Finite Group Databasebeta

← #80773

Subgroups — #80773

Order 384.

526 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 125 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×#919C211yesmaximal, characteristic, complemented
2C22×D24C211yesmaximal, characteristic, complemented
3#5630C212yesmaximal, complemented
4C3×#467C211yesmaximal, characteristic, complemented
5#5191C211yesmaximal, characteristic, complemented
6C8×#67C211yesmaximal, characteristic, complemented
7#5046C212yesmaximal, complemented
8#4883C211yesmaximal, characteristic, complemented
9#4872C211yesmaximal, characteristic, complemented
10#4819C212yesmaximal, complemented
11#4810C212yesmaximal, complemented
12#3280—31nomaximal, Sylow
13C2×#239C2212yescomplemented
14C22×D12C2212yes—
15C2×C3×Q16C2211yescharacteristic, complemented
16C22×C3×C8C2211yescharacteristic
17C3×C22⋊Q8C2212yes—
18#947C2212yescomplemented
19C2×#215C2212yescomplemented
20C2×#215C2212yescomplemented
21C4×#67C2211yescharacteristic
22#919—24no—
23C2×D24—24no—
24C2×D24C2211yescharacteristic
25C2×D24C2212yes—
26C2×D24C2211yescharacteristic
27C2×D3×C8C2211yescharacteristic, complemented
28#895—22no—
29#895C2212yescomplemented
30#894—22no—
31#893C2212yescomplemented
32#882—22no—
33D3×C22⋊C4—22no—
34C3×#107C2211yescharacteristic, complemented
35C3×#103C2212yescomplemented
36#830C2211yescharacteristic
37(C2×D3)⋊C8C2211yescharacteristic, complemented
38Dic3⋊C8C2211yescharacteristic, complemented
39C8×Dic3C2211yescharacteristic, complemented
40#809C2212yescomplemented
41#809C2212yescomplemented
42C2×#135—31no—
43C22×D8—31no—
44#546—32no—
45#495—31no—
46#467D311yescharacteristic, complemented
47#466—31no—
48#463—32no—
49#450—32no—
50#449—32no—
51C4×D8—31no—
52C8×D4—31no—
53C23×D3—22no—
54C2×C3×Q8C2312yes—
55C22×C3×C4C2311yescharacteristic
56C2×#67D411yescharacteristic
57#239—24no—
58#239—42no—
59C2×D12—24no—
60C2×D12—22no—
61C2×D12—22no—
62C2×D12—22no—
63C2×D12—42no—
64C2×D12C2312yes—
65C2×D12C2312yes—
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3C2311yescharacteristic
69C3×Q16—22no—
70C2×C3×C8—22no—
71C2×C3×C8C2311yescharacteristic
72C2×C3×C8C2311yescharacteristic
73C3×C4⋊C4—22no—
74C3×C4⋊C4C2312yes—
75C3×C22⋊C4—22no—
76#217D411yescharacteristic
77#215—24no—
78#215—42no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—22no—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4D411yescharacteristic
84Dic3⋊C4D411yescharacteristic
85C4×Dic3C2311yescharacteristic
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88D24—24no—
89D24—22no—
90D24D414yescomplemented
91D3×C8—22no—
92C2×#41—32no—
93C22×D4—32no—
94#135—64no—
95C2×Q16C2×D311yescharacteristic, complemented
96C2×SD16—32no—
97C2×SD16—32no—
98C2×D8—31no—
99C2×D8—32no—
100C2×D8—31no—
101C2×D8—64no—
102C22×C8—31no—
103C22×C8C2×D311yescharacteristic
104#124—32no—
105#123—62no—
106C22⋊Q8C2×D312yes—
107#121—32no—
108#121—62no—
109#120—62no—
110C4×D4—31no—
111C4×D4—32no—
112C2×C22⋊C4—62no—
113#107C2×D311yescharacteristic, complemented
114C4⋊C8—31no—
115#103C2×D312yescomplemented
116#102—32no—
117#102—32no—
118C22⋊C8—31no—
119C22⋊C8—31no—
120C4×C8—31no—
121C23×C3C2×D411yescharacteristic
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—44no—
125C22×D3—44no—
126C22×D3—82no—
127C22×D3C2×D411yescharacteristic
128C3×Q8—22no—
129C3×Q8—22no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2411yescharacteristic
136#67—22no—
137C2×Dic3C2×D411yescharacteristic
138C2×Dic3C2×D411yescharacteristic
139C2×Dic3C2×D411yescharacteristic
140D12—22no—
141D12—22no—
