Finite Group Databasebeta

← #10595317

Subgroups — #10595317

Order 576.

554 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#239C211yesmaximal, characteristic, complemented
2D3×#235C211yesmaximal, characteristic, complemented
3#64151C211yesmaximal, characteristic, complemented
4C3×#942C211yesmaximal, characteristic, complemented
5#63949C212yesmaximal, complemented
6#63947C212yesmaximal, complemented
7#63943C212yesmaximal, complemented
8#63939C212yesmaximal, complemented
9#63830C211yesmaximal, characteristic, complemented
10#63823C211yesmaximal, characteristic, complemented
11D3×#208C211yesmaximal, characteristic, complemented
12#5822—31nomaximal
13#5777—31nomaximal
14C2×C32×Q8C2211yescharacteristic, complemented
15#3828C2211yescharacteristic
16C3×#239—21no—
17C3×#239—21no—
18C3×#239C2212yes—
19C3×#239C2212yescomplemented
20C3×#235C2211yescharacteristic
21C2×C3×D12—21no—
22C2×C3×D12C2211yescharacteristic, complemented
23C2×C3×C4×D3—21no—
24C2×C3×C4×D3C2211yescharacteristic
25D3×#67—21no—
26#3804—21no—
27D3×D12C2211yescharacteristic, complemented
28C4×D32—21no—
29#3799C2211yescharacteristic, complemented
30#3798—21no—
31#3795C2211yescharacteristic, complemented
32D3×Dic6C2211yescharacteristic, complemented
33#3756C2212yescomplemented
34#3755C2212yescomplemented
35#3748C2211yescharacteristic, complemented
36C3×#216C2212yescomplemented
37C3×#215C2212yescomplemented
38C3×#208C2211yescharacteristic, complemented
39#3716C2212yescomplemented
40#3715C2212yescomplemented
41#3714C2212yescomplemented
42#3713C2212yescomplemented
43#3711C2212yescomplemented
44D3×C3⋊C8C2212yescomplemented
45C2×C3×#41—31no—
46D3×#41—31no—
47C2×#239D311yescharacteristic, complemented
48C2×#235—31no—
49C3×#137—31no—
50#951—32no—
51#950—32no—
52#950—32no—
53#949—31no—
54#949—32no—
55#942—31no—
56#942—31no—
57#942—61no—
58#942D311yescharacteristic, complemented
59#932—31no—
60#932—31no—
61C2×#208—31no—
62#919—32no—
63#918—32no—
64#916—32no—
65#914—32no—
66#908—31no—
67D3×M16—31no—
68C22×C3×D3—21no—
69C22×C3×D3D411yescharacteristic
70C2×D32—21no—
71C32×Q8—21no—
72C32×Q8C2312yes—
73C2×C32×C4—21no—
74C2×C32×C4C2311yescharacteristic
75#638—21no—
76C4⋊(C3⋊D3)C2311yescharacteristic
77C4×C3⋊D3—21no—
78#634C2311yescharacteristic
79C3×#67—21no—
80C2×C3×Dic3D411yescharacteristic
81C3×D12—22no—
82C3×D12—22no—
83C3×D12—21no—
84C3×D12C2311yescharacteristic
85C3×D12C2312yes—
86C3×C4×D3—21no—
87C3×C4×D3—22no—
88C3×C4×D3—22no—
89C3×C4×D3D411yescharacteristic
90C3×C4×D3D411yescharacteristic
91C3×C4×D3C2311yescharacteristic
92C3×C4×D3C2311yescharacteristic
93C3×Dic6C2311yescharacteristic
94#627—21no—
95#626—21no—
96#626—21no—
97#625—21no—
98#624—21no—
99D3×Dic3—21no—
100D3×Dic3—21no—
101C3⋊(C3⋊C8)C2312yes—
102C3×C3⋊C8C2312yes—
103#575—91noSylow
104C3×#41—32no—
105C3×#41—32no—
106C3×#41—31no—
107C3×#41—61no—
108C3×#41—61no—
109C2×C3×Q8—21no—
110C2×C3×Q8C2×D311yescharacteristic, complemented
111C2×C3×Q8C2×D311yescharacteristic, complemented
112C2×C3×D4—31no—
113C2×C3×D4—61no—
114C22×C3×C4—31no—
115C22×C3×C4—61no—
116C2×#67—61no—
117#239—21no—
118#239—21no—
119#239—31no—
120#239C2×D312yes—
121#239C2×D312yescomplemented
122D3×Q8—31no—
123#237—31no—
124#237—61no—
125D3×D4—31no—
126D3×D4—61no—
127#235—31no—
128#235—31no—
129#235—31no—
130#235—31no—
131#235—61no—
132#235—61no—
