Finite Group Databasebeta

← #86498

Subgroups — #86498

Order 384.

1050 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#541C211yesmaximal, characteristic, complemented
2#5726C212yesmaximal, complemented
3#5711C214yesmaximal, complemented
4#5709C214yesmaximal, complemented
5#5697C214yesmaximal, complemented
6#5695C212yesmaximal, complemented
7#5692C214yesmaximal, complemented
8#5629C212yesmaximal, complemented
9#5627C212yesmaximal, complemented
10#5581C212yesmaximal, complemented
11#5577C212yesmaximal, complemented
12#5535C211yesmaximal, characteristic, complemented
13#5524C211yesmaximal, characteristic, complemented
14#3539—31nomaximal, Sylow
15C2×#239C2212yes—
16C2×#239C2212yescomplemented
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#930C2212yes—
29#898C2214yescomplemented
30#897C2212yes—
31#896C2214yescomplemented
32#895C2214yes—
33#895C2214yes—
34#895C2212yes—
35#895C2212yes—
36#895C2214yes—
37#895C2212yescomplemented
38#895C2214yescomplemented
39#895C2214yescomplemented
40#894C2214yes—
41#894C2214yes—
42#894C2214yes—
43#894C2214yes—
44#894C2214yescomplemented
45#894C2214yescomplemented
46#893C2214yes—
47#893C2214yescomplemented
48#892C2212yes—
49D3×C4⋊C4C2214yescomplemented
50D3×C4⋊C4C2212yescomplemented
51#889C2214yescomplemented
52#886C2212yes—
53#884C2214yes—
54#884C2214yes—
55#884C2214yes—
56#882C2212yes—
57#882C2214yes—
58#67⋊C4C2214yes—
59#876C2214yes—
60#874C2214yes—
61#874C2211yescharacteristic, complemented
62C4×D12C2212yes—
63C4×D12C2212yes—
64C4×D12C2214yescomplemented
65C4×D12C2212yescomplemented
66#872C2214yescomplemented
67#870C2211yescharacteristic, complemented
68#560—34no—
69#553—34no—
70#553—32no—
71#552—32no—
72#550—32no—
73#547—32no—
74#541D311yescharacteristic, complemented
75#540—34no—
76#540—31no—
77#539—34no—
78#537—32no—
79#525—32no—
80#525—31no—
81C23×D3C2311yescharacteristic
82C2×C3×Q8C2312yes—
83C22×C3×C4C2311yescharacteristic
84C2×#67C2314yes—
85C2×#67C2312yes—
86C22×Dic3C2311yescharacteristic
87#239—44no—
88#239—44no—
89#235—42no—
90C2×D12C2312yes—
91C2×D12C2312yes—
92C2×D12C2312yes—
93C2×D12C2314yes—
94C2×D12C2314yes—
95C2×D12C2314yes—
96C2×D12C2314yes—
97C2×D12C2314yes—
98C2×D12C2311yescharacteristic
99C2×C4×D3—42no—
100C2×C4×D3C2312yes—
101C2×C4×D3C2312yes—
102C2×C4×D3C2314yes—
103C2×C4×D3C2314yes—
104C2×C4×D3C2314yes—
105C2×C4×D3C2314yes—
106C2×C4×D3C2312yes—
107C2×Dic6C2311yescharacteristic
108C3×C4⋊C4C2314yes—
109C3×C4⋊C4C2312yes—
110C3×C4⋊C4C2314yes—
111C3×C4⋊C4C2314yes—
112C3×C4⋊C4C2314yes—
113C3×C4⋊C4C2312yes—
114C3×C22⋊C4C2314yes—
115C3×C22⋊C4C2312yes—
116C3×C42C2311yescharacteristic
117C3×C42C2314yes—
118C3×C42C2311yescharacteristic
119(C2×D3)⋊C4C2312yes—
120(C2×D3)⋊C4C2314yes—
121(C2×D3)⋊C4C2312yes—
122(C2×D3)⋊C4C2312yes—
123(C2×D3)⋊C4C2312yes—
124(C2×D3)⋊C4C2314yes—
