Finite Group Databasebeta

← #30617101

Subgroups — #30617101

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462591C212yesmaximal, complemented
2C2×#16840804C211yesmaximal, characteristic, complemented
3#131×#6923C211yesmaximal, characteristic, complemented
4#21461976C211yesmaximal, characteristic, complemented
5#21461939C211yesmaximal, characteristic, complemented
6#21461907C212yesmaximal, complemented
7#21461856C212yesmaximal, complemented
8#21461706C211yesmaximal, characteristic, complemented
9#21461624C211yesmaximal, characteristic, complemented
10#21461576C212yesmaximal, complemented
11#21455537C211yesmaximal, characteristic
12D61⋊#131C2211yescharacteristic
13C2×#14519864C2212yes—
14#41×#6923C2211yescharacteristic
15#16840809—22no—
16#16840807—22no—
17#16840805—22no—
18#16840804—22no—
19#16840804—21no—
20#16840804—21no—
21#16840804—21no—
22#16840804—21no—
23#16840777C2212yes—
24C2×#14519848C2211yescharacteristic
25#16840772C2211yescharacteristic
26#16840760C2212yes—
27#16840759C2212yes—
28C2×#14519841C2212yes—
29#16840742C2212yes—
30#16840728—21no—
31#16840728C2212yes—
32#16840728C2211yescharacteristic
33#16840725—21no—
34#16840725C2211yescharacteristic
35M16×#6923—21no—
36M16×#6923C2211yescharacteristic
37C2×C8×#6923—21no—
38C2×C8×#6923C2211yescharacteristic
39#16840645C2212yes—
40#16840645C2212yes—
41#16840634C2212yes—
42#16840558C2212yes—
43C2×#14519796C2211yescharacteristic
44#131×D61C2211yescharacteristic
45#16840038C411yescharacteristic, complemented
46#16840037C411yescharacteristic, complemented
47#16840029C411yescharacteristic, complemented
48#16840024C412yescomplemented
49#16840023C412yescomplemented
50#108×D61C2211yescharacteristic
51#16839743C2212yes—
52C61×#472C411yescharacteristic, complemented
53#14519865D411yescharacteristic
54#14519865D411yescharacteristic
55#14519864—21no—
56#14519864—21no—
57#14519864—22no—
58#14519864—22no—
59#14519864—22no—
60#14519864—22no—
61#14519864D411yescharacteristic
62#14519864D411yescharacteristic
63C2×#11758213C2312yes—
64D61⋊M16—21no—
65D61⋊M16C2311yescharacteristic
66C2×#11758217—22no—
67C2×D61⋊C8—21no—
68C2×D61⋊C8C2311yescharacteristic
69C22×#92420—22no—
70Q8×#6923—21no—
71D4×#6923—21no—
72D4×#6923—22no—
73#14519856—21no—
74#14519856C2312yes—
75#14519856C2311yescharacteristic
76C2×C4×#6923—21no—
77C2×C4×#6923C2311yescharacteristic
78#14519848—22no—
79#14519848—21no—
80#14519848—21no—
81#14519847—22no—
82#14519846—21no—
83#14519845—21no—
84#14519842—22no—
85#14519841—22no—
86#14519841—22no—
87#14519841—22no—
88#14519841—22no—
89#14519841—22no—
90#14519835—24no—
91#14519831—21no—
92#14519831—21no—
93#14519831—22no—
94#14519831—21no—
95C8×#6923—21no—
96C8×#6923—21no—
97C8×#6923—22no—
98C8×#6923—21no—
99C8×#6923D411yescharacteristic
100C8×#6923D411yescharacteristic
101C8×#6923D411yescharacteristic
102C8×#6923D411yescharacteristic