142D12—22no—
143D12—44no—
144D12—44no—
145D12—82no—
146D12C2×D414yes—
147C4×D3—21no—
148C4×D3—21no—
149C4×D3—44no—
150C4×D3—82no—
151C3×C8—21no—
152C3×C8C2×D412yes—
153C3⋊C8—21no—
154C3⋊C8—21no—
155C24—62no—
156#41—64no—
157#41—122no—
158C2×Q8C22×D312yes—
159C2×D4—31no—
160C2×D4—32no—
161C2×D4—32no—
162C2×D4—64no—
163C2×D4—62no—
164C2×D4—62no—
165C2×D4—62no—
166C2×D4—122no—
167C22×C4—31no—
168C22×C4—62no—
169C22×C4—62no—
170C22×C4C22×D311yescharacteristic
171Q16—22no—
172SD16—64no—
173SD16—122no—
174D8—34no—
175D8—64no—
176D8—62no—
177C2×C8—22no—
178C2×C8—31no—
179C2×C8—31no—
180C2×C8—62no—
181C2×C8C22×D311yescharacteristic
182C2×C8C22×D311yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—31no—
185C4⋊C4C22×D312yes—
186C22⋊C4—22no—
187C22⋊C4—31no—
188C22⋊C4—31no—
189C22⋊C4—62no—
190C22⋊C4—62no—
191C22⋊C4—62no—
192C22⋊C4—62no—
193C42—31no—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3#13512yescomplemented
197C22×C3C22×D411yescharacteristic
198C2×D3—21no—
199C2×D3—24no—
200C2×D3—21no—
201C2×D3—21no—
202C2×D3—42no—
203C2×D3—42no—
204C2×D3—42no—
205C2×D3—42no—
206C2×D3—42no—
207C2×D3—88no—
208C3×C4—21no—
209C3×C4—42no—
210C3×C4—42no—
211C3×C4C22×D411yescharacteristic
212C3×C4H(5,𝔽2)11yescharacteristic
213Dic3—21no—
214Dic3—21no—
215Dic3—21no—
216C23—31no—
217C23—62no—
218C23—62no—
219C23—124no—
220C23—124no—
221C23—242no—
222C23D3×D411yescharacteristic
223Q8—22no—
224Q8—22no—
225D4—34no—
226D4—62no—
227D4—62no—
228D4—62no—
229D4—62no—
230D4—124no—
231D4—124no—
232D4—242no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—124no—
243C2×C4—242no—
244C2×C4D3×D411yescharacteristic
245C2×C4C23×D311yescharacteristic
246C8—21no—
247C8—61no—
248C8—61no—
249C8D3×D412yes—
250C2×C3—22no—
251C2×C3D4211yescharacteristic
252C2×C3C2×#13511yescharacteristic
253C2×C3#57611yescharacteristic
254D3—41no—
255D3—44no—
256D3—82no—
257C22—21no—
258C22—21no—
259C22—61no—
260C22—61no—
261C22—61no—
262C22—64no—
263C22—122no—
264C22—122no—
265C22—122no—
266C22—122no—
267C22—122no—
268C22—248no—
269C22#91912yescomplemented
270C22C2×D3×D411yescharacteristic
271C4—21no—
272C4—42no—
273C4—42no—
274C4—61no—
275C4—61no—
276C4—61no—
277C4C2×D3×D411yescharacteristic
278C4#100911yescharacteristic
279C3#328011yescharacteristic, complemented, Sylow
280C2—22no—
281C2—121no—
282C2—124no—
283C2—242no—
284C2#560911yescharacteristic
285C2C2×#91911yescharacteristic
286C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#467C2
socleC22×C3C22×D4
hypercenter#467D3
hypocenterC3#3280
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#919C2semidirect
2. C22×D24C2semidirect
3. #5630C2semidirect
4. C3×#467C2semidirect
5. #5191C2semidirect
6. C8×#67C2semidirect
7. #5046C2semidirect
8. #4883C2semidirect
9. #4872C2semidirect
10. #4819C2semidirect
11. #4810C2semidirect
13. C2×#239C22semidirect
15. C2×C3×Q16C22semidirect
18. #947C22semidirect
19. C2×#215C22semidirect
20. C2×#215C22semidirect
27. C2×D3×C8C22semidirect
29. #895C22semidirect
31. #893C22semidirect
34. C3×#107C22semidirect
35. C3×#103C22semidirect
37. (C2×D3)⋊C8C22semidirect
38. Dic3⋊C8C22semidirect
39. C8×Dic3C22semidirect
40. #809C22semidirect
41. #809C22semidirect
46. #467D3semidirect
90. D24D4semidirect
95. C2×Q16C2×D3semidirect
113. #107C2×D3semidirect
115. #103C2×D3semidirect
196. C22×C3#135semidirect
269. C22#919semidirect
279. C3#3280semidirect

Maximal subgroups

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

subgroupindex
C2×#9192
C22×D242
#56302
C3×#4672
#51912
C8×#672
#50462
#48832
#48722
#48192
#48102
#32803