Finite Group Databasebeta

← #11887152

Subgroups — #11887152

Order 1152.

983 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5730C211yesmaximal, characteristic, complemented
2D122C211yesmaximal, characteristic
3#10592787C211yesmaximal, characteristic, complemented
4C3×#5084C211yesmaximal, characteristic, complemented
5#10590596C212yesmaximal, complemented
6D3×#809C212yesmaximal, complemented
7#10590577C212yesmaximal, complemented
8#10590423C212yesmaximal, complemented
9#10590367C211yesmaximal, characteristic, complemented
10C4×#3715C211yesmaximal, characteristic, complemented
11#10590329C211yesmaximal, characteristic, complemented
12#83634—31nomaximal
13#81075—31nomaximal
14C2×D3×D12—21no—
15C2×D3×D12C2212yes—
16C32×C4⋊Q8C2211yescharacteristic, complemented
17#64083C2211yescharacteristic
18C3×#946—22no—
19C3×#897C2212yescomplemented
20C3×#893—21no—
21C3×D3×C4⋊C4C2212yescomplemented
22C3×#890—21no—
23C3×C4×D12C2211yescharacteristic
24C3×C4×D12C2211yescharacteristic
25C3×C4×Dic6C2211yescharacteristic, complemented
26#63923—22no—
27#63909—21no—
28#63909C2212yes—
29C2×#3717C2212yescomplemented
30C2×#3715C2211yescharacteristic
31C2×D3×C3⋊C8C2212yescomplemented
32#63636C2212yescomplemented
33#63636C2212yescomplemented
34C4×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
35C3×#809C2212yes—
36C3×#808C2212yescomplemented
37C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
38#63574C2211yescharacteristic, complemented
39#63565C2211yescharacteristic, complemented
40#63564C2212yescomplemented
41#63562C2211yescharacteristic
42(C2×D3)⋊(C3⋊C8)C2212yescomplemented
43C3×#554—31no—
44#5730D311yescharacteristic, complemented
45D4×D12—31no—
46D4×D12—31no—
47C3×#476—31no—
48#5088—31no—
49#5088—31no—
50#5088—61no—
51#5084D311yescharacteristic, complemented
52#5048—32no—
53#5046—32no—
54#5027—31no—
55C4×#215—31no—
56C4×#214—31no—
57D4×C3⋊C8—31no—
58#4968—32no—
59C2×#809—32no—
60#4864—32no—
61#4863—32no—
62#4848—31no—
63#4834—31no—
64#4776—32no—
65D3×#102—32no—
66C22×D32—22no—
67C2×C32×Q8—21no—
68C2×C4⋊(C3⋊D3)—21no—
69C2×C4⋊(C3⋊D3)—21no—
70C2×C4⋊(C3⋊D3)C2312yes—
71C2×C3×D12D411yescharacteristic
72C2×C3×D12C2311yescharacteristic
73C2×C3×C4×D3—21no—
74C2×C3×C4×D3—22no—
75C2×C3×C4×D3C2312yes—
76C2×C3×Dic6C2311yescharacteristic
77C2×#626—22no—
78C2×#626—22no—
79D3×D12—24no—
80D3×D12—42no—
81D3×D12—44no—
82C32×C4⋊C4—21no—
83C32×C4⋊C4C2312yes—
84C32×C42C2311yescharacteristic
85C2×C3⋊(C3⋊C8)C2311yescharacteristic
86C2×C3⋊(C3⋊C8)C2311yescharacteristic
87C3×(C2×D3)⋊C4—22no—
88C3×(C2×D3)⋊C4—21no—
89C3×(C2×D3)⋊C4D412yes—
90C3×C4⋊Dic3—22no—
91C3×C4⋊Dic3D411yescharacteristic
92C3×C4⋊Dic3C2311yescharacteristic
