Finite Group Databasebeta

← #80729

Subgroups — #80729

Order 384.

620 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×D8C211yesmaximal, characteristic, complemented
2C22×D24C211yesmaximal, characteristic, complemented
3#5609C212yesmaximal, complemented
4C3×#466C211yesmaximal, characteristic, complemented
5#5151C211yesmaximal, characteristic, complemented
6C8×#67C211yesmaximal, characteristic, complemented
7#5036C212yesmaximal, complemented
8#4883C211yesmaximal, characteristic, complemented
9#4872C211yesmaximal, characteristic, complemented
10#4786C212yesmaximal, complemented
11#4773C212yesmaximal, complemented
12D4×D8—31nomaximal, Sylow
13C2×D3×D4—22no—
14C2×D3×D4C2212yescomplemented
15C22×D12C2212yescomplemented
16C2×C3×D8C2211yescharacteristic, complemented
17C22×C3×C8C2211yescharacteristic, complemented
18C3×#121C2212yescomplemented
19#940C2212yescomplemented
20#937—22no—
21C2×#213C2212yescomplemented
22C2×#213C2212yescomplemented
23C4×#67C2211yescharacteristic
24D3×D8—24no—
25C2×D24—24no—
26C2×D24C2211yescharacteristic
27C2×D24C2211yescharacteristic, complemented
28C2×D24C2212yescomplemented
29C2×D3×C8C2211yescharacteristic, complemented
30#895C2212yescomplemented
31#893C2212yescomplemented
32#884—22no—
33#882—22no—
34C3×#107C2211yescharacteristic, complemented
35C3×#102C2212yescomplemented
36#830C2211yescharacteristic, complemented
37(C2×D3)⋊C8C2211yescharacteristic, complemented
38Dic3⋊C8C2211yescharacteristic, complemented
39C8×Dic3C2211yescharacteristic, complemented
40#809C2212yescomplemented
41#809C2212yescomplemented
42C22×D8—31no—
43C22×D8—31no—
44D42—32no—
45#493—31no—
46#466—31no—
47#466D311yescharacteristic, complemented
48#459—32no—
49#447—32no—
50#447—32no—
51C4×D8—31no—
52C8×D4—31no—
53C23×D3—22no—
54C23×D3—22no—
55C2×C3×D4—22no—
56C2×C3×D4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C2×#67—22no—
59C2×#67—22no—
60C2×#67—22no—
61C2×#67D411yescharacteristic
62D3×D4—24no—
63D3×D4—42no—
64D3×D4—48no—
65C2×D12—24no—
66C2×D12—22no—
67C2×D12—22no—
68C2×D12—42no—
69C2×D12C2312yes—
70C2×D12C2312yes—
71C2×C4×D3—22no—
72C2×C4×D3C2311yescharacteristic
73C3×D8—22no—
74C2×C3×C8—22no—
75C2×C3×C8C2311yescharacteristic
76C2×C3×C8C2311yescharacteristic
77C3×C4⋊C4C2312yes—
78C3×C22⋊C4—22no—
79#217D411yescharacteristic
80#213—24no—
81#213—42no—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4D411yescharacteristic
85Dic3⋊C4D411yescharacteristic
86C4×Dic3C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88C2×C3⋊C8C2311yescharacteristic
89D24—24no—
90D24—22no—
91D24D414yescomplemented
92D3×C8—22no—
93C22×D4—32no—
94C22×D4—32no—
95C22×D4—62no—
96C2×D8—32no—
97C2×D8—32no—
98C2×D8—31no—
99C2×D8—32no—
100C2×D8—31no—
101C2×D8—64no—
102C2×D8—64no—
103C2×D8C2×D311yescharacteristic, complemented
104C22×C8—31no—
105C22×C8C2×D311yescharacteristic, complemented
106#127—32no—
107#121—32no—
108#121—62no—
109#121C2×D312yescomplemented
110#120—62no—
111#120—62no—
112C4×D4—31no—
113C4×D4—32no—
114#107C2×D311yescharacteristic, complemented
115C4⋊C8—31no—
116#102—32no—
117#102—32no—
118#102C2×D312yescomplemented
119C22⋊C8—31no—
120C22⋊C8—31no—
121C4×C8—31no—
122C23×C3—22no—
123C23×C3C2×D411yescharacteristic
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—44no—
127C22×D3—44no—
128C22×D3—44no—
129C22×D3—44no—
130C22×D3—82no—
131C22×D3—82no—
132C22×D3C2×D411yescharacteristic
133C3×D4—22no—
134C3×D4—22no—
135C3×D4—44no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2411yescharacteristic
141#67—44no—
142#67—44no—
143#67—82no—
144#67D814yescomplemented
145C2×Dic3C2×D411yescharacteristic
146C2×Dic3C2×D411yescharacteristic
147C2×Dic3C2×D411yescharacteristic
148D12—22no—
149D12—22no—
150D12—22no—
151D12—44no—
152D12—44no—
153D12C2×D414yescomplemented
154C4×D3—21no—
155C4×D3—21no—
156C4×D3—44no—
157C3×C8—21no—
158C3×C8C2×D412yescomplemented
159C3⋊C8—21no—