103C2×#11758224C2312yes—
104#41×D61C2311yescharacteristic
105#14519810C2×C412yes—
106#14519807C2×C412yes—
107#14519806—22no—
108#14519803C2×C412yes—
109#14519801—22no—
110#14519798—22no—
111#14519796—22no—
112#14519796—21no—
113#14519796—21no—
114#14519796—21no—
115#14519796—21no—
116C2×Dic244C2×C411yescharacteristic, complemented
117C2×#11758200C2×C412yescomplemented
118#14519785—21no—
119#14519785C2×C411yescharacteristic
120#14519784C2×C411yescharacteristic
121#14519783—21no—
122#14519783C2×C411yescharacteristic
123M16×D61—21no—
124M16×D61C2311yescharacteristic
125M16×D61C2312yes—
126C2×C8×D61—21no—
127C2×C8×D61C2311yescharacteristic
128#14519713C2×C411yescharacteristic, complemented
129#14519712C2×C412yescomplemented
130#14519695C2×C412yescomplemented
131#14519695C2×C412yescomplemented
132#14519694C2×C412yescomplemented
133AGL(1,ℤ/8)×C61C2×C412yes—
134C2×D8×C61C2×C411yescharacteristic, complemented
135#131×C61C2×C411yescharacteristic
136#108×C61C2×C411yescharacteristic, complemented
137#100×C61C2×C412yescomplemented
138C22×#6923—22no—
139C22×#6923—21no—
140#41×C61C22×C411yescharacteristic
141C2×D4×C61C22×C412yes—
142C2×C22⋊D61—22no—
143C22×Dic61—22no—
144#11758226C2×D411yescharacteristic
145Q8×D61#4111yescharacteristic
146#11758224—21no—
147#11758224—22no—
148#11758224—22no—
149#11758224—22no—
150#11758224—22no—
151#11758224C2×D411yescharacteristic
152D4×D61—21no—
153D4×D61#4111yescharacteristic
154#11758222—21no—
155#11758222C22×C411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61C2411yescharacteristic
158C2×Dic122C22×C412yes—
159C22⋊#6923—22no—
160C22⋊#6923—22no—
161C22⋊#6923#4112yes—
162#11758217—21no—
163#11758217—21no—
164#11758217—21no—
165#11758217—42no—
166#11758217—42no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—22no—
170C2×#92420—42no—
171C2×#92420—42no—
172C2×#92420#4111yescharacteristic
173C2×#92420#4111yescharacteristic
174C4⋊#6923—22no—
175C4⋊#6923—21no—
176C4⋊#6923—22no—
177C4×#6923—22no—
178C4×#6923—21no—
179C4×#6923—22no—
180C4×#6923—21no—
181C4×#6923C2×D411yescharacteristic
182C4×#6923C2×D411yescharacteristic
183#11758213—21no—
184#11758213—21no—
185#11758213—22no—
186#11758213—22no—
187#11758213C2×D411yescharacteristic
188#11758213C2×D411yescharacteristic
189D61⋊C8—21no—
190D61⋊C8—21no—
191D61⋊C8—21no—
192D61⋊C8C2×D411yescharacteristic
193D61⋊C8C2×D411yescharacteristic
194D61⋊C8C2×D411yescharacteristic
195D61⋊C8C2×D411yescharacteristic
196SD16×C61—22no—
197D8×C61—22no—
198D8×C61—21no—
199D8×C61—21no—
200M16×C61—21no—
201M16×C61C22×C412yes—
202M16×C61C22×C411yescharacteristic
203C2×C8×C61—21no—
204C2×C8×C61C22×C411yescharacteristic
205#11758202—22no—
206#11758200—22no—
207#11758200—22no—
208#11758200—22no—
209#11758200—22no—
210#11758199—22no—
211#11758194—21no—
212#11758194C22×C411yescharacteristic
213#11758194C22×C412yes—
214C2×#92418—21no—