93C3×C4⋊Dic3C2311yescharacteristic
94C3×Dic3⋊C4—21no—
95C3×Dic3⋊C4—22no—
96C3×C4×Dic3—21no—
97C2×C3×C3⋊C8C2312yes—
98#3722—22no—
99#3717—42no—
100#3715—22no—
101D3×C3⋊C8—42no—
102#3295—91noSylow
103C2×D3×D4—32no—
104C2×D3×D4—61no—
105C22×D12—32no—
106C22×D12—61no—
107C3×C4⋊Q8—21no—
108C3×C4⋊Q8C2×D311yescharacteristic, complemented
109C3×C4⋊Q8C2×D311yescharacteristic, complemented
110C3×#125—61no—
111C3×C22⋊Q8—32no—
112C3×C22⋊Q8—62no—
113C3×C4×Q8—31no—
114C3×C4×D4—31no—
115C3×C4×D4—31no—
116C3×C4×D4—61no—
117C2×C3×C4⋊C4—32no—
118#946—22no—
119C2×#215—31no—
120C2×#215—32no—
121C2×#214—31no—
122#930—32no—
123#930—61no—
124C22×C3⋊C8—32no—
125C2×#204—32no—
126C2×D3×C8—32no—
127#897C2×D312yescomplemented
128#895—32no—
129#895—61no—
130#893—21no—
131D3×C4⋊C4C2×D312yescomplemented
132#890—21no—
133#882—62no—
134#882—62no—
135#874—31no—
136#874—31no—
137#874—61no—
138C4×D12C2×D311yescharacteristic
139C4×D12C2×D311yescharacteristic
140C4×Dic6C2×D311yescharacteristic, complemented
141C3×#106—32no—
142C3×C4⋊C8—31no—
143C3×#102—32no—
144#835—31no—
145#832—31no—
146#830—32no—
147#821—32no—
148(C2×D3)⋊C8—32no—
149#810—31no—
150#809—31no—
151#809—32no—
152#809—32no—
153#809—32no—
154#809—32no—
155#809—32no—
156#809—62no—
157#809—62no—
158#809C2×D312yes—
159#808—31no—
160#808—31no—
161#808C2×D312yescomplemented
162C4⋊(C3⋊C8)C2×D311yescharacteristic, complemented
163C4×C3⋊C8—31no—
164C4×C3⋊C8—31no—
165C4×C3⋊C8—61no—
166C22×C3⋊D3—22no—
167C22×C3×D3—21no—
168C22×C3×D3C2×D412yes—
169C2×D32—44no—
170C2×D32—44no—
171C2×D32—82no—
172C32×Q8—81no—
173C2×C32×C4—22no—
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2×D411yescharacteristic
176C2×C32×C4C2411yescharacteristic
177C4⋊(C3⋊D3)—22no—
178C4⋊(C3⋊D3)—22no—
179C4⋊(C3⋊D3)—42no—
180C4⋊(C3⋊D3)—81no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3C2×D412yes—
185C3×D12—21no—
186C3×D12SD1614yescomplemented
187C3×D12#4112yes—
188C3×C4×D3—22no—
189C3×C4×D3—22no—
190C3×C4×D3—42no—
191C3×C4×D3—82no—
192C3×Dic6—21no—
193C3×Dic6—21no—
194#626—44no—
195#626—44no—
196C3⋊(C3⋊C8)—21no—
197C3⋊(C3⋊C8)—21no—
198C3×C3⋊C8—22no—
199#554—31no—
200D42—91no—
201#486—91no—
202#481—92no—
203#476—31no—
204#465—92no—
205#460—91no—
206#450—92no—
207C4×SD16—91no—
208C8×D4—91no—
209C2×#102—92no—
210C23×D3—62no—
211C23×D3—62no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—31no—
215C2×C3×Q8—41no—
216C2×C3×D4—31no—
217C2×C3×D4—31no—
218C22×C3×C4—32no—
219C22×C3×C4—62no—
220C22×C3×C4—61no—
221C2×#67—62no—
222C2×#67—62no—
223D3×D4—64no—
224D3×D4—122no—
225D3×D4—124no—
226C2×D12—32no—
227C2×D12—32no—
228C2×D12—62no—
229C2×D12—64no—
230C2×D12—62no—
231C2×D12—62no—