160C3⋊C8—21no—
161C24—62no—
162C24—62no—
163C2×D4—22no—
164C2×D4—31no—
165C2×D4—32no—
166C2×D4—32no—
167C2×D4—64no—
168C2×D4—62no—
169C2×D4—62no—
170C2×D4—62no—
171C2×D4—62no—
172C2×D4—64no—
173C2×D4—62no—
174C2×D4—122no—
175C2×D4—122no—
176C2×D4—128no—
177C2×D4C22×D312yes—
178C22×C4—31no—
179C22×C4—62no—
180C22×C4C22×D311yescharacteristic
181D8—22no—
182D8—34no—
183D8—64no—
184D8—64no—
185D8—62no—
186D8—122no—
187C2×C8—22no—
188C2×C8—31no—
189C2×C8—31no—
190C2×C8—62no—
191C2×C8C22×D311yescharacteristic
192C2×C8C22×D311yescharacteristic
193C4⋊C4—31no—
194C4⋊C4C22×D312yes—
195C22⋊C4—22no—
196C22⋊C4—31no—
197C22⋊C4—31no—
198C22⋊C4—62no—
199C22⋊C4—62no—
200C42—31no—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3—42no—
206C22×C3C2×D812yescomplemented
207C22×C3C22×D411yescharacteristic
208C2×D3—21no—
209C2×D3—21no—
210C2×D3—24no—
211C2×D3—42no—
212C2×D3—42no—
213C2×D3—42no—
214C2×D3—42no—
215C2×D3—42no—
216C2×D3—88no—
217C2×D3—88no—
218C2×D3C2×D812yescomplemented
219C3×C4—21no—
220C3×C4—42no—
221C3×C4C22×D411yescharacteristic
222C3×C4H(5,𝔽2)11yescharacteristic
223Dic3—21no—
224Dic3—21no—
225Dic3C2×D812yescomplemented
226C23—22no—
227C23—31no—
228C23—62no—
229C23—62no—
230C23—124no—
231C23—124no—
232C23—124no—
233C23—124no—
234C23—242no—
235C23—242no—
236C23D3×D411yescharacteristic
237D4—22no—
238D4—22no—
239D4—34no—
240D4—34no—
241D4—44no—
242D4—62no—
243D4—62no—
244D4—62no—
245D4—124no—
246D4—124no—
247D4—124no—
248D4—124no—
249D4—242no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—124no—
259C2×C4D3×D411yescharacteristic
260C2×C4C23×D311yescharacteristic
261C8—21no—
262C8—61no—
263C8—61no—
264C8D3×D412yescomplemented
265C2×C3—22no—
266C2×C3—82no—
267C2×C3D4211yescharacteristic
268C2×C3C22×D811yescharacteristic
269C2×C3#57611yescharacteristic
270D3—22no—
271D3—44no—
272D3—82no—
273C22—21no—
274C22—21no—
275C22—32no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—61no—
280C22—61no—
281C22—64no—
282C22—122no—
283C22—122no—
284C22—122no—
285C22—122no—
286C22—122no—
287C22—248no—
288C22—248no—
289C22D3×D812yescomplemented
290C22C2×D3×D411yescharacteristic
291C4—21no—
292C4—32no—
293C4—42no—
294C4—61no—
295C4—61no—
296C4C2×D3×D411yescharacteristic
297C4#100911yescharacteristic
298C3D4×D811yescharacteristic, complemented, Sylow
299C2—22no—
300C2—62no—
301C2—82no—
302C2—124no—
303C2—242no—
304C2#560911yescharacteristic
305C2C2×D3×D811yescharacteristic
306C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#466C2
socleC22×C3C22×D4
hypercenter#466D3
hypocenterC3D4×D8
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×D3×D8C2semidirect
2. C22×D24C2semidirect
3. #5609C2semidirect
4. C3×#466C2semidirect
5. #5151C2semidirect
6. C8×#67C2semidirect
7. #5036C2semidirect
8. #4883C2semidirect
9. #4872C2semidirect
10. #4786C2semidirect
11. #4773C2semidirect
14. C2×D3×D4C22semidirect
15. C22×D12C22semidirect
16. C2×C3×D8C22semidirect
17. C22×C3×C8C22semidirect
18. C3×#121C22semidirect
19. #940C22semidirect
21. C2×#213C22semidirect
22. C2×#213C22semidirect
27. C2×D24C22semidirect
28. C2×D24C22semidirect
29. C2×D3×C8C22semidirect
30. #895C22semidirect
31. #893C22semidirect
34. C3×#107C22semidirect
35. C3×#102C22semidirect
36. #830C22semidirect
37. (C2×D3)⋊C8C22semidirect
38. Dic3⋊C8C22semidirect
39. C8×Dic3C22semidirect
40. #809C22semidirect
41. #809C22semidirect
47. #466D3semidirect
91. D24D4semidirect
103. C2×D8C2×D3semidirect
105. C22×C8C2×D3semidirect
109. #121C2×D3semidirect
114. #107C2×D3semidirect
118. #102C2×D3semidirect
144. #67D8semidirect
153. D12C2×D4semidirect
158. C3×C8C2×D4semidirect
206. C22×C3C2×D8semidirect
218. C2×D3C2×D8semidirect
225. Dic3C2×D8semidirect
264. C8D3×D4semidirect
289. C22D3×D8semidirect
298. C3D4×D8semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D82
C22×D242
#56092
C3×#4662
#51512
C8×#672
#50362
#48832
#48722
#47862
#47732
D4×D83