Finite Group Databasebeta

← #86515

Subgroups — #86515

Order 384.

1014 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#541C211yesmaximal, characteristic, complemented
2#5727C212yesmaximal, complemented
3#5712C214yesmaximal, complemented
4#5705C214yesmaximal, complemented
5#5694C212yesmaximal, complemented
6#5692C214yesmaximal, complemented
7#5689C214yesmaximal, complemented
8#5628C212yesmaximal, complemented
9D3×C22⋊Q8C212yesmaximal, complemented
10#5581C212yesmaximal, complemented
11Q8×D12C212yesmaximal, complemented
12#5524C211yesmaximal, characteristic
13#5518C211yesmaximal, characteristic, complemented
14#3538—31nomaximal, Sylow
15C2×D3×Q8C2212yes—
16C2×D3×Q8C2212yescomplemented
17C2×#235C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic, complemented
20C3×#126C2214yes—
21C3×#125C2214yescomplemented
22C3×#125C2211yescharacteristic, complemented
23C3×C22⋊Q8C2212yes—
24C3×C4×Q8C2212yescomplemented
25C3×#117C2211yescharacteristic
26#946C2212yes—
27#946C2212yescomplemented
28#924C2212yes—
29#898C2214yescomplemented
30#897C2212yes—
31#897C2214yes—
32#897C2212yes—
33#897C2214yescomplemented
34#897C2212yescomplemented
35#896C2214yes—
36#896C2214yes—
37#896C2214yescomplemented
38#895C2212yes—
39#894C2214yes—
40#894C2214yes—
41#894C2214yescomplemented
42#893C2214yes—
43#892C2214yes—
44#892C2212yescomplemented
45D3×C4⋊C4C2214yes—
46D3×C4⋊C4C2212yes—
47#890C2214yescomplemented
48#889C2214yescomplemented
49#889C2214yescomplemented
50#888C2214yescomplemented
51#888C2214yescomplemented
52#886C2212yes—
53#885C2214yes—
54#883C2214yes—
55#882C2212yes—
56D3×C22⋊C4C2214yes—
57#879C2214yes—
58#878C2214yes—
59#876C2214yescomplemented
60C4×D12C2214yes—
61C4×D12C2212yes—
62C4×D12C2212yes—
63#872C2214yes—
64#870C2211yescharacteristic, complemented
65#869C2214yescomplemented
66#869C2211yescharacteristic, complemented
67C4×Dic6C2212yescomplemented
68#563—34no—
69#552—32no—
70#552—32no—
71#551—34no—
72D4×Q8—32no—
73D4×Q8—32no—
74#544—34no—
75#541—31no—
76#541D311yescharacteristic, complemented
77#539—34no—
78#536—32no—
79#525—31no—
80C2×C22⋊Q8—32no—
81C23×D3C2311yescharacteristic
82C2×C3×Q8C2312yes—
83C22×C3×C4C2311yescharacteristic
84C2×#67C2312yes—
85C22×Dic3C2311yescharacteristic
86D3×Q8—44no—
87D3×Q8—44no—
88#235—42no—
89C2×D12C2312yes—
90C2×D12C2312yes—
91C2×D12C2311yescharacteristic
92C2×C4×D3—42no—
93C2×C4×D3C2314yes—
94C2×C4×D3C2312yes—
95C2×C4×D3C2312yes—
96C2×C4×D3C2314yes—
97C2×C4×D3C2314yes—
98C2×C4×D3C2312yes—
99C2×Dic6C2312yes—
100C2×Dic6C2314yes—
101C2×Dic6C2314yes—
102C2×Dic6C2311yescharacteristic
103C3×C4⋊C4C2314yes—
104C3×C4⋊C4C2314yes—
105C3×C4⋊C4C2312yes—
106C3×C4⋊C4C2314yes—
107C3×C4⋊C4C2314yes—
108C3×C4⋊C4C2312yes—
109C3×C22⋊C4C2314yes—
110C3×C22⋊C4C2312yes—
111C3×C42C2311yescharacteristic
112C3×C42C2311yescharacteristic
113C3×C42C2314yes—
114#217C2314yes—