215C2×#92418C22×C411yescharacteristic
216Dic244—22no—
217Dic244—21no—
218Dic244—21no—
219#11758190—22no—
220#11758189—21no—
221#11758189—22no—
222#11758189—21no—
223#11758189—21no—
224C8×D61—21no—
225C8×D61—22no—
226C8×D61—21no—
227C8×D61—21no—
228C8×D61C2×D411yescharacteristic
229C8×D61C2×D411yescharacteristic
230C8×D61#4111yescharacteristic
231C8×D61#4111yescharacteristic
232C23×C61—22no—
233C22×D61—21no—
234C22×D61C2×#4111yescharacteristic
235C2×#6923—21no—
236C2×#6923—22no—
237C2×#6923—21no—
238C2×#6923—42no—
239C2×#6923—44no—
240C2×#6923—42no—
241Q8×C61C4×D411yescharacteristic
242D4×C61—21no—
243D4×C61—22no—
244D4×C61—22no—
245D4×C61C4×D411yescharacteristic
246C2×C4×C61—21no—
247C2×C4×C61C23×C411yescharacteristic
248C22⋊D61—21no—
249C22⋊D61—21no—
250C22⋊D61—21no—
251C22⋊D61—42no—
252C22⋊D61—42no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61C2×#4111yescharacteristic
258D244—21no—
259D244C4×D411yescharacteristic
260C4×D61—21no—
261C4×D61—21no—
262C4×D61—21no—
263C4×D61C22×D411yescharacteristic
264C4×D61C22×D411yescharacteristic
265C4×D61C2×#4111yescharacteristic
266C4×D61C2×#4111yescharacteristic
267Dic122—21no—
268Dic122—22no—
269Dic122—22no—
270Dic122C4×D411yescharacteristic
271#92420—21no—
272#92420—21no—
273#92420—42no—
274#92420#12111yescharacteristic
275#92420#12111yescharacteristic
276#92420#12111yescharacteristic
277#92420#12111yescharacteristic
278C8×C61—21no—
279C8×C61—22no—
280C8×C61C4×D411yescharacteristic
281C8×C61C4×D411yescharacteristic
282#92418—21no—
283#92418—22no—
284#92418C4×D411yescharacteristic
285#92418C4×D411yescharacteristic
286#21558—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—42no—
289C22×C61—82no—
290C22×C61C4×#4111yescharacteristic
291C2×D61—21no—
292C2×D61—41no—
293C2×D61—42no—
294C2×D61#52511yescharacteristic
295C2×D61#53211yescharacteristic
296#6923—22no—
297#6923—42no—
298#6923—41no—
299#6923—82no—
300C4×C61—21no—
301C4×C61C2×C4×D411yescharacteristic
302C4×C61C2×C4×D411yescharacteristic
303Dic61—21no—
304Dic61—42no—
305Dic61C2×#12111yescharacteristic
306Dic61#52911yescharacteristic
307#3067—611no—
308#2955—612no—
309C2×#443—611no—
310C4×#131—611no—
311#1976—611no—
312#1953—611no—
313#1932—612no—
314#1906—612no—
315#1814—611no—
316#1772—611no—
317#1757—612no—
318C2×C61—21no—
319C2×C61—41no—
320C2×C61—82no—
321C2×C61C4×#12111yescharacteristic
322D61—21no—
323D61—21no—
324D61—41no—
325#575—612no—
326C2×#135—611no—
327C2×#131—611no—
328C2×#131—611no—
329C2×#131—612no—
330C4×#41—611no—
331#473—611no—
332#473—611no—
333#472—611no—
334#472#692311yescharacteristic, complemented
335#471—612no—
336#471—612no—
337#444—1222no—
338#444—1222no—
339#443—1221no—
340#443—1221no—
341#443—1221no—
342#443—1221no—
343#443—1222no—
344#443—1222no—
345#430—612no—
346C2×#108—611no—
347C2×#108—611no—
348#427—611no—
349#421—612no—
350#421—612no—
351C2×#104—612no—
352#413—612no—