125(C2×D3)⋊C4C2314yes—
126(C2×D3)⋊C4C2314yes—
127(C2×D3)⋊C4C2314yes—
128(C2×D3)⋊C4C2314yes—
129(C2×D3)⋊C4C2314yes—
130C4⋊Dic3C2312yes—
131C4⋊Dic3C2312yes—
132C4⋊Dic3C2312yes—
133Dic3⋊C4C2314yes—
134Dic3⋊C4C2314yes—
135Dic3⋊C4C2314yes—
136Dic3⋊C4C2314yes—
137C4×Dic3C2314yes—
138C2×#41—32no—
139C2×#41—32no—
140C2×#41—31no—
141C23×C4—31no—
142C4⋊Q8C2×D311yescharacteristic, complemented
143#127—31no—
144#127—34no—
145#126—34no—
146#126—34no—
147#126C2×D314yes—
148#125—34no—
149#125—31no—
150#125C2×D314yescomplemented
151#125C2×D311yescharacteristic, complemented
152#123—34no—
153#123—34no—
154#123—34no—
155#123—34no—
156#123—34no—
157#123—34no—
158#123—32no—
159C22⋊Q8—32no—
160C22⋊Q8—32no—
161C22⋊Q8—32no—
162C22⋊Q8—34no—
163C22⋊Q8C2×D312yes—
164#121—32no—
165#121—32no—
166#121—32no—
167#121—34no—
168#121—32no—
169#121—34no—
170#121—34no—
171#121—34no—
172#121—34no—
173#121—34no—
174#121—34no—
175#121—34no—
176#120—34no—
177#120—32no—
178C4×Q8C2×D312yescomplemented
179C4×D4—34no—
180C4×D4—34no—
181C4×D4—34no—
182C4×D4—32no—
183C4×D4—32no—
184C4×D4—34no—
185C4×D4—32no—
186#117—32no—
187#117—34no—
188#117C2×D311yescharacteristic
189C2×C4⋊C4—32no—
190C2×C4⋊C4—34no—
191C23×C3C2411yescharacteristic
192C22×D3—41no—
193C22×D3—42no—
194C22×D3C2412yes—
195C22×D3C2412yes—
196C22×D3C2414yes—
197C3×Q8—24no—
198C2×C3×C4—22no—
199C2×C3×C4C2414yes—
200C2×C3×C4C2412yes—
201C2×C3×C4C2414yes—
202C2×C3×C4C2412yes—
203C2×C3×C4C2411yescharacteristic
204C2×C3×C4C2411yescharacteristic
205#67—42no—
206#67—44no—
207C2×Dic3—41no—
208C2×Dic3C2412yes—
209C2×Dic3C2414yes—
210C2×Dic3C2412yes—
211D12—24no—
212D12—42no—
213D12—44no—
214D12—44no—
215D12—42no—
216D12—44no—
217D12—41no—
218D12—44no—
219D12—44no—
220C4×D3—24no—
221C4×D3—24no—
222C4×D3—42no—
223C4×D3—44no—
224C4×D3—44no—
225C4×D3—44no—
226C4×D3—44no—
227Dic6—41no—
228C24—31no—
229#41—122no—
230#41—124no—
231#41—124no—
232C2×Q8—31no—
233C2×Q8C22×D312yes—
234C2×D4—34no—
235C2×D4—32no—
236C2×D4—32no—
237C2×D4—32no—
238C2×D4—34no—
239C2×D4—32no—
240C2×D4—34no—
241C2×D4—34no—
242C2×D4—34no—
243C2×D4—31no—
244C2×D4—34no—
245C22×C4—31no—
246C22×C4—32no—
247C22×C4—32no—
248C22×C4—34no—
249C22×C4—34no—
250C22×C4—34no—
251C22×C4—34no—
252C22×C4—32no—
253C22×C4—122no—
254C22×C4C22×D311yescharacteristic
255C4⋊C4—32no—
256C4⋊C4—32no—
257C4⋊C4—34no—
258C4⋊C4—34no—
259C4⋊C4—34no—
260C4⋊C4—34no—
261C4⋊C4—32no—
262C4⋊C4C22×D312yes—
263C4⋊C4C22×D314yes—
264C4⋊C4C22×D312yes—
265C4⋊C4C22×D314yes—
266C4⋊C4C22×D314yes—
267C4⋊C4C22×D314yes—
268C22⋊C4—32no—
269C22⋊C4—34no—
270C22⋊C4—34no—
271C22⋊C4—34no—
272C22⋊C4—32no—
273C22⋊C4—32no—
274C22⋊C4—34no—
275C22⋊C4—32no—
276C22⋊C4—34no—
277C22⋊C4—34no—
278C22⋊C4—34no—
279C22⋊C4C22×D312yes—
280C22⋊C4C22×D314yes—
281C42—34no—
282C42C22×D314yes—
283C42C22×D311yescharacteristic