133#235—61no—
134#235—61no—
135#235C2×D311yescharacteristic
136C2×D12—21no—
137C2×D12—31no—
138C2×D12C2×D311yescharacteristic, complemented
139C2×C4×D3—21no—
140C2×C4×D3—61no—
141C2×C4×D3—61no—
142C2×C4×D3C2×D311yescharacteristic
143C2×Dic6—31no—
144C3×Q16—32no—
145C3×SD16—32no—
146C3×M16—31no—
147#216—32no—
148#216—32no—
149#216—62no—
150#216C2×D312yescomplemented
151#215—32no—
152#215—32no—
153#215—32no—
154#215—62no—
155#215C2×D312yescomplemented
156#214—32no—
157#214—32no—
158#213—32no—
159#213—32no—
160#213—32no—
161#208—31no—
162#208—31no—
163#208—32no—
164#208—61no—
165#208C2×D311yescharacteristic, complemented
166C2×C3⋊C8—32no—
167D24—32no—
168#204—32no—
169ΓL(1,𝔽25)—32no—
170D3×C8—32no—
171C22×C32C2×D411yescharacteristic
172C2×C3⋊D3—21no—
173C2×C3×D3—21no—
174C2×C3×D3—22no—
175C2×C3×D3—41no—
176C2×C3×D3—42no—
177C2×C3×D3C2×D411yescharacteristic
178C2×C3×D3C2×D411yescharacteristic
179D32—42no—
180C32×C4—22no—
181C32×C4C2×D411yescharacteristic
182C32×C4C2411yescharacteristic
183C3⋊Dic3—21no—
184C3×Dic3—21no—
185C3×Dic3C2×D411yescharacteristic
186C3×Dic3C2×D411yescharacteristic
187C2×#41—31no—
188C2×#41—91no—
189#137—31no—
190#137—92no—
191#137—91no—
192AGL(1,ℤ/8)—91no—
193AGL(1,ℤ/8)—92no—
194AGL(1,ℤ/8)—91no—
195#135—92no—
196#135—92no—
197C2×M16—91no—
198C23×C3—31no—
199C23×C3—61no—
200C22×D3—21no—
201C22×D3—61no—
202C22×D3—61no—
203C22×D3#6711yescharacteristic
204C3×Q8—21no—
205C3×Q8—21no—
206C3×Q8—22no—
207C3×Q8—31no—
208C3×Q8—41no—
209C3×Q8C22×D312yes—
210C3×Q8C22×D312yes—
211C3×D4—32no—
212C3×D4—31no—
213C3×D4—62no—
214C3×D4—62no—
215C3×D4—61no—
216C3×D4—61no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—41no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—61no—
229C2×C3×C4C22×D311yescharacteristic
230C2×C3×C4C22×D311yescharacteristic
231#67—21no—
232#67—61no—
233#67—61no—
234#67—61no—
235#67—61no—
236#67—61no—
237#67—61no—
238#67—121no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3#6711yescharacteristic
242D12—22no—
243D12—22no—
244D12—21no—
245D12—31no—
246D12—31no—
247D12—61no—
248D12—61no—
249D12—61no—
250D12C22×D312yes—
251D12C22×D311yescharacteristic
252C4×D3—21no—
253C4×D3—22no—
254C4×D3—22no—
255C4×D3—61no—
256C4×D3—61no—
257C4×D3—61no—
258C4×D3—61no—
259C4×D3—61no—
260C4×D3—61no—
261C4×D3—121no—
262C4×D3#6711yescharacteristic
263C4×D3#6711yescharacteristic
264C4×D3C22×D311yescharacteristic
265C4×D3C22×D311yescharacteristic
266Dic6—31no—
267Dic6—31no—
268Dic6—61no—
269Dic6—61no—
270Dic6—61no—
271Dic6C22×D311yescharacteristic
272C3×C8—32no—
273C3⋊C8—32no—
274C3⋊C8—32no—
275C3⋊C8—62no—
276C3⋊C8C22×D312yes—
277C2×C32—21no—
278C2×C32C22×D411yescharacteristic
279C3⋊D3—41no—
280C3×D3—21no—
281C3×D3—21no—
282C3×D3—41no—
283C3×D3—42no—
284#41—32no—
285#41—32no—
286#41—31no—
287#41—61no—
288#41—61no—
289#41—91no—
290#41—91no—
291#41—91no—
292#41—181no—
293#41—181no—
294C2×Q8—91no—
295C2×Q8D3211yescharacteristic, complemented
296C2×D4—31no—
297C2×D4—61no—
298C2×D4—91no—
299C2×D4—181no—
300C22×C4—31no—
301C22×C4—61no—
302C22×C4—181no—
303Q16—32no—
304Q16—92no—
305SD16—32no—
306SD16—92no—
307SD16—92no—
308SD16—92no—
309D8—92no—
310D8—92no—
311M16—31no—
312M16—92no—
313M16—91no—
314C2×C8—92no—
315C22×C3—21no—
316C22×C3—31no—
317C22×C3—31no—
318C22×C3—62no—
319C22×C3—61no—
320C22×C3—121no—
321C22×C3—122no—
