Finite Group Databasebeta

← #11889173

Subgroups — #11889173

Order 1152.

1206 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#556C211yesmaximal, characteristic, complemented
2#10596021C211yesmaximal, characteristic
3#10592788C211yesmaximal, characteristic, complemented
4#10592783C211yesmaximal, characteristic, complemented
5#10592751C211yesmaximal, characteristic, complemented
6#10592747C211yesmaximal, characteristic, complemented
7#10592746C211yesmaximal, characteristic, complemented
8#10592745C211yesmaximal, characteristic, complemented
9#10592727C211yesmaximal, characteristic, complemented
10#10592725C211yesmaximal, characteristic, complemented
11#10592710C211yesmaximal, characteristic, complemented
12#10592702C211yesmaximal, characteristic, complemented
13#10592701C211yesmaximal, characteristic, complemented
14#10592700C211yesmaximal, characteristic, complemented
15#10592695C211yesmaximal, characteristic, complemented
16#83665—34nomaximal
17C2×#3828C2211yescharacteristic
18C22×C4⋊(C3⋊D3)C2211yescharacteristic
19C32×C4⋊Q8C2211yescharacteristic, complemented
20C32×#126—21no—
21C32×#126—21no—
22C32×#124—21no—
23C32×C22⋊Q8C2211yescharacteristic
24C32×C22⋊Q8C2211yescharacteristic, complemented
25C32×C4×Q8—21no—
26C32×C4×Q8C2211yescharacteristic, complemented
27C32×C4×D4C2211yescharacteristic
28C32×#117C2211yescharacteristic
29C32×#117C2211yescharacteristic, complemented
30C2×#3756C2211yescharacteristic, complemented
31C2×#3755C2211yescharacteristic
32C2×#3755C2211yescharacteristic
33#64146—21no—
34#64139C2211yescharacteristic
35#64133C2211yescharacteristic
36C2×#3748C2211yescharacteristic
37C22×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
38#64103—21no—
39#64093—21no—
40C22⋊C4×C3⋊D3—21no—
41#64084C2211yescharacteristic
42C4×C4⋊(C3⋊D3)C2211yescharacteristic
43#63662C2211yescharacteristic, complemented
44#63657C2211yescharacteristic
45#63657C2211yescharacteristic, complemented
46#63637C2211yescharacteristic, complemented
47#63637C2211yescharacteristic, complemented
48#63637C2211yescharacteristic, complemented
49#63636C2211yescharacteristic
50#63636C2211yescharacteristic
51#63636C2211yescharacteristic
52#63636C2211yescharacteristic
53#63636C2211yescharacteristic
54#63636C2211yescharacteristic
55#63635C2211yescharacteristic, complemented
56#63635C2211yescharacteristic, complemented
57#63634C2211yescharacteristic, complemented
58C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
59#63630C2211yescharacteristic, complemented
60C3×#556D314yescomplemented
61#5550—34no—
62#5089—34no—
63#5084—34no—
64#5052—34no—
65#5048—34no—
66#5047—34no—
67#5046—34no—
68#5028—34no—
69#5026—34no—
70#5011—34no—
71#5003—34no—
72#5002—34no—
73#5001—34no—
74#4996—34no—
75C23×C3⋊D3—21no—
76C2×C32×Q8—21no—
77C2×C32×Q8C2311yescharacteristic
78C2×C32×D4D411yescharacteristic
79C22×C32×C4C2311yescharacteristic
80C22×C32×C4C2311yescharacteristic
81C2×#638—21no—
82C2×#638—21no—
83#3828—22no—
84#3828—41no—
85C2×C4⋊(C3⋊D3)—22no—
86C2×C4⋊(C3⋊D3)—21no—
87C2×C4⋊(C3⋊D3)—41no—
88C2×C4⋊(C3⋊D3)C2311yescharacteristic
89C2×C4⋊(C3⋊D3)C2311yescharacteristic
90C2×C4×C3⋊D3—21no—
91C2×C4×C3⋊D3—21no—
92C2×#634C2311yescharacteristic
93C32×C4⋊C4—21no—
94C32×C4⋊C4—21no—
95C32×C4⋊C4—21no—
96C32×C4⋊C4—21no—
