Finite Group Databasebeta

← #87115

Subgroups — #87115

Order 384.

1058 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#563C211yesmaximal, characteristic, complemented
2#5735C212yesmaximal, complemented
3#5725C212yesmaximal, complemented
4#5711C214yesmaximal, complemented
5#5706C214yesmaximal, complemented
6#5700C212yesmaximal, complemented
7#5691C212yesmaximal, complemented
8#5683C211yesmaximal, characteristic, complemented
9#5668C211yesmaximal, characteristic, complemented
10#5666C212yesmaximal, complemented
11#5656C212yesmaximal, complemented
12#5647C214yesmaximal, complemented
13#5627C214yesmaximal, complemented
14#3529—31nomaximal, Sylow
15C2×#239C2212yes—
16C2×#239C2211yescharacteristic, complemented
17C2×#237C2211yescharacteristic
18C22×C4×D3C2212yescomplemented
19C3×C4⋊Q8C2212yescomplemented
20C3×#126C2214yescomplemented
21C3×#125C2212yescomplemented
22C3×#124C2211yescharacteristic
23C3×#123C2212yes—
24C3×C22⋊Q8C2214yescomplemented
25#947C2212yes—
26#947C2212yescomplemented
27#946C2214yescomplemented
28#944C2211yescharacteristic, complemented
29#940C2211yescharacteristic
30#937C2212yes—
31#930C2214yes—
32#929C2212yescomplemented
33#898C2214yescomplemented
34#897C2212yescomplemented
35#896C2214yescomplemented
36#896C2214yescomplemented
37#895C2212yes—
38#895C2214yes—
39#895C2214yescomplemented
40#895C2214yescomplemented
41#894C2214yes—
42#894C2214yescomplemented
43#894C2214yescomplemented
44#894C2214yescomplemented
45#893C2212yes—
46#893C2214yescomplemented
47#893C2214yescomplemented
48#892C2214yescomplemented
49#889C2214yescomplemented
50#887C2212yescomplemented
51#885C2214yes—
52#884C2214yes—
53#884C2214yes—
54#884C2214yescomplemented
55#883C2214yescomplemented
56#882C2214yes—
57#882C2212yescomplemented
58#882C2214yescomplemented
59#67⋊C4C2214yes—
60#67⋊C4C2214yescomplemented
61#877C2212yescomplemented
62#876C2214yescomplemented
63#875C2212yes—
64#875C2211yescharacteristic, complemented
65#874C2212yes—
66C42×D3C2211yescharacteristic, complemented
67C42×D3C2212yescomplemented
68#563—32no—
69#563D311yescharacteristic, complemented
70#562—32no—
71#562—34no—
72H(3,𝔽4)—31no—
73#560—34no—
74#560—32no—
75#532—32no—
76#532—31no—
77#529—34no—
78#529—32no—
79#525—32no—
80#525—34no—
81C23×D3C2312yes—
82C2×C3×Q8C2311yescharacteristic
83C2×C3×Q8C2312yes—
84C2×C3×D4C2311yescharacteristic
85C22×C3×C4C2312yes—
86C2×#67C2314yes—
87C2×#67C2314yes—
88C2×#67C2312yes—
89C22×Dic3C2312yes—
90#239—44no—
91#239—42no—
92#237—42no—
93C2×D12C2312yes—
94C2×D12C2314yes—
95C2×D12C2314yes—
96C2×D12C2312yes—
97C2×D12C2312yes—
98C2×D12C2311yescharacteristic
99C2×D12C2314yes—
100C2×C4×D3—44no—
101C2×C4×D3C2312yes—
102C2×C4×D3C2311yescharacteristic
103C2×C4×D3C2314yes—
104C2×C4×D3C2314yes—
105C2×C4×D3C2312yes—
106C2×C4×D3C2314yes—
107C2×Dic6C2312yes—
108C2×Dic6C2311yescharacteristic
109C3×C4⋊C4C2312yes—
110C3×C4⋊C4C2314yes—
111C3×C4⋊C4C2314yes—
112C3×C4⋊C4C2312yes—
113C3×C4⋊C4C2314yes—
114C3×C22⋊C4C2312yes—
115C3×C22⋊C4C2314yes—
116C3×C22⋊C4C2314yes—
117C3×C42C2312yes—
118C3×C42C2311yescharacteristic
119#217C2312yes—
120(C2×D3)⋊C4C2314yes—
121(C2×D3)⋊C4C2314yes—
122(C2×D3)⋊C4C2314yes—
123(C2×D3)⋊C4C2314yes—
124(C2×D3)⋊C4C2314yes—
125(C2×D3)⋊C4C2314yes—
126(C2×D3)⋊C4C2314yes—
127(C2×D3)⋊C4C2314yes—
128(C2×D3)⋊C4C2314yes—
129C4⋊Dic3C2314yes—
130Dic3⋊C4C2314yes—
131Dic3⋊C4C2314yes—
132Dic3⋊C4C2314yes—
133Dic3⋊C4C2314yes—
134C4×Dic3C2314yes—
135C4×Dic3C2312yes—
136C4×Dic3C2312yes—
137C4×Dic3C2311yescharacteristic
138C2×#41—32no—
139C2×#41—31no—
140C2×#41—31no—
141C23×C4—32no—
142C4⋊Q8—31no—
143C4⋊Q8C2×D312yescomplemented
144#127—31no—
145#127—32no—
146#126—34no—
147#126—34no—
148#126C2×D314yescomplemented
149#125—34no—
150#125C2×D312yescomplemented
151#124—32no—
152#124—32no—
153#124—34no—
154#124—32no—
155#124—31no—
156#124C2×D311yescharacteristic
