C61C22×C411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61C22×C412yes—
152C22×C4×C61C22×C411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D412yes—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D412yes—
160C2×Dic122—24no—
161C2×Dic122—21no—
162C2×Dic122—22no—
163C2×Dic122C22×C412yes—
164#11758217—82no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—22no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—22no—
171C2×#92420—44no—
172C2×#92420—42no—
173C2×#92420—81no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C22×C412yes—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊Dic61—22no—
180C22⋊Dic61—21no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—21no—
184C22⋊Dic61—22no—
185C22⋊Dic61—21no—
186C22⋊Dic61C22⋊C412yes—
187C22⋊Dic61C22⋊C412yes—
188C22⋊Dic61C22⋊C412yes—
189C4⋊Dic61—21no—
190C4⋊Dic61—22no—
191C4⋊Dic61—21no—
192C4⋊Dic61C22⋊C412yes—
193C4⋊Dic61C22⋊C411yescharacteristic
194C4⋊Dic61C22⋊C411yescharacteristic
195C4⋊Dic61C22⋊C412yes—
196C4⋊Dic61C22×C411yescharacteristic
197C4⋊Dic61C22×C411yescharacteristic
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—21no—
211C4×Dic61C22×C411yescharacteristic
212C4×Dic61C22×C412yes—
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D412yes—
216C4×Dic61C2×D412yes—
217C4×Dic61C2411yescharacteristic
218C23×C61—21no—
219C23×C61C2×C22⋊C412yes—
220C23×C61C2×C22⋊C411yescharacteristic
221Q8×C61—22no—
222Q8×C61—22no—
223D4×C61—24no—
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—42no—
227D4×C61—82no—
228C2×C4×C61—24no—
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—42no—
232C2×C4×C61—81no—
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61#13114yes—
236C2×C4×C61C23×C411yescharacteristic
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—42no—
245C2×Dic61—41no—
246C2×Dic61—42no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—81no—
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C412yes—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C22×D411yescharacteristic
255C2×Dic61C22×D411yescharacteristic
256Dic122—24no—
257Dic122—42no—
258Dic122—82no—
259#92420—22no—
260#92420—41no—
261#92420—42no—
262#92420—42no—
263#92420—41no—
264#92420—41no—
265#17484—611nomaximal, Sylow
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61#40812yes—
275C22×C61C22×C22⋊C411yescharacteristic
276C4×C61—22no—
277C4×C61—41no—
278C4×C61—42no—
279C4×C61#40812yes—
280C4×C61C2×#13112yes—
281Dic61—21no—
282Dic61—22no—
283Dic61—21no—
284Dic61—22no—
285Dic61—21no—
286Dic61—42no—
287Dic61—41no—
288Dic61—41no—
289Dic61#57711yescharacteristic
290Dic61#57811yescharacteristic
291#3461—611no—
292#2989—611no—
293C8×#41—611no—
294#1497—612no—
295#1495—612no—
296#1494—612no—
297#1493—612no—
298#1479—611no—
299#1477—611no—
300#1476—611no—
301C2×Q8⋊C8—611no—
302C2×C61—22no—
303C2×C61—42no—
304C2×C61—81no—
305C2×C61C2×#40811yescharacteristic
306C2×C61#289811yescharacteristic
307C2×C61#290111yescharacteristic
308#544—611no—
309#542—612no—
310#542—1221no—
311#541—612no—
312#541—1221no—
313#539—1222no—
314#537#692311yescharacteristic
315#536—611no—
316#536—611no—
317#533—612no—
318#533—1221no—
319#529—612no—
320#529—1221no—
321#529—1222no—
322C4×#41—611no—
323C4×#41—611no—
324C2×C4×Q8—611no—
325#446—1221no—
326C8×Q8—1221no—
327#436—1221no—
328#435—1222no—
329C8×D4—1222no—
330C8×D4—1221no—
331#432—612no—
332#431—611no—
333#431—612no—
334#431—611no—
335#423—612no—
336C2×C4⋊C8—611no—
337C2×C4×C8—611no—
338C2×C4×C8—612no—
339Q8⋊C8—611no—
340Q8⋊C8—611no—
341Q8⋊C8—612no—
342Q8⋊C8—612no—
343Q8⋊C8—612no—
344Q8⋊C8—612no—
345Q8⋊C8—612no—
346D4⋊C8—611no—
347D4⋊C8—611no—
348D4⋊C8—612no—
349C61#1748411yescharacteristic, complemented, Sylow
350C2×#41C2×#692311yescharacteristic
351C2×#41C2×#692311yescharacteristic
352C22×Q8—611no—
353C2×M16—1222no—
354C22×C8—1221no—
355C22×C8—1221no—
356C22×C8—1222no—
357C4⋊Q8—612no—
358C4⋊Q8—612no—
359C4⋊Q8—1221no—
360C4⋊Q8—1221no—
361#126—1222no—
362#126—1222no—
363#126—1222no—
364#126—1222no—
365#125—612no—
366#125—1222no—
367#125—1222no—
368#125—1221no—
369#125—1222no—
370#124—612no—
371#124—1222no—
372#124—1221no—
373#123—1222no—
374#123—1222no—
375#123—1222no—
376#123—1222no—
377C22⋊Q8—21no—
378C22⋊Q8—612no—
379C22⋊Q8—612no—