97C32×C4⋊C4D411yescharacteristic
98C32×C4⋊C4C2311yescharacteristic
99C32×C4⋊C4C2311yescharacteristic
100C32×C4⋊C4C2311yescharacteristic
101C32×C22⋊C4—21no—
102C32×C22⋊C4—21no—
103C32×C22⋊C4—21no—
104C32×C22⋊C4—21no—
105C32×C22⋊C4D411yescharacteristic
106C32×C22⋊C4D411yescharacteristic
107C32×C42—21no—
108C32×C42—21no—
109C32×C42—21no—
110C32×C42C2311yescharacteristic
111#3757—21no—
112#3756—41no—
113#3755—22no—
114#3755—41no—
115#3752—21no—
116#3752—21no—
117#3752—21no—
118#3752—21no—
119C4⋊(C3⋊Dic3)C2311yescharacteristic
120#3750—21no—
121#3748—41no—
122C2×C3⋊(C3⋊C8)—41no—
123C2×C3⋊(C3⋊C8)C2311yescharacteristic
124C2×C3⋊(C3⋊C8)C2311yescharacteristic
125C2×C3⋊(C3⋊C8)C2311yescharacteristic
126C2×C3⋊(C3⋊C8)C2311yescharacteristic
127#3311—91noSylow
128C2×#235—34no—
129C22×D12—34no—
130C3×C4⋊Q8C2×D314yescomplemented
131C3×#126—24no—
132C3×#126—24no—
133C3×#124—24no—
134C3×C22⋊Q8C2×D314yes—
135C3×C22⋊Q8C2×D314yescomplemented
136C3×C4×Q8—24no—
137C3×C4×Q8C2×D314yescomplemented
138C3×C4×D4C2×D314yes—
139C3×#117C2×D314yes—
140C3×#117C2×D314yescomplemented
141C2×#216—34no—
142C2×#215—34no—
143C2×#215—34no—
144#937—64no—
145#930—34no—
146#924—34no—
147C2×#208—34no—
148C22×C3⋊C8—34no—
149#894—64no—
150#884—64no—
151D3×C22⋊C4—64no—
152#875—34no—
153C4×D12—34no—
154#835—34no—
155#830—34no—
156#830—34no—
157#810—34no—
158#810—34no—
159#810—34no—
160#809—34no—
161#809—34no—
162#809—34no—
163#809—34no—
164#809—34no—
165#809—34no—
166#808—34no—
167#808—34no—
168#807—34no—
169C4⋊(C3⋊C8)—34no—
170#803—34no—
171C23×C32C2×D411yescharacteristic
172C23×C32C2×D411yescharacteristic
173C22×C3⋊D3—21no—
174C22×C3⋊D3—21no—
175C22×C3⋊D3—42no—
176C22×C3⋊D3—42no—
177C22×C3⋊D3—81no—
178C32×Q8—21no—
179C32×Q8—21no—
180C32×Q8—42no—
181C32×D4—41no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4C2×D411yescharacteristic
191C2×C32×C4C2×D411yescharacteristic
192C2×C32×C4C2×D411yescharacteristic
193C2×C32×C4C2×D411yescharacteristic
194C2×C32×C4C2411yescharacteristic
195#638—42no—
196#638—81no—
197C2×C3⋊Dic3—21no—
198C2×C3⋊Dic3—21no—
199C4⋊(C3⋊D3)—21no—
200C4⋊(C3⋊D3)—21no—
201C4⋊(C3⋊D3)—21no—
202C4⋊(C3⋊D3)—42no—
203C4⋊(C3⋊D3)#4112yes—
204C4×C3⋊D3—42no—
205C4×C3⋊D3—81no—
206#634—21no—
207#634—21no—
208C3⋊(C3⋊C8)—21no—
209C3⋊(C3⋊C8)—21no—
210C3⋊(C3⋊C8)—21no—
211C3⋊(C3⋊C8)—21no—
212#556C3⋊D311yescharacteristic, complemented
213#546—91no—
214#489—91no—
215#485—91no—
216#482—91no—
217#476—91no—
218#470—91no—
219#465—91no—
220#464—91no—
221#452—91no—
222#450—91no—
223#440—91no—
224#435—91no—
225#418—91no—
226#416—91no—
227C23×D3—64no—
228C2×C3×Q8—24no—
229C2×C3×Q8C22×D314yes—
230C2×C3×D4#6714yes—
231C22×C3×C4C22×D314yes—
232C22×C3×C4C22×D314yes—
233C2×#67—64no—
234C2×#67—64no—
235#235—68no—
236#235—124no—
237C2×D12—34no—
238C2×D12—34no—
239C2×D12—68no—
240C2×D12—64no—
241C2×D12—124no—
242C2×C4×D3—64no—
243C2×C4×D3—64no—
244C2×Dic6—34no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4#6714yes—
250C3×C4⋊C4C22×D314yes—
251C3×C4⋊C4C22×D314yes—
252C3×C4⋊C4C22×D314yes—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—24no—
257C3×C22⋊C4#6714yes—
258C3×C22⋊C4#6714yes—
259C3×C42—24no—
260C3×C42—24no—
261C3×C42—24no—
262C3×C42C22×D314yes—
263#217—64no—
264#216—124no—
265#215—68no—
266#215—124no—
