Finite Group Databasebeta

← #81576

Subgroups — #81576

Order 384.

494 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 123 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#5726C211yesmaximal, characteristic, complemented
2#5717C211yesmaximal, characteristic, complemented
3C3×#486C211yesmaximal, characteristic, complemented
4#5087C211yesmaximal, characteristic, complemented
5#5072C211yesmaximal, characteristic, complemented
6#4775C212yesmaximal, complemented
7#4774C212yesmaximal, complemented
8#4773C212yesmaximal, complemented
9D3×#102C212yesmaximal, complemented
10C4×D24C211yesmaximal, characteristic, complemented
11C8×D12C211yesmaximal, characteristic, complemented
12#3294—31nomaximal, Sylow
13C2×#239—21no—
14C2×D3×D4—21no—
15C2×D3×D4C2212yes—
16C3×C4⋊Q8C2211yescharacteristic, complemented
17C3×#127C2211yescharacteristic
18#946—21