115(C2×D3)⋊C4C2312yes—
116(C2×D3)⋊C4C2314yes—
117(C2×D3)⋊C4C2312yes—
118(C2×D3)⋊C4C2314yes—
119(C2×D3)⋊C4C2312yes—
120(C2×D3)⋊C4C2314yes—
121(C2×D3)⋊C4C2312yes—
122(C2×D3)⋊C4C2314yes—
123(C2×D3)⋊C4C2314yes—
124C4⋊Dic3C2312yes—
125C4⋊Dic3C2314yes—
126C4⋊Dic3C2312yes—
127C4⋊Dic3C2314yes—
128C4⋊Dic3C2314yes—
129C4⋊Dic3C2312yes—
130Dic3⋊C4C2314yes—
131Dic3⋊C4C2314yes—
132Dic3⋊C4C2314yes—
133Dic3⋊C4C2314yes—
134Dic3⋊C4C2314yes—
135Dic3⋊C4C2314yes—
136C4×Dic3C2314yes—
137C4×Dic3C2314yes—
138C2×#41—31no—
139C22×Q8—32no—
140C22×Q8—32no—
141C23×C4—31no—
142C4⋊Q8—34no—
143C4⋊Q8—34no—
144C4⋊Q8—31no—
145C4⋊Q8—34no—
146C4⋊Q8C2×D311yescharacteristic, complemented
147#126—34no—
148#126—34no—
149#126—34no—
150#126C2×D314yes—
151#125—34no—
152#125—34no—
153#125—34no—
154#125—31no—
155#125C2×D314yescomplemented
156#125C2×D311yescharacteristic, complemented
157#124—34no—
158#123—34no—
159#123—34no—
160#123—34no—
161#123—32no—
162#123—34no—
163C22⋊Q8—32no—
164C22⋊Q8—32no—
165C22⋊Q8—32no—
166C22⋊Q8—34no—
167C22⋊Q8—32no—
168C22⋊Q8—32no—
169C22⋊Q8—32no—
170C22⋊Q8—34no—
171C22⋊Q8—34no—
172C22⋊Q8—34no—
173C22⋊Q8—34no—
174C22⋊Q8—34no—
175C22⋊Q8C2×D312yes—
176#121—32no—
177#120—32no—
178C4×Q8—32no—
179C4×Q8C2×D312yescomplemented
180C4×D4—34no—
181C4×D4—32no—
182C4×D4—32no—
183C4×D4—34no—
184#117—32no—
185#117—34no—
186#117—34no—
187#117C2×D311yescharacteristic
188C2×C4⋊C4—34no—
189C2×C4⋊C4—32no—
190C2×C22⋊C4—34no—
191C23×C3C2411yescharacteristic
192C22×D3—24no—
193C22×D3—41no—
194C22×D3C2412yes—
195C22×D3C2412yes—
196C3×Q8—24no—
197C2×C3×C4—22no—
198C2×C3×C4C2412yes—
199C2×C3×C4C2414yes—
200C2×C3×C4C2411yescharacteristic
201C2×C3×C4C2412yes—
202C2×C3×C4C2414yes—
203C2×C3×C4C2411yescharacteristic
204#67—42no—
205C2×Dic3—41no—
206C2×Dic3C2414yes—
207C2×Dic3C2412yes—
208C2×Dic3C2414yes—
209C2×Dic3C2412yes—
210D12—24no—
211D12—24no—
212D12—41no—
213C4×D3—28no—
214C4×D3—24no—
215C4×D3—24no—
216C4×D3—42no—
217C4×D3—44no—
218C4×D3—44no—
219Dic6—24no—
220Dic6—41no—
221Dic6—44no—
222Dic6—44no—
223C24—31no—
224#41—122no—
225C2×Q8—32no—
226C2×Q8—34no—
227C2×Q8—34no—
228C2×Q8—31no—
229C2×Q8—124no—
230C2×Q8—124no—
231C2×Q8C22×D312yes—
232C2×D4—32no—
233C2×D4—32no—
234C2×D4—32no—
235C2×D4—31no—
236C22×C4—31no—
237C22×C4—34no—
238C22×C4—32no—
239C22×C4—32no—
240C22×C4—34no—
241C22×C4—32no—
242C22×C4—34no—
243C22×C4—122no—
244C22×C4C22×D311yescharacteristic
245C4⋊C4—32no—
246C4⋊C4—34no—
247C4⋊C4—34no—
248C4⋊C4—34no—
249C4⋊C4—34no—
250C4⋊C4—34no—
251C4⋊C4—34no—
252C4⋊C4—34no—
253C4⋊C4—34no—
254C4⋊C4—32no—
255C4⋊C4—32no—
256C4⋊C4—34no—
257C4⋊C4C22×D312yes—
258C4⋊C4C22×D314yes—
259C4⋊C4C22×D314yes—
260C4⋊C4C22×D312yes—
261C4⋊C4C22×D314yes—
262C4⋊C4C22×D314yes—
263C22⋊C4—34no—
264C22⋊C4—34no—
265C22⋊C4—34no—
266C22⋊C4—32no—
267C22⋊C4—32no—
268C22⋊C4—34no—
269C22⋊C4—32no—
270C22⋊C4—32no—
271C22⋊C4—34no—
272C22⋊C4—34no—