380C22⋊Q8—1222no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1221no—
384C22⋊Q8—1221no—
385C22⋊Q8C2×#692312yes—
386#121—22no—
387#121—21no—
388#121C2×#692312yes—
389C4×Q8—611no—
390C4×Q8—611no—
391C4×Q8—611no—
392C4×Q8—611no—
393C4×Q8—612no—
394C4×Q8—612no—
395C4×Q8—1221no—
396C4×Q8—1221no—
397C4×Q8—1222no—
398C4×D4—612no—
399C4×D4—611no—
400C4×D4—612no—
401C4×D4—611no—
402C4×D4—611no—
403C4×D4—611no—
404C4×D4—1221no—
405C4×D4—1221no—
406C4×D4—1222no—
407#117—611no—
408#117—612no—
409#117—611no—
410#117—1222no—
411#117—1221no—
412#117—1222no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C2×#692311yescharacteristic
415C2×C42—611no—
416C2×C42—612no—
417C2×C42—1221no—
418C2×C42—1221no—
419C4⋊C8—612no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—1221no—
423C4⋊C8—1221no—
424C4⋊C8—1222no—
425C4⋊C8—1222no—
426C22⋊C8—1221no—
427C22⋊C8—1222no—
428C22⋊C8—1221no—
429C22⋊C8—1222no—
430#97—1222no—
431C4×C8—611no—
432C4×C8—611no—
433C4×C8—612no—
434C4×C8—1221no—
435C4×C8—1222no—
436C4×C8—1221no—
437#41—24no—
438#41—24no—
439C2×Q8—612no—
440C2×Q8—1222no—
441C2×Q8—1221no—
442C2×Q8—1224no—
443C2×Q8C22⋊#692311yescharacteristic
444C2×Q8C22×#692311yescharacteristic
445C2×D4—21no—
446C2×D4—22no—
447C2×D4C22⋊#692312yes—
448C2×D4C22⋊#692311yescharacteristic
449C2×D4C22×#692311yescharacteristic
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—611no—
453C22×C4—612no—
454C22×C4—611no—
455C22×C4—612no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4C22×#692312yes—
460C22×C4C22×#692311yescharacteristic
461M16—4882no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1222no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1222no—
468C2×C8—4881no—
469C2×C8—2442no—
470C2×C8—2444no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—612no—
474C4⋊C4—611no—
475C4⋊C4—611no—
476C4⋊C4—612no—
477C4⋊C4—611no—
478C4⋊C4—611no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4C22×#692312yes—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—612no—
494C22⋊C4—612no—
495C22⋊C4—612no—
496C22⋊C4—1221no—
497C22⋊C4—1221no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—1221no—
501C22⋊C4—1222no—
502C22⋊C4—1222no—
503C42—611no—
504C42—611no—
505C42—612no—
506C42—612no—
507C42—611no—
508C42—611no—
509C42—612no—
510C42—1221no—
511C42—1222no—
512C42—1221no—
513C42—1222no—
514C42—1221no—
515C23—21no—
516C23C2×C22⋊#692312yes—
517C23C2×C22⋊#692311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—1224no—
521Q8—4882no—
522Q8—2442no—
523D4—24no—
524D4—22no—
525D4—22no—
526D4—42no—
527D4—82no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—42no—
532C2×C4—81no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—611no—
539C2×C4—1221no—
540C2×C4—1222no—
541C2×C4—1221no—
542C2×C4—1222no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1222no—
546C2×C4—4881no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4C2×C22⋊#692312yes—
553C2×C4C2×C22⋊#692311yescharacteristic
554C2×C4#1451986414yes—
555C2×C4C23×#692311yescharacteristic
556C8—1222no—
557C8—2441no—
558C8—2442no—
559C8—2441no—
560C8—2441no—
561C8—2442no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22#1684073512yes—
571C22C22×C22⋊#692311yescharacteristic
572C4—22no—
573C4—41no—
574C4—42no—
575C4—611no—
576C4—611no—
577C4—1221no—
578C4—1222no—
579C4—1221no—
580C4—1221no—
581C4—1222no—
582C4—2442no—
583C4—2441no—
584C4—2441no—
585C4#1684073512yes—
586C4C2×#1451986412yes—
587C2—22no—
588C2—42no—
589C2—81no—
590C2C2×#1684073511yescharacteristic
591C2#2146244111yescharacteristic
592C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
Fitting, prime radicalC61×#537C4
socleC22×C61C22×C22⋊C4
cosocleC4×Dic61C24
hypercenter#537#6923
hypocenterC61#17484
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #21462738C2semidirect
2. #41×#92420C2semidirect
3. #21461181C2semidirect
4. #21461174C2semidirect
5. #21461170C2semidirect
6. #21461167C2semidirect
7. #21461123C2semidirect
8. #21461114C2semidirect
9. #21461110C2semidirect
10. C2×#16840612C2semidirect
349. C61#17484semidirect

Maximal subgroups

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

subgroupindex
#214627382
#41×#924202
#214611812
#214611742
#214611702
#214611672
#214611232
#214611142
#214611102
C2×#168406122
#214592472
#1748461