232C2×D12—61no—
233C2×D12—61no—
234C2×D12—61no—
235C2×D12—61no—
236C2×D12—121no—
237C2×D12—121no—
238C2×D12—122no—
239C2×D12—124no—
240C2×D12#6711yescharacteristic
241C2×D12C22×D311yescharacteristic
242C2×C4×D3—21no—
243C2×C4×D3—22no—
244C2×C4×D3C22×D312yes—
245C2×Dic6C22×D311yescharacteristic
246C2×C3×C8—32no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—31no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—41no—
254C3×C4⋊C4—62no—
255C3×C4⋊C4—62no—
256C3×C4⋊C4—61no—
257C3×C4⋊C4C22×D312yes—
258C3×C4⋊C4C22×D312yes—
259C3×C22⋊C4—32no—
260C3×C22⋊C4—62no—
261C3×C22⋊C4—61no—
262C3×C42—21no—
263C3×C42—61no—
264C3×C42C22×D311yescharacteristic
265C3×C42C22×D311yescharacteristic
266#215—62no—
267#215—122no—
268#214—62no—
269(C2×D3)⋊C4—22no—
270(C2×D3)⋊C4—21no—
271(C2×D3)⋊C4—62no—
272(C2×D3)⋊C4—62no—
273(C2×D3)⋊C4#6712yes—
274C4⋊Dic3—22no—
275C4⋊Dic3#6711yescharacteristic
276C4⋊Dic3C22×D311yescharacteristic
277C4⋊Dic3C22×D311yescharacteristic
278Dic3⋊C4—21no—
279Dic3⋊C4—22no—
280C4×Dic3—21no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—31no—
283C2×C3⋊C8—31no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—61no—
286C2×C3⋊C8—61no—
287C2×C3⋊C8—122no—
288C2×C3⋊C8C22×D312yes—
289#204—122no—
290D3×C8—122no—
291C22×C32C22×D411yescharacteristic
292C2×C3⋊D3—42no—
293C2×C3⋊D3—42no—
294C2×C3⋊D3—42no—
295C2×C3×D3—22no—
296C2×C3×D3—22no—
297C2×C3×D3—22no—
298C2×C3×D3—41no—
299C2×C3×D3—41no—
300C2×C3×D3C2×SD1614yes—
301D32—88no—
302C32×C4—21no—
303C32×C4—42no—
304C32×C4C2×SD1612yes—
305C32×C4C2×#4111yescharacteristic
306C32×C4H(5,𝔽2)11yescharacteristic
307C3×Dic3—22no—
308C3×Dic3—41no—
309C3×Dic3—41no—
310C3×Dic3—41no—
311C22×D4—92no—
312C22×D4—181no—
313C2×SD16—92no—
314C2×SD16—91no—
315C22×C8—92no—
316C4⋊Q8D3211yescharacteristic, complemented
317#127—91no—
318#125—61no—
319C22⋊Q8—32no—
320C22⋊Q8—62no—
321#121—92no—
322#121—181no—
323#120—182no—
324C4×Q8—31no—
325C4×D4—31no—
326C4×D4—31no—
327C4×D4—61no—
328C2×C4⋊C4—32no—
329#106—32no—
330#106—91no—
331C4⋊C8—31no—
332#103—91no—
333#102—32no—
334#102—92no—
335#102—91no—
336#102—92no—
337#102—92no—
338C22⋊C8—92no—
339C4×C8—91no—
340C23×C3—32no—
341C23×C3—61no—
342C22×D3—21no—
343C22×D3—62no—
344C22×D3—62no—
345C22×D3—122no—
346C22×D3—124no—
347C22×D3—124no—
348C22×D3—124no—
349C22×D3—124no—
350C22×D3—242no—
351C22×D3—242no—
352C22×D3C2×#6712yes—
353C3×Q8—61no—
354C3×Q8—61no—
355C3×Q8—81no—
356C3×Q8—81no—
357C3×Q8—82no—
358C3×D4—34no—
359C3×D4—32no—
360C3×D4—61no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—32no—
367C2×C3×C4—42no—
368C2×C3×C4—62no—
369C2×C3×C4—62no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—61no—