157#123—32no—
158#123—34no—
159#123—34no—
160#123—34no—
161#123—34no—
162#123—34no—
163#123C2×D312yes—
164C22⋊Q8—34no—
165C22⋊Q8—34no—
166C22⋊Q8—34no—
167C22⋊Q8—32no—
168C22⋊Q8C2×D314yescomplemented
169#121—34no—
170#121—34no—
171#121—32no—
172#121—34no—
173#121—34no—
174#121—34no—
175#121—34no—
176#121—34no—
177#120—32no—
178#120—34no—
179#120—32no—
180#120—34no—
181C4×Q8—32no—
182C4×D4—32no—
183C4×D4—34no—
184C4×D4—34no—
185C4×D4—34no—
186C4×D4—34no—
187#117—34no—
188#117—32no—
189C2×C42—32no—
190C2×C42—31no—
191C23×C3C2412yes—
192C22×D3—42no—
193C22×D3—44no—
194C22×D3C2414yes—
195C22×D3C2412yes—
196C22×D3C2411yescharacteristic
197C3×Q8—41no—
198C3×Q8—42no—
199C3×D4—41no—
200C2×C3×C4—42no—
201C2×C3×C4C2414yes—
202C2×C3×C4C2412yes—
203C2×C3×C4C2412yes—
204C2×C3×C4C2411yescharacteristic
205C2×C3×C4C2414yes—
206#67—42no—
207#67—44no—
208#67—44no—
209C2×Dic3—24no—
210C2×Dic3C2412yes—
211C2×Dic3C2414yes—
212C2×Dic3C2412yes—
213C2×Dic3C2411yescharacteristic
214D12—42no—
215D12—44no—
216D12—44no—
217D12—41no—
218D12—44no—
219D12—42no—
220D12—42no—
221C4×D3—28no—
222C4×D3—22no—
223C4×D3—24no—
224C4×D3—44no—
225C4×D3—44no—
226C4×D3—42no—
227Dic6—42no—
228Dic6—41no—
229C24—32no—
230#41—122no—
231#41—124no—
232#41—122no—
233C2×Q8—32no—
234C2×Q8—31no—
235C2×Q8C22×D312yes—
236C2×Q8C22×D311yescharacteristic
237C2×D4—34no—
238C2×D4—34no—
239C2×D4—32no—
240C2×D4—32no—
241C2×D4—34no—
242C2×D4—34no—
243C2×D4—32no—
244C2×D4—34no—
245C2×D4—31no—
246C2×D4—32no—
247C2×D4C22×D311yescharacteristic
248C22×C4—32no—
249C22×C4—31no—
250C22×C4—32no—
251C22×C4—34no—
252C22×C4—34no—
253C22×C4—32no—
254C22×C4—34no—
255C22×C4—124no—
256C22×C4C22×D312yes—
257C4⋊C4—34no—
258C4⋊C4—34no—
259C4⋊C4—34no—
260C4⋊C4—34no—
261C4⋊C4—34no—
262C4⋊C4C22×D314yes—
263C4⋊C4C22×D314yes—
264C4⋊C4C22×D314yes—
265C4⋊C4C22×D312yes—
266C4⋊C4C22×D312yes—
267C22⋊C4—32no—
268C22⋊C4—34no—
269C22⋊C4—34no—
270C22⋊C4—34no—
271C22⋊C4—34no—
272C22⋊C4—34no—
273C22⋊C4—34no—
274C22⋊C4—34no—
275C22⋊C4—34no—
276C22⋊C4—34no—
277C22⋊C4C22×D312yes—
278C22⋊C4C22×D314yes—
279C22⋊C4C22×D314yes—
280C42—34no—
281C42—32no—
282C42—32no—
283C42—31no—
284C42C22×D311yescharacteristic
285C42C22×D312yes—
286C22×C3—22no—
287C22×C3—24no—
288C22×C3C2511yescharacteristic
289C2×D3—24no—
290C2×D3—22no—
291C2×D3—21no—
292C2×D3—22no—
293C2×D3—24no—
294C2×D3—24no—
295C2×D3—24no—
296C2×D3—48no—
297C3×C4—21no—
298C3×C4—24no—
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—24no—
302Dic3—22no—
303Dic3—24no—
304Dic3—22no—
305Dic3H(5,𝔽2)12yes—
306C23—34no—
307C23—32no—
308C23—31no—
309C23—122no—
310C23—124no—
311C23C23×D312yes—
312Q8—42no—
313Q8—41no—
314Q8—121no—
315Q8—122no—
316D4—41no—
317D4—124no—
318D4—121no—
319D4—122no—
320D4—122no—
321D4—124no—
322D4—124no—
323D4—122no—
324D4—122no—
325D4—124no—
326D4—124no—
327C2×C4—31no—
328C2×C4—34no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—42no—
332C2×C4—68no—
333C2×C4—62no—
334C2×C4—64no—
335C2×C4—64no—
336C2×C4—124no—
337C2×C4—124no—
338C2×C4—122no—
339C2×C4C23×D312yes—
340C2×C4C23×D311yescharacteristic
341C2×C4C23×D314yes—
342C2×C4C23×D312yes—
343C2×C4C23×D314yes—
344C2×C3—42no—
345C2×C3C2×H(5,𝔽2)11yescharacteristic
346C2×C3#58512yes—
347D3—42no—
348D3—41no—
349D3—44no—
350C22—22no—
351C22—24no—
352C22—64no—
353C22—62no—
354C22—61no—
355C22—62no—
356C22—64no—
357C22—64no—
358C22—64no—
359C22—128no—
360C22C24×D311yescharacteristic
361C4—24no—
362C4—21no—
363C4—22no—
364C4—24no—
365C4—22no—
366C4—32no—
367C4—62no—
368C4—62no—
369C4—64no—
370C3#352911yescharacteristic, complemented, Sylow
371C2—42no—
372C2—124no—
373C2—122no—
374C2—121no—
375C2D3×H(5,𝔽2)11yescharacteristic
376C2#596812yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#563C2
hypercenter#563D3
hypocenterC3#3529
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #563