267(C2×D3)⋊C4—64no—
268(C2×D3)⋊C4—64no—
269(C2×D3)⋊C4—64no—
270(C2×D3)⋊C4—64no—
271C4⋊Dic3—34no—
272Dic3⋊C4—64no—
273#208—124no—
274C2×C3⋊C8—34no—
275C2×C3⋊C8—34no—
276C2×C3⋊C8—34no—
277C2×C3⋊C8—34no—
278C2×C3⋊C8—124no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32C22×D411yescharacteristic
286C2×C3⋊D3—22no—
287C2×C3⋊D3—41no—
288C2×C3⋊D3—41no—
289C2×C3⋊D3—41no—
290C2×C3⋊D3—41no—
291C2×C3⋊D3—41no—
292C2×C3⋊D3—84no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4C2×#4111yescharacteristic
302C32×C4H(5,𝔽2)11yescharacteristic
303C3⋊Dic3—41no—
304C3⋊Dic3—41no—
305C2×#41—91no—
306C22×D4—91no—
307C2×Q16—91no—
308C2×SD16—91no—
309C2×SD16—91no—
310C2×M16—91no—
311C22×C8—91no—
312C4⋊Q8C2×C3⋊D311yescharacteristic, complemented
313#126—21no—
314#126—21no—
315#124—21no—
316#124—91no—
317#123—181no—
318C22⋊Q8—91no—
319C22⋊Q8C2×C3⋊D311yescharacteristic
320C22⋊Q8C2×C3⋊D311yescharacteristic, complemented
321#121—91no—
322#121—181no—
323#120—181no—
324C4×Q8—21no—
325C4×Q8C2×C3⋊D311yescharacteristic, complemented
326C4×D4—91no—
327C4×D4C2×C3⋊D311yescharacteristic
328#117C2×C3⋊D311yescharacteristic
329#117C2×C3⋊D311yescharacteristic, complemented
330C2×C22⋊C4—181no—
331#107—91no—
332#106—91no—
333#106—91no—
334C4⋊C8—91no—
335#103—91no—
336#103—91no—
337#103—91no—
338#103—91no—
339#102—91no—
340#102—91no—
341#102—91no—
342#102—91no—
343#102—91no—
344#102—91no—
345C22⋊C8—91no—
346C22⋊C8—91no—
347#97—91no—
348C23×C3C2×#6714yes—
349C23×C3C2×#6714yes—
350C22×D3—64no—
351C22×D3—64no—
352C22×D3—128no—
353C22×D3—128no—
354C22×D3—244no—
355C3×Q8—24no—
356C3×Q8—24no—
357C3×Q8—48no—
358C3×D4—44no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4C2×#6714yes—
368C2×C3×C4C2×#6714yes—
369C2×C3×C4C2×#6714yes—
370C2×C3×C4C2×#6714yes—
371C2×C3×C4C23×D314yes—
372#67—128no—
373#67—244no—
374C2×Dic3—64no—
375C2×Dic3—64no—
376D12—38no—
377D12—64no—
378D12—64no—
379D12—64no—
380D12—128no—
381C4×D3—128no—
382C4×D3—244no—
383Dic6—64no—
384Dic6—64no—
385C3⋊C8—64no—
386C3⋊C8—64no—
387C3⋊C8—64no—
388C3⋊C8—64no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32#54611yescharacteristic
392C2×C32#57511yescharacteristic
393C2×C32#57711yescharacteristic
394C3⋊D3—42no—
395C3⋊D3—81no—
396C24—181no—
397#41—182no—
398#41—361no—
399C2×Q8—21no—
400C2×Q8—91no—
401C2×Q8C22×C3⋊D311yescharacteristic
402C2×D4—91no—
403C2×D4—91no—
404C2×D4—182no—
405C2×D4—181no—
406C2×D4—181no—
407C2×D4—181no—
408C2×D4—361no—
409C2×D4#63811yescharacteristic
410C22×C4—181no—
411C22×C4—181no—
412C22×C4C22×C3⋊D311yescharacteristic
413C22×C4C22×C3⋊D311yescharacteristic
414Q16—361no—
415SD16—182no—
416SD16—361no—
417M16—361no—
418C2×C8—91no—
419C2×C8—91no—
420C2×C8—91no—
421C2×C8—91no—
422C2×C8—361no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—91no—
428C4⋊C4—181no—
429C4⋊C4#63811yescharacteristic
430C4⋊C4C22×C3⋊D311yescharacteristic
431C4⋊C4C22×C3⋊D311yescharacteristic
432C4⋊C4C22×C3⋊D311yescharacteristic
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—21no—
436C22⋊C4—21no—
437C22⋊C4—181no—
438C22⋊C4—181no—
439C22⋊C4—181no—
440C22⋊C4—181no—
441C22⋊C4—181no—
442C22⋊C4#63811yescharacteristic
443C22⋊C4#63811yescharacteristic
444C42—21no—
445C42—21no—
446C42—21no—
447C42C22×C3⋊D311yescharacteristic
448C22×C3—24no—
449C22×C3—24no—