353#410—612no—
354#405—611no—
355#405—611no—
356#405—612no—
357#405—1221no—
358#405—1221no—
359C4×M16—611no—
360C4×M16—1221no—
361C2×C4×C8—611no—
362C2×C4×C8—1221no—
363#344—612no—
364#344—612no—
365#339—612no—
366C61#2155811yescharacteristic, complemented, Sylow
367C2×#41—611no—
368C2×#41—612no—
369#137—612no—
370#137—612no—
371#137—1222no—
372AGL(1,ℤ/8)—612no—
373AGL(1,ℤ/8)—1222no—
374AGL(1,ℤ/8)C2×#692312yes—
375#135—1221no—
376#135—1221no—
377#135—1221no—
378#135—1221no—
379#135—1222no—
380#135—1222no—
381C2×Q16—611no—
382C2×SD16—612no—
383C2×D8C2×#692311yescharacteristic, complemented
384#131—611no—
385#131—611no—
386#131—611no—
387#131—611no—
388#131—611no—
389#131—611no—
390#131—611no—
391#131—1221no—
392#131—1221no—
393#131—1221no—
394#131—1221no—
395#131—1222no—
396#131—1222no—
397#131—1222no—
398#131—1222no—
399#131C2×#692311yescharacteristic
400C2×M16—611no—
401C2×M16—611no—
402C2×M16—612no—
403C2×M16—612no—
404C2×M16—1221no—
405C2×M16—1221no—
406C2×M16—1222no—
407C22×C8—611no—
408C22×C8—611no—
409C22×C8—1221no—
410C22×C8—1221no—
411C22×C8—1222no—
412C4×Q8—1221no—
413C4×D4—1222no—
414C4×D4—1221no—
415#117—611no—
416#117—612no—
417#117—1221no—
418C2×C42—611no—
419C2×C42—1221no—
420#108—612no—
421#108—611no—
422#108—1221no—
423#108—1221no—
424#108—1222no—
425#108—1222no—
426#108C2×#692311yescharacteristic, complemented
427#107—1221no—
428#106—1221no—
429#104—1222no—
430#104—1222no—
431#104—1222no—
432#104—1222no—
433#104—1222no—
434#104—1222no—
435#101—612no—
436#101—612no—
437#100—612no—
438#100C2×#692312yescomplemented
439#99—1224no—
440#97—1221no—
441#97—1221no—
442#97—1221no—
443#97—1222no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8—611no—
448C4×C8—1221no—
449C4×C8—1221no—
450C4×C8—1221no—
451C4×C8—1222no—
452#41—611no—
453#41—611no—
454#41—611no—
455#41—1221no—
456#41—1221no—
457#41—1222no—
458#41—1222no—
459#41—1222no—
460#41—1222no—
461#41C22×#692311yescharacteristic
462C2×Q8—611no—
463C2×Q8—612no—
464C2×D4—611no—
465C2×D4—1221no—
466C2×D4—1222no—
467C2×D4C22×#692312yes—
468C22×C4—611no—
469C22×C4—1221no—
470C22×C4—1222no—
471C22×C4—1221no—
472C22×C4—1222no—
473Q16—1222no—
474Q16—1222no—
475Q16—1221no—
476Q16—1221no—
477SD16—22no—
478SD16—1222no—
479SD16—1222no—
480SD16—1222no—
481SD16—1222no—
482SD16—1222no—
483D8—22no—
484D8—21no—
485D8—21no—
486D8—1222no—
487M16—21no—
488M16—611no—
489M16—611no—
490M16—611no—
491M16—612no—
492M16—1221no—
493M16—1221no—
494M16—1222no—
495M16—1221no—
496M16—1221no—
497M16—1221no—
498M16—1221no—
499M16—1221no—
500M16—1221no—
501M16—1221no—
502M16—1222no—
503M16—1222no—
504M16—2442no—
505M16—2442no—
506M16C22×#692311yescharacteristic
507M16C22×#692312yes—
508C2×C8—21no—
509C2×C8—611no—
510C2×C8—611no—
511C2×C8—611no—
512C2×C8—611no—
513C2×C8—611no—
514C2×C8—611no—
515C2×C8—611no—
516C2×C8—611no—
517C2×C8—611no—
518C2×C8—611no—