Complements

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

NH ≅ G/Ntype
1. C3×#563C2semidirect
2. #5735C2semidirect
3. #5725C2semidirect
4. #5711C2semidirect
5. #5706C2semidirect
6. #5700C2semidirect
7. #5691C2semidirect
8. #5683C2semidirect
9. #5668C2semidirect
10. #5666C2semidirect
11. #5656C2semidirect
12. #5647C2semidirect
13. #5627C2semidirect
16. C2×#239C22semidirect
18. C22×C4×D3C22semidirect
19. C3×C4⋊Q8C22semidirect
20. C3×#126C22semidirect
21. C3×#125C22semidirect
24. C3×C22⋊Q8C22semidirect
26. #947C22semidirect
27. #946C22semidirect
28. #944C22semidirect
32. #929C22semidirect
33. #898C22semidirect
34. #897C22semidirect
35. #896C22semidirect
36. #896C22semidirect
39. #895C22semidirect
40. #895C22semidirect
42. #894C22semidirect
43. #894C22semidirect
44. #894C22semidirect
46. #893C22semidirect
47. #893C22semidirect
48. #892C22semidirect
49. #889C22semidirect
50. #887C22semidirect
54. #884C22semidirect
55. #883C22semidirect
57. #882C22semidirect
58. #882C22semidirect
60. #67⋊C4C22semidirect
61. #877C22semidirect
62. #876C22semidirect
64. #875C22semidirect
66. C42×D3C22semidirect
67. C42×D3C22semidirect
69. #563D3semidirect
143. C4⋊Q8C2×D3semidirect
148. #126C2×D3semidirect
150. #125C2×D3semidirect
168. C22⋊Q8C2×D3semidirect
370. C3#3529semidirect

Maximal subgroups

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

subgroupindex
C3×#5632
#57352
#57252
#57112
#57062
#57002
#56912
#56832
#56682
#56662
#56562
#56472
#56272
#35293