450C22×C3—24no—
451C22×C3—24no—
452C22×C3—24no—
453C22×C3—24no—
454C22×C3C22×#6714yes—
455C2×D3—68no—
456C2×D3—124no—
457C2×D3—124no—
458C2×D3—124no—
459C2×D3—124no—
460C2×D3—124no—
461C2×D3—2416no—
462C3×C4—24no—
463C3×C4—24no—
464C3×C4—24no—
465C3×C4—24no—
466C3×C4—44no—
467C3×C4—44no—
468C3×C4—44no—
469C3×C4—44no—
470C3×C4D3×#4114yes—
471C3×C4#100914yes—
472Dic3—124no—
473Dic3—124no—
474C32#331111yescharacteristic, complemented, Sylow
475C23—181no—
476C23—181no—
477C23—362no—
478C23—362no—
479C23—721no—
480C23C2×#63811yescharacteristic
481C23C2×#63811yescharacteristic
482Q8—21no—
483Q8—21no—
484Q8—42no—
485Q8—181no—
486Q8—181no—
487D4—41no—
488D4—92no—
489D4—181no—
490D4—181no—
491D4—181no—
492D4—362no—
493D4—362no—
494D4—721no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—181no—
504C2×C4—181no—
505C2×C4—362no—
506C2×C4—721no—
507C2×C4C2×#63811yescharacteristic
508C2×C4C2×#63811yescharacteristic
509C2×C4C2×#63811yescharacteristic
510C2×C4C2×#63811yescharacteristic
511C2×C4C23×C3⋊D311yescharacteristic
512C8—181no—
513C8—181no—
514C8—181no—
515C8—181no—
516C2×C3—44no—
517C2×C3—44no—
518C2×C3#582214yes—
519C2×C3#582914yes—
520C2×C3#583714yes—
521D3—128no—
522D3—244no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—182no—
530C22—361no—
531C22—361no—
532C22—361no—
533C22—361no—
534C22—361no—
535C22—724no—
536C22C22×#63811yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—361no—
546C4—361no—
547C4#41×C3⋊D311yescharacteristic
548C4#6436611yescharacteristic
549C3#8366514yescomplemented
550C2—41no—
551C2—41no—
552C2—362no—
553C2—721no—
554C2#1059629311yescharacteristic
555C2#1059630011yescharacteristic
556C2#1059630811yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#556C2
socleC22×C32C22×D4
prime radical#10596021C2
hypercenter#556C3⋊D3
hypocenterC32#3311
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
1. C32×#556C2semidirect
3. #10592788C2semidirect
4. #10592783C2semidirect
5. #10592751C2semidirect
6. #10592747C2semidirect
7. #10592746C2semidirect
8. #10592745C2semidirect
9. #10592727C2semidirect
10. #10592725C2semidirect
11. #10592710C2semidirect
12. #10592702C2semidirect
13. #10592701C2semidirect
14. #10592700C2semidirect
15. #10592695C2semidirect
19. C32×C4⋊Q8C22semidirect
24. C32×C22⋊Q8C22semidirect
26. C32×C4×Q8C22semidirect
29. C32×#117C22semidirect
30. C2×#3756C22semidirect
37. C22×C3⋊(C3⋊C8)C22semidirect
43. #63662C22semidirect
45. #63657C22semidirect
46. #63637C22semidirect
47. #63637C22semidirect
48. #63637C22semidirect
55. #63635C22semidirect
56. #63635C22semidirect
57. #63634C22semidirect
58. C4⋊(C3⋊(C3⋊C8))C22semidirect
59. #63630C22semidirect
60. C3×#556D3semidirect
130. C3×C4⋊Q8C2×D3semidirect
135. C3×C22⋊Q8C2×D3semidirect
137. C3×C4×Q8C2×D3semidirect
140. C3×#117C2×D3semidirect
212. #556C3⋊D3semidirect
312. C4⋊Q8C2×C3⋊D3semidirect
320. C22⋊Q8C2×C3⋊D3semidirect
325. C4×Q8C2×C3⋊D3semidirect
329. #117C2×C3⋊D3semidirect
474. C32#3311semidirect
549. C3#83665semidirect

Maximal subgroups

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

subgroupindex
C32×#5562
#105960212
#105927882
#105927832
#105927512
#105927472
#105927462
#105927452
#105927272
#105927252
#105927102
#105927022
#105927012
#105927002
#105926952
#836653