Finite Group Databasebeta

← #79606

Subgroups — #79606

Order 384.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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×#912C211yesmaximal, characteristic, complemented
2C2×D3×D8C211yesmaximal, characteristic, complemented
3#5554C212yesmaximal, complemented
4C3×C4×D8C211yesmaximal, characteristic, complemented
5#5016C212yesmaximal, complemented
6#4887C211yesmaximal, characteristic, complemented
7#4883C211yesmaximal, characteristic, complemented
8#4781C212yesmaximal, complemented
9#4776C212yesmaximal, complemented
10#4696C211yesmaximal, characteristic, complemented
11C8×D12C211yesmaximal, characteristic, complemented
12#3280—31nomaximal, Sylow
13C2×#237C2212yescomplemented
14C2×D3×D4C2212yes—
15C2×C3×D8C2211yescharacteristic
16C3×C4×D4C2212yescomplemented
17C2×#214C2212yescomplemented
18C2×#213C2212yes—
19#930—22no—
20C2×(C2×D3)⋊C4—22no—
21#912—24no—
22D3×D8—24no—
23C2×Dic12C2211yescharacteristic, complemented
24C2×D24C2211yescharacteristic
25C2×#204C2212yescomplemented
26C2×D3×C8C2211yescharacteristic
27C2×D3×C8C2211yescharacteristic, complemented
28#897C2212yes—
29#895C2212yescomplemented
30#886—22no—
31#882—22no—
32#875C2212yescomplemented
33C4×D12C2211yescharacteristic
34C3×#107C2211yescharacteristic, complemented
35C3×#102C2212yescomplemented
36C3×C4×C8C2211yescharacteristic, complemented
37(C2×D3)⋊C8C2211yescharacteristic
38(C2×D3)⋊C8C2211yescharacteristic, complemented
39#810C2212yescomplemented
40#809C2212yescomplemented
41C4⋊(C3⋊C8)C2211yescharacteristic, complemented
42C2×#135—31no—
43C22×D8—31no—
44#546—32no—
45#495—31no—
46#467—31no—
47#466—31no—
48#463—32no—
49#450—32no—
50#449—32no—
51C4×D8D311yescharacteristic, complemented
52C8×D4—31no—
53C23×D3—22no—
54C2×C3×D4C2312yes—
55C22×C3×C4—22no—
56C2×#67—22no—
57C2×#67—22no—
58C22×Dic3—22no—
59#237—24no—
60#237—42no—
61D3×D4—24no—
62D3×D4—42no—
63C2×D12—22no—
64C2×D12D411yescharacteristic
65C2×D12C2312yes—
66C2×C4×D3C2311yescharacteristic
67C2×C4×D3C2311yescharacteristic
68C2×Dic6C2312yes—
69C3×D8D414yescomplemented
70C2×C3×C8C2311yescharacteristic
71C2×C3×C8C2311yescharacteristic
72C3×C4⋊C4C2312yes—
73C3×C22⋊C4—22no—
74C3×C42C2311yescharacteristic
75#214—24no—
76#213—24no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4D411yescharacteristic
82(C2×D3)⋊C4D411yescharacteristic
83C4⋊Dic3—22no—
84C4⋊Dic3D411yescharacteristic
85C2×C3⋊C8C2311yescharacteristic
86C2×C3⋊C8C2311yescharacteristic
87Dic12—22no—
88D24—22no—
89#204—42no—
90D3×C8—22no—
91D3×C8—22no—
92C2×#41—32no—
93C22×D4—32no—
94#135—64no—
95C2×Q16—31no—
96C2×SD16—32no—
97C2×SD16—32no—
98C2×D8—32no—
99C2×D8—31no—
100C2×D8—64no—
101C2×D8C2×D311yescharacteristic
102C22×C8—31no—
103C22×C8—31no—
104#124—32no—
105#123—62no—
106C22⋊Q8—32no—
107#121—32no—
108#121—62no—
109#120—62no—
110C4×D4—31no—
111C4×D4C2×D312yescomplemented
112C2×C22⋊C4—62no—
113#107C2×D311yescharacteristic, complemented
114C4⋊C8—31no—
115#103—32no—
116#102—32no—
117#102C2×D312yescomplemented
118C22⋊C8—31no—
119C22⋊C8—31no—
120C4×C8C2×D311yescharacteristic, complemented
121C23×C3—22no—
122C22×D3—22no—
123C22×D3—44no—
124C22×D3—44no—
125C22×D3—82no—
126C22×D3C2×D411yescharacteristic
127C22×D3C2×D411yescharacteristic
128C3×D4—22no—
129C3×D4C2×D414yes—
130C2×C3×C4—22no—
131C2×C3×C4—82no—
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2411yescharacteristic
135#67—44no—
136#67—44no—
137C2×Dic3—22no—
138C2×Dic3—44no—
139C2×Dic3C2×D411yescharacteristic
140C2×Dic3C2×D411yescharacteristic
141D12—22no—
142D12—22no—
143D12—22no—
144D12—82no—
145C4×D3—21no—