322C22×C3D3×D411yescharacteristic
323C22×C3C2×#6711yescharacteristic
324C2×D3—21no—
325C2×D3—22no—
326C2×D3—41no—
327C2×D3—42no—
328C2×D3—61no—
329C2×D3—61no—
330C2×D3—121no—
331C2×D3—122no—
332C2×D3—122no—
333C2×D3C2×#6711yescharacteristic
334C2×D3C2×#6711yescharacteristic
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—31no—
340C3×C4—31no—
341C3×C4—42no—
342C3×C4—61no—
343C3×C4D3×D411yescharacteristic
344C3×C4C2×#6711yescharacteristic
345C3×C4C23×D311yescharacteristic
346C3×C4C23×D311yescharacteristic
347Dic3—21no—
348Dic3—61no—
349Dic3—61no—
350Dic3—121no—
351Dic3C2×#6711yescharacteristic
352Dic3C2×#6711yescharacteristic
353C32#57511yescharacteristic, complemented, Sylow
354C23—31no—
355C23—61no—
356C23—181no—
357Q8—21no—
358Q8—31no—
359Q8—91no—
360Q8—181no—
361Q8C2×D3212yes—
362D4—31no—
363D4—32no—
364D4—62no—
365D4—62no—
366D4—61no—
367D4—61no—
368D4—91no—
369D4—181no—
370D4—181no—
371D4—181no—
372D4—181no—
373C2×C4—21no—
374C2×C4—31no—
375C2×C4—31no—
376C2×C4—31no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—61no—
382C2×C4—181no—
383C2×C4—181no—
384C2×C4—181no—
385C2×C4—181no—
386C2×C4C2×D3211yescharacteristic
387C8—32no—
388C8—92no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—41no—
393C2×C3—61no—
394C2×C3—61no—
395C2×C3—122no—
396C2×C3—121no—
397C2×C3C2×D3×D411yescharacteristic
398C2×C3C22×#6711yescharacteristic
399D3—21no—
400D3—21no—
401D3—41no—
402D3—42no—
403D3—121no—
404D3—121no—
405D3—241no—
406C22—31no—
407C22—31no—
408C22—62no—
409C22—61no—
410C22—121no—
411C22—122no—
412C22—181no—
413C22—362no—
414C22D3×#6711yescharacteristic
415C4—22no—
416C4—31no—
417C4—31no—
418C4—61no—
419C4—181no—
420C4D3×#6711yescharacteristic
421C4C22×D3211yescharacteristic
422C3—21no—
423C3#577711yescharacteristic, complemented
424C3#582211yescharacteristic, complemented
425C2—21no—
426C2—61no—
427C2—61no—
428C2—122no—
429C2—121no—
430C2—361no—
431C2C2×D3×#6711yescharacteristic
subgroupG/H
centerC2C2×D3×#67
commutator, cosocleC32×C4C24
FrattiniC4C22×D32
FittingC2×C32×Q8C22
socleC2×C32C22×D4
hypercenterC2×Q8D32
hypocenterC32#575
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#239C2semidirect
2. D3×#235C2semidirect
3. #64151C2semidirect
4. C3×#942C2semidirect
5. #63949C2semidirect
6. #63947C2semidirect
7. #63943C2semidirect
8. #63939C2semidirect
9. #63830C2semidirect
10. #63823C2semidirect
11. D3×#208C2semidirect
14. C2×C32×Q8C22semidirect
19. C3×#239C22semidirect
22. C2×C3×D12C22semidirect
27. D3×D12C22semidirect
29. #3799C22semidirect
31. #3795C22semidirect
32. D3×Dic6C22semidirect
33. #3756C22semidirect
34. #3755C22semidirect
35. #3748C22semidirect
36. C3×#216C22semidirect
37. C3×#215C22semidirect
38. C3×#208C22semidirect
39. #3716C22semidirect
40. #3715C22semidirect
41. #3714C22semidirect
42. #3713C22semidirect
43. #3711C22semidirect
44. D3×C3⋊C8C22semidirect
47. C2×#239D3semidirect
58. #942D3semidirect
110. C2×C3×Q8C2×D3semidirect
111. C2×C3×Q8C2×D3semidirect
121. #239C2×D3semidirect
138. C2×D12C2×D3semidirect
150. #216C2×D3semidirect
155. #215C2×D3semidirect
165. #208C2×D3semidirect
295. C2×Q8D32semidirect
353. C32#575semidirect
423. C3#5777semidirect
424. C3#5822semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2392
D3×#2352
#641512
C3×#9422
#639492
#639472
#639432
#639392
#638302
#638232
D3×#2082
#58223
#57773