284C42C22×D311yescharacteristic
285C22×C3—21no—
286C22×C3—22no—
287C22×C3C2511yescharacteristic
288C2×D3—22no—
289C2×D3—22no—
290C2×D3—24no—
291C2×D3—22no—
292C2×D3—24no—
293C2×D3—22no—
294C2×D3—24no—
295C2×D3—22no—
296C2×D3—22no—
297C2×D3—44no—
298C3×C4—21no—
299C3×C4—24no—
300C3×C4—22no—
301C3×C4—24no—
302C3×C4—22no—
303C3×C4H(5,𝔽2)12yes—
304Dic3—22no—
305Dic3—22no—
306Dic3—24no—
307C23—32no—
308C23—32no—
309C23—34no—
310C23—121no—
311C23—122no—
312C23C23×D311yescharacteristic
313Q8—24no—
314Q8—121no—
315D4—64no—
316D4—121no—
317D4—124no—
318D4—124no—
319D4—122no—
320D4—124no—
321D4—124no—
322D4—122no—
323D4—124no—
324D4—122no—
325D4—124no—
326C2×C4—22no—
327C2×C4—32no—
328C2×C4—34no—
329C2×C4—32no—
330C2×C4—64no—
331C2×C4—64no—
332C2×C4—122no—
333C2×C4—124no—
334C2×C4—124no—
335C2×C4—124no—
336C2×C4—124no—
337C2×C4—121no—
338C2×C4C23×D314yes—
339C2×C4C23×D312yes—
340C2×C4C23×D312yes—
341C2×C4C23×D314yes—
342C2×C4C23×D311yescharacteristic
343C2×C4C23×D311yescharacteristic
344C2×C3—41no—
345C2×C3C2×H(5,𝔽2)11yescharacteristic
346C2×C3#58512yes—
347D3—42no—
348D3—44no—
349D3—42no—
350C22—21no—
351C22—22no—
352C22—62no—
353C22—62no—
354C22—64no—
355C22—62no—
356C22—64no—
357C22—62no—
358C22—64no—
359C22—62no—
360C22—62no—
361C22—124no—
362C22C24×D311yescharacteristic
363C4—21no—
364C4—24no—
365C4—22no—
366C4—24no—
367C4—22no—
368C4—62no—
369C4—62no—
370C4—64no—
371C4#100912yes—
372C3#353911yescharacteristic, complemented, Sylow
373C2—41no—
374C2—122no—
375C2—122no—
376C2—124no—
377C2C2×#100911yescharacteristic
378C2#596812yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#541C2
hypercenter#541D3
hypocenterC3#3539
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. #5726C2semidirect
3. #5711C2semidirect
4. #5709C2semidirect
5. #5697C2semidirect
6. #5695C2semidirect
7. #5692C2semidirect
8. #5629C2semidirect
9. #5627C2semidirect
10. #5581C2semidirect
11. #5577C2semidirect
12. #5535C2semidirect
13. #5524C2semidirect
16. C2×#239C22semidirect
19. C3×C4⋊Q8C22semidirect
21. C3×#125C22semidirect
22. C3×#125C22semidirect
24. C3×C4×Q8C22semidirect
27. #946C22semidirect
29. #898C22semidirect
31. #896C22semidirect
37. #895C22semidirect
38. #895C22semidirect
39. #895C22semidirect
44. #894C22semidirect
45. #894C22semidirect
47. #893C22semidirect
49. D3×C4⋊C4C22semidirect
50. D3×C4⋊C4C22semidirect
51. #889C22semidirect
61. #874C22semidirect
64. C4×D12C22semidirect
65. C4×D12C22semidirect
66. #872C22semidirect
67. #870C22semidirect
74. #541D3semidirect
142. C4⋊Q8C2×D3semidirect
150. #125C2×D3semidirect
151. #125C2×D3semidirect
178. C4×Q8C2×D3semidirect
372. C3#3539semidirect

Maximal subgroups

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

subgroupindex
C3×#5412
#57262
#57112
#57092
#56972
#56952
#56922
#56292
#56272
#55812
#55772
#55352
#55242
#35393