273C22⋊C4C22×D314yes—
274C22⋊C4C22×D312yes—
275C42—34no—
276C42—34no—
277C42C22×D314yes—
278C42C22×D311yescharacteristic
279C42C22×D311yescharacteristic
280C22×C3—21no—
281C22×C3—22no—
282C22×C3C2511yescharacteristic
283C2×D3—22no—
284C2×D3—22no—
285C2×D3—22no—
286C2×D3—22no—
287C2×D3—22no—
288C2×D3—44no—
289C2×D32−1+414yes—
290C3×C4—24no—
291C3×C4—21no—
292C3×C4—24no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4H(5,𝔽2)12yes—
296Dic3—24no—
297Dic3—22no—
298Dic3—22no—
299Dic3—24no—
300C23—32no—
301C23—32no—
302C23—64no—
303C23—121no—
304C23C23×D311yescharacteristic
305Q8—24no—
306Q8—64no—
307Q8—124no—
308Q8—124no—
309Q8—121no—
310D4—64no—
311D4—64no—
312D4—121no—
313D4—122no—
314C2×C4—22no—
315C2×C4—34no—
316C2×C4—32no—
317C2×C4—34no—
318C2×C4—32no—
319C2×C4—68no—
320C2×C4—64no—
321C2×C4—64no—
322C2×C4—122no—
323C2×C4—124no—
324C2×C4—124no—
325C2×C4—121no—
326C2×C4C23×D312yes—
327C2×C4C23×D314yes—
328C2×C4C23×D312yes—
329C2×C4C23×D314yes—
330C2×C4C23×D311yescharacteristic
331C2×C4C23×D311yescharacteristic
332C2×C3—41no—
333C2×C3C2×H(5,𝔽2)11yescharacteristic
334C2×C3C2×2−1+412yes—
335D3—24no—
336D3—42no—
337C22—21no—
338C22—22no—
339C22—34no—
340C22—62no—
341C22—62no—
342C22—62no—
343C22—62no—
344C22—62no—
345C22—124no—
346C22C24×D311yescharacteristic
347C4—21no—
348C4—24no—
349C4—24no—
350C4—22no—
351C4—22no—
352C4—62no—
353C4—62no—
354C4—64no—
355C4—64no—
356C4#100912yes—
357C3#353811yescharacteristic, complemented, Sylow
358C2—41no—
359C2—64no—
360C2—122no—
361C2C2×#100911yescharacteristic
362C2D3×2−1+412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#541C2
prime radical#5524C2
hypercenter#541D3
hypocenterC3#3538
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
1. C3×#541C2semidirect
2. #5727C2semidirect
3. #5712C2semidirect
4. #5705C2semidirect
5. #5694C2semidirect
6. #5692C2semidirect
7. #5689C2semidirect
8. #5628C2semidirect
9. D3×C22⋊Q8C2semidirect
10. #5581C2semidirect
11. Q8×D12C2semidirect
13. #5518C2semidirect
16. C2×D3×Q8C22semidirect
19. C3×C4⋊Q8C22semidirect
21. C3×#125C22semidirect
22. C3×#125C22semidirect
24. C3×C4×Q8C22semidirect
27. #946C22semidirect
29. #898C22semidirect
33. #897C22semidirect
34. #897C22semidirect
37. #896C22semidirect
41. #894C22semidirect
44. #892C22semidirect
47. #890C22semidirect
48. #889C22semidirect
49. #889C22semidirect
50. #888C22semidirect
51. #888C22semidirect
59. #876C22semidirect
64. #870C22semidirect
65. #869C22semidirect
66. #869C22semidirect
67. C4×Dic6C22semidirect
76. #541D3semidirect
146. C4⋊Q8C2×D3semidirect
155. #125C2×D3semidirect
156. #125C2×D3semidirect
179. C4×Q8C2×D3semidirect
357. C3#3538semidirect

Maximal subgroups

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

subgroupindex
C3×#5412
#57272
#57122
#57052
#56942
#56922
#56892
#56282
D3×C22⋊Q82
#55812
Q8×D122
#55242
#55182
#35383