373C2×C3×C4—122no—
374C2×C3×C4—242no—
375C2×C3×C4D3×D411yescharacteristic
376C2×C3×C4D3×D411yescharacteristic
377C2×C3×C4C2×#6711yescharacteristic
378C2×C3×C4C2×#6711yescharacteristic
379C2×C3×C4C23×D311yescharacteristic
380C2×C3×C4C23×D311yescharacteristic
381#67—124no—
382#67—124no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3C2×#6712yes—
387D12—21no—
388D12—62no—
389D12—62no—
390D12—62no—
391D12—62no—
392D12—122no—
393D12—122no—
394D12—124no—
395D12—124no—
396D12—122no—
397D12—122no—
398D12—242no—
399D12—241no—
400D12—241no—
401D12—242no—
402D12#21514yescomplemented
403D12#23912yes—
404C4×D3—22no—
405C4×D3—22no—
406C4×D3—42no—
407C4×D3—82no—
408Dic6—21no—
409Dic6—21no—
410C3×C8—62no—
411C3⋊C8—22no—
412C3⋊C8—61no—
413C3⋊C8—61no—
414C3⋊C8—61no—
415C3⋊C8—61no—
416C3⋊C8—121no—
417C3⋊C8—121no—
418C2×C32#54611yescharacteristic
419C2×C32C22×SD1611yescharacteristic
420C2×C32#57611yescharacteristic
421C3⋊D3—82no—
422C3×D3—24no—
423C3×D3—42no—
424C24—182no—
425C2×Q8—21no—
426C2×Q8—31no—
427C2×D4—31no—
428C2×D4—31no—
429C2×D4—92no—
430C2×D4—182no—
431C2×D4—184no—
432C2×D4—182no—
433C2×D4—181no—
434C2×D4—181no—
435C2×D4—362no—
436C2×D4—364no—
437C22×C4—32no—
438C22×C4—62no—
439C22×C4—61no—
440SD16—182no—
441SD16—362no—
442C2×C8—32no—
443C2×C8—91no—
444C2×C8—91no—
445C2×C8—362no—
446C4⋊C4—21no—
447C4⋊C4—31no—
448C4⋊C4—31no—
449C4⋊C4—31no—
450C4⋊C4—62no—
451C4⋊C4—62no—
452C4⋊C4—61no—
453C4⋊C4C2×D3212yes—
454C22⋊C4—32no—
455C22⋊C4—62no—
456C22⋊C4—61no—
457C22⋊C4—182no—
458C42—61no—
459C42C2×D3211yescharacteristic
460C22×C3—21no—
461C22×C3—34no—
462C22×C3—62no—
463C22×C3—62no—
464C22×C3—62no—
465C22×C3—121no—
466C22×C3—121no—
467C22×C3C2×D3×D411yescharacteristic
468C22×C3C22×#6711yescharacteristic
469C2×D3—22no—
470C2×D3—22no—
471C2×D3—22no—
472C2×D3—41no—
473C2×D3—41no—
474C2×D3—122no—
475C2×D3—122no—
476C2×D3—122no—
477C2×D3—122no—
478C2×D3—122no—
479C2×D3—122no—
480C2×D3—242no—
481C2×D3—242no—
482C2×D3—242no—
483C2×D3—248no—
484C2×D3—248no—
485C2×D3C2×#21514yes—
486C3×C4—22no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—22no—
492C3×C4—42no—
493C3×C4—42no—
494C3×C4—62no—
495C3×C4—82no—
496C3×C4—121no—
497C3×C4—121no—
498C3×C4—121no—
499C3×C4D3×SD1612yes—
500C3×C4C2×#21512yes—
501C3×C4C2×#23911yescharacteristic
502C3×C4D3×#4111yescharacteristic
503C3×C4#100911yescharacteristic
504C3×C4#100911yescharacteristic
505Dic3—22no—
506Dic3—41no—
507Dic3—41no—
508Dic3—41no—
509C32#329511yescharacteristic, complemented, Sylow
510C23—32no—
511C23—61no—
512C23—182no—
513C23—364no—
514C23—364no—
515C23—722no—
516Q8—61no—
517Q8—61no—
518Q8—81no—
519D4—32no—
520D4—34no—
521D4—61no—