519C2×C8—611no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1222no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1222no—
531C2×C8—2442no—
532C2×C8—2442no—
533C2×C8C22×#692311yescharacteristic
534C4⋊C4—1221no—
535C4⋊C4—1222no—
536C4⋊C4—1222no—
537C22⋊C4—612no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C42—611no—
541C42—611no—
542C42—1221no—
543C42—1221no—
544C42—1222no—
545C42—1222no—
546C23—22no—
547C23—611no—
548C23—1221no—
549Q8—611no—
550Q8—1221no—
551Q8—1222no—
552Q8—1222no—
553Q8D4×#692311yescharacteristic
554D4—21no—
555D4—22no—
556D4—22no—
557D4—611no—
558D4—1221no—
559D4—1221no—
560D4—1221no—
561D4—1221no—
562D4—2442no—
563D4—2442no—
564D4D4×#692311yescharacteristic
565C2×C4—21no—
566C2×C4—611no—
567C2×C4—611no—
568C2×C4—611no—
569C2×C4—611no—
570C2×C4—611no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2444no—
583C2×C4—2442no—
584C2×C4C23×#692311yescharacteristic
585C8—21no—
586C8—22no—
587C8—611no—
588C8—611no—
589C8—611no—
590C8—611no—
591C8—611no—
592C8—611no—
593C8—1221no—
594C8—1221no—
595C8—1221no—
596C8—1222no—
597C8—2442no—
598C8D4×#692311yescharacteristic
599C8D4×#692311yescharacteristic
600C22—21no—
601C22—42no—
602C22—82no—
603C22—611no—
604C22—611no—
605C22—1221no—
606C22—2441no—
607C22—2442no—
608C22#41×#692311yescharacteristic
609C4—21no—
610C4—611no—
611C4—611no—
612C4—1222no—
613C4—1221no—
614C4—4882no—
615C4—2441no—
616C4—2442no—
617C4—2442no—
618C4C2×D4×#692311yescharacteristic
619C4C2×D4×#692311yescharacteristic
620C2—21no—
621C2—41no—
622C2—82no—
623C2—1221no—
624C2—1221no—
625C2—2441no—
626C2#121×#692311yescharacteristic
subgroupG/H
centerC2#121×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#472C4
socleC2×C61C4×#121
prime radical#21455537C2
cosocleC2×C4×D61C24
hypercenter#472#6923
hypocenterC61#21558
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #21462591C2semidirect
2. C2×#16840804C2semidirect
3. #131×#6923C2semidirect
4. #21461976C2semidirect
5. #21461939C2semidirect
6. #21461907C2semidirect
7. #21461856C2semidirect
8. #21461706C2semidirect
9. #21461624C2semidirect
10. #21461576C2semidirect
45. #16840038C4semidirect
46. #16840037C4semidirect
47. #16840029C4semidirect
48. #16840024C4semidirect
49. #16840023C4semidirect
52. C61×#472C4semidirect
116. C2×Dic244C2×C4semidirect
117. C2×#11758200C2×C4semidirect
128. #14519713C2×C4semidirect
129. #14519712C2×C4semidirect
130. #14519695C2×C4semidirect
131. #14519695C2×C4semidirect
132. #14519694C2×C4semidirect
134. C2×D8×C61C2×C4semidirect
136. #108×C61C2×C4semidirect
137. #100×C61C2×C4semidirect
334. #472#6923semidirect
366. C61#21558semidirect
383. C2×D8C2×#6923semidirect
426. #108C2×#6923semidirect
438. #100C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214625912
C2×#168408042
#131×#69232
#214619762
#214619392
#214619072
#214618562
#214617062
#214616242
#214615762
#214555372
#2155861