146C4×D3—21no—
147C4×D3—21no—
148C4×D3—21no—
149Dic6—22no—
150Dic6—22no—
151C3×C8—21no—
152C3×C8C2×D412yes—
153C3⋊C8—21no—
154C3⋊C8—21no—
155C24—62no—
156#41—64no—
157#41—122no—
158C2×Q8—32no—
159C2×D4—31no—
160C2×D4—32no—
161C2×D4—62no—
162C2×D4—62no—
163C2×D4—64no—
164C2×D4—62no—
165C2×D4—122no—
166C2×D4C22×D312yes—
167C22×C4—22no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—62no—
171Q16—62no—
172SD16—64no—
173SD16—122no—
174D8—64no—
175D8—62no—
176D8D1214yescomplemented
177C2×C8—31no—
178C2×C8—31no—
179C2×C8—62no—
180C2×C8—62no—
181C2×C8C22×D311yescharacteristic
182C2×C8C22×D311yescharacteristic
183C4⋊C4—31no—
184C4⋊C4—62no—
185C4⋊C4C22×D312yes—
186C22⋊C4—22no—
187C22⋊C4—31no—
188C22⋊C4—31no—
189C22⋊C4—62no—
190C22⋊C4—62no—
191C22⋊C4—62no—
192C22⋊C4—62no—
193C42C22×D311yescharacteristic
194C22×C3—24no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3C22×D411yescharacteristic
198C2×D3—21no—
199C2×D3—21no—
200C2×D3—21no—
201C2×D3—21no—
202C2×D3—21no—
203C2×D3—42no—
204C2×D3—42no—
205C2×D3—42no—
206C2×D3—88no—
207C2×D3#13512yescomplemented
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—42no—
211C3×C4C22×D411yescharacteristic
212C3×C4H(5,𝔽2)11yescharacteristic
213Dic3—21no—
214Dic3—21no—
215Dic3—42no—
216C23—22no—
217C23—31no—
218C23—31no—
219C23—62no—
220C23—124no—
221C23—124no—
222C23—242no—
223Q8—62no—
224Q8—62no—
225D4—22no—
226D4—62no—
227D4—62no—
228D4—62no—
229D4—124no—
230D4—124no—
231D4—242no—
232D4C2×D1214yes—
233C2×C4—22no—
234C2×C4—31no—
235C2×C4—31no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—62no—
241C2×C4—82no—
242C2×C4—124no—
243C2×C4D3×D411yescharacteristic
244C2×C4D3×D411yescharacteristic
245C2×C4C23×D311yescharacteristic
246C8—21no—
247C8—61no—
248C8—61no—
249C8C2×D1212yes—
250C2×C3—44no—
251C2×C3D4211yescharacteristic
252C2×C3C2×#13511yescharacteristic
253C2×C3#57611yescharacteristic
254D3—22no—
255D3—41no—
256D3—82no—
257C22—24no—
258C22—32no—
259C22—42no—
260C22—42no—
261C22—61no—
262C22—61no—
263C22—61no—
264C22—61no—
265C22—61no—
266C22—122no—
267C22—122no—
268C22—122no—
269C22—248no—
270C22C2×D3×D411yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—61no—
275C4—61no—
276C4—122no—
277C4C22×D1211yescharacteristic
278C4#100911yescharacteristic
279C3#328011yescharacteristic, complemented, Sylow
280C2—44no—
281C2—62no—
282C2—121no—
283C2—242no—
284C2D4×D1211yescharacteristic
285C2#575711yescharacteristic
286C2D3×#13511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
hypercenterC4×D8D3
hypocenterC3#3280
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#912C2semidirect
2. C2×D3×D8C2semidirect
3. #5554C2semidirect
4. C3×C4×D8C2semidirect
5. #5016C2semidirect
6. #4887C2semidirect
7. #4883C2semidirect
8. #4781C2semidirect
9. #4776C2semidirect
10. #4696C2semidirect
11. C8×D12C2semidirect
13. C2×#237C22semidirect
16. C3×C4×D4C22semidirect
17. C2×#214C22semidirect
23. C2×Dic12C22semidirect
25. C2×#204C22semidirect
27. C2×D3×C8C22semidirect
29. #895C22semidirect
32. #875C22semidirect
34. C3×#107C22semidirect
35. C3×#102C22semidirect
36. C3×C4×C8C22semidirect
38. (C2×D3)⋊C8C22semidirect
39. #810C22semidirect
40. #809C22semidirect
41. C4⋊(C3⋊C8)C22semidirect
51. C4×D8D3semidirect
69. C3×D8D4semidirect
111. C4×D4C2×D3semidirect
113. #107C2×D3semidirect
117. #102C2×D3semidirect
120. C4×C8C2×D3semidirect
176. D8D12semidirect
207. C2×D3#135semidirect
279. C3#3280semidirect

Maximal subgroups

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

subgroupindex
C2×#9122
C2×D3×D82
#55542
C3×C4×D82
#50162
#48872
#48832
#47812
#47762
#46962
C8×D122
#32803