Finite Group Databasebeta

← #79608

Subgroups — #79608

Order 384.

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

normal subgroups only

This group has 494 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
1#5551C211yesmaximal, characteristic, complemented
2D4×D12C211yesmaximal, characteristic, complemented
3C3×C4×D8C211yesmaximal, characteristic, complemented
4#5156C212yesmaximal, complemented
5#5015C211yesmaximal, characteristic, complemented
6#5007C211yesmaximal, characteristic, complemented
7#4882C212yesmaximal, complemented
8#4775C212yesmaximal, complemented
9D3×#102C212yesmaximal, complemented
10#4694C211yesmaximal, characteristic, complemented
11C8×D12C211yesmaximal, characteristic, complemented
12#3294—31nomaximal, Sylow
13C2×D3×D4C2212yes—
14C2×#235—21no—
15C22×D12—21no—
16C2×C3×D8C2211yescharacteristic
17C3×C4×D4C2211yescharacteristic
18C3×C4×D4C2211yescharacteristic, complemented
19#938C2212yescomplemented
20C2×#213C2212yescomplemented
21#930—21no—
22#924—21no—
23C2×D3×C8C2212yescomplemented
24#895C2212yes—
25D3×C4⋊C4C2212yescomplemented
26#883—22no—
27#882—22no—
28#874C2211yescharacteristic
29C4×D12C2211yescharacteristic
30#869C2211yescharacteristic, complemented
31C3×#107C2211yescharacteristic, complemented
32C3×#102C2211yescharacteristic
33C3×#102C2211yescharacteristic, complemented
34C3×C4×C8C2211yescharacteristic, complemented
35#832C2212yes—
36#821C2212yescomplemented
37#821C2212yescomplemented
38(C2×D3)⋊C8C2212yescomplemented
39#809C2212yescomplemented
40#807C2212yescomplemented
41C4⋊(C3⋊C8)C2211yescharacteristic, complemented
42#547—31no—
43D42—31no—
44#486—31no—
45#480—32no—
46#474—31no—
47#466—32no—
48#459—31no—
49#447—32no—
50C4×D8D311yescharacteristic, complemented
51C8×D4—31no—
52C2×#102—32no—
53C23×D3—22no—
54C2×C3×D4C2311yescharacteristic
55C2×C3×D4C2311yescharacteristic
56C22×C3×C4—21no—
57C22×C3×C4—21no—
58C2×#67—22no—
59C2×#67—22no—
60D3×D4—24no—
61D3×D4—42no—
62#235—44no—
63C2×D12—22no—
64C2×D12—21no—
65C2×D12—21no—
66C2×D12—44no—
67C2×D12D411yescharacteristic
68C2×D12C2312yes—
69C2×C4×D3—22no—
70C2×C4×D3C2312yes—
71C2×Dic6—21no—
72C3×D8—22no—
73C2×C3×C8C2311yescharacteristic
74C2×C3×C8C2311yescharacteristic
75C3×C4⋊C4C2311yescharacteristic
76C3×C4⋊C4C2311yescharacteristic
77C3×C22⋊C4—21no—
78C3×C22⋊C4—21no—
79C3×C42C2311yescharacteristic
80#217—22no—
81#213—42no—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4D412yes—
84C4⋊Dic3—21no—
85C4⋊Dic3D411yescharacteristic
86C4⋊Dic3C2312yes—
87Dic3⋊C4—22no—
88C2×C3⋊C8C2312yes—
89D3×C8—42no—
90C2×#41—61no—
91C22×D4—32no—
92C22×D4—61no—
93C2×D8—32no—
94C2×D8C2×D311yescharacteristic
95C22×C8—32no—
96C4⋊Q8—31no—
97#127—31no—
98#123—62no—
99C22⋊Q8—61no—
100#121—32no—
101#121—32no—
102#121—61no—
103#120—62no—
104C4×D4—31no—
105C4×D4C2×D311yescharacteristic
106C4×D4C2×D311yescharacteristic, complemented
107C2×C4⋊C4—32no—
108#107—32no—
109#107C2×D311yescharacteristic, complemented
110C4⋊C8—31no—
111#102—32no—
112#102—32no—
113#102—32no—
114#102—32no—
115#102C2×D311yescharacteristic
116#102C2×D311yescharacteristic, complemented
117C22⋊C8—32no—
118C4×C8C2×D311yescharacteristic, complemented
119C23×C3—21no—
120C23×C3—21no—
121C22×D3—22no—
122C22×D3—44no—
123C22×D3—44no—
124C22×D3—82no—
125C22×D3C2×D412yes—
126C3×D4—21no—
127C3×D4—21no—
128C3×D4—21no—
129C3×D4#4112yescomplemented
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2411yescharacteristic
137#67—44no—
138#67—44no—
139C2×Dic3—22no—
140C2×Dic3C2×D412yes—
141D12—22no—
142D12—22no—
143D12—44no—
144D12—42no—
145D12—81no—