522D4—182no—
523D4—182no—
524D4—364no—
525D4—364no—
526D4—362no—
527D4—721no—
528C2×C4—22no—
529C2×C4—32no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—122no—
536C2×C4—242no—
537C2×C4D3×#6711yescharacteristic
538C2×C4D3×#6711yescharacteristic
539C2×C4C22×D3211yescharacteristic
540C8—62no—
541C8—181no—
542C8—181no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—64no—
547C2×C3—122no—
548C2×C3#563011yescharacteristic
549C2×C3C2×D3×SD1611yescharacteristic
550C2×C3#577411yescharacteristic
551C2×C3C22×#21511yescharacteristic
552C2×C3#582911yescharacteristic
553C2×C3#583511yescharacteristic
554D3—24no—
555D3—42no—
556D3—242no—
557D3—242no—
558D3—482no—
559C22—34no—
560C22—62no—
561C22—62no—
562C22—62no—
563C22—121no—
564C22—121no—
565C22—362no—
566C22—362no—
567C22—362no—
568C22—728no—
569C22C2×D3×#6711yescharacteristic
570C4—21no—
571C4—42no—
572C4—62no—
573C4—121no—
574C4—121no—
575C4—121no—
576C4D3×#21512yes—
577C4D3×#23911yescharacteristic
578C4#6432011yescharacteristic
579C3—21no—
580C3#8107511yescharacteristic, complemented
581C3#8363411yescharacteristic, complemented
582C2—64no—
583C2—122no—
584C2—722no—
585C2#1059509711yescharacteristic
586C2C2×D3×#21511yescharacteristic
587C2#1059542711yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radicalD122C2
hypercenterC4⋊Q8D32
hypocenterC32#3295
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5730C2semidirect
3. #10592787C2semidirect
4. C3×#5084C2semidirect
5. #10590596C2semidirect
6. D3×#809C2semidirect
7. #10590577C2semidirect
8. #10590423C2semidirect
9. #10590367C2semidirect
10. C4×#3715C2semidirect
11. #10590329C2semidirect
16. C32×C4⋊Q8C22semidirect
19. C3×#897C22semidirect
21. C3×D3×C4⋊C4C22semidirect
25. C3×C4×Dic6C22semidirect
29. C2×#3717C22semidirect
31. C2×D3×C3⋊C8C22semidirect
32. #63636C22semidirect
33. #63636C22semidirect
34. C4×C3⋊(C3⋊C8)C22semidirect
36. C3×#808C22semidirect
37. C3×C4⋊(C3⋊C8)C22semidirect
38. #63574C22semidirect
39. #63565C22semidirect
40. #63564C22semidirect
42. (C2×D3)⋊(C3⋊C8)C22semidirect
44. #5730D3semidirect
51. #5084D3semidirect
108. C3×C4⋊Q8C2×D3semidirect
109. C3×C4⋊Q8C2×D3semidirect
127. #897C2×D3semidirect
131. D3×C4⋊C4C2×D3semidirect
140. C4×Dic6C2×D3semidirect
161. #808C2×D3semidirect
162. C4⋊(C3⋊C8)C2×D3semidirect
186. C3×D12SD16semidirect
316. C4⋊Q8D32semidirect
402. D12#215semidirect
509. C32#3295semidirect
580. C3#81075semidirect
581. C3#83634semidirect

Maximal subgroups

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

subgroupindex
C3×#57302
D1222
#105927872
C3×#50842
#105905962
D3×#8092
#105905772
#105904232
#105903672
C4×#37152
#105903292
#836343
#810753