146D12D814yescomplemented
147C4×D3—22no—
148C4×D3—22no—
149C4×D3—44no—
150Dic6—42no—
151C3×C8—21no—
152C3×C8—21no—
153C3⋊C8—22no—
154C24—62no—
155#41—124no—
156C2×Q8—61no—
157C2×D4—31no—
158C2×D4—32no—
159C2×D4—62no—
160C2×D4—62no—
161C2×D4—64no—
162C2×D4—62no—
163C2×D4—61no—
164C2×D4—61no—
165C2×D4—124no—
166C2×D4—122no—
167C2×D4C22×D311yescharacteristic
168C2×D4C22×D311yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—32no—
172C22×C4—62no—
173D8—22no—
174D8—122no—
175C2×C8—32no—
176C2×C8—122no—
177C2×C8C22×D311yescharacteristic
178C2×C8C22×D311yescharacteristic
179C4⋊C4—31no—
180C4⋊C4—32no—
181C4⋊C4—62no—
182C4⋊C4—61no—
183C4⋊C4C22×D311yescharacteristic
184C4⋊C4C22×D311yescharacteristic
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—32no—
188C22⋊C4—62no—
189C22⋊C4—62no—
190C42C22×D311yescharacteristic
191C22×C3—22no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3C22×D411yescharacteristic
198C2×D3—22no—
199C2×D3—22no—
200C2×D3—42no—
201C2×D3—42no—
202C2×D3—42no—
203C2×D3—88no—
204C2×D3C2×D814yes—
205C3×C4—21no—
206C3×C4—41no—
207C3×C4—41no—
208C3×C4C2×D812yes—
209C3×C4C2×#4111yescharacteristic
210C3×C4H(5,𝔽2)11yescharacteristic
211Dic3—22no—
212Dic3—42no—
213C23—21no—
214C23—21no—
215C23—32no—
216C23—62no—
217C23—124no—
218C23—124no—
219C23—242no—
220Q8—122no—
221D4—21no—
222D4—21no—
223D4—21no—
224D4—34no—
225D4—62no—
226D4—62no—
227D4—124no—
228D4—124no—
229D4—124no—
230D4—122no—
231D4—241no—
232D4#23512yescomplemented
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—32no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—124no—
242C2×C4D3×D411yescharacteristic
243C2×C4D3×D411yescharacteristic
244C2×C4C23×D311yescharacteristic
245C8—21no—
246C8—21no—
247C8—62no—
248C2×C3—42no—
249C2×C3—81no—
250C2×C3#54611yescharacteristic
251C2×C3C22×D811yescharacteristic
252C2×C3#57711yescharacteristic
253D3—24no—
254D3—82no—
255C22—22no—
256C22—34no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—62no—
263C22—62no—
264C22—122no—
265C22—122no—
266C22—122no—
267C22—248no—
268C22C2×D3×D411yescharacteristic
269C4—21no—
270C4—41no—
271C4—41no—
272C4—62no—
273C4—122no—
274C4D3×D812yes—
275C4C2×#23511yescharacteristic
276C4#100911yescharacteristic
277C3#329411yescharacteristic, complemented, Sylow
278C2—42no—
279C2—64no—
280C2—81no—
281C2—242no—
282C2#555011yescharacteristic
283C2C2×D3×D811yescharacteristic
284C2#577011yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
hypercenterC4×D8D3
hypocenterC3#3294
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. #5551C2semidirect
2. D4×D12C2semidirect
3. C3×C4×D8C2semidirect
4. #5156C2semidirect
5. #5015C2semidirect
6. #5007C2semidirect
7. #4882C2semidirect
8. #4775C2semidirect
9. D3×#102C2semidirect
10. #4694C2semidirect
11. C8×D12C2semidirect
18. C3×C4×D4C22semidirect
19. #938C22semidirect
20. C2×#213C22semidirect
23. C2×D3×C8C22semidirect
25. D3×C4⋊C4C22semidirect
30. #869C22semidirect
31. C3×#107C22semidirect
33. C3×#102C22semidirect
34. C3×C4×C8C22semidirect
36. #821C22semidirect
37. #821C22semidirect
38. (C2×D3)⋊C8C22semidirect
39. #809C22semidirect
40. #807C22semidirect
41. C4⋊(C3⋊C8)C22semidirect
50. C4×D8D3semidirect
106. C4×D4C2×D3semidirect
109. #107C2×D3semidirect
116. #102C2×D3semidirect
118. C4×C8C2×D3semidirect
129. C3×D4#41semidirect
146. D12D8semidirect
232. D4#235semidirect
277. C3#3294semidirect

Maximal subgroups

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

subgroupindex
#55512
D4×D122
C3×C4×D82
#51562
#50152
#50072
#48822
#47752
D3×#1022
#46942
C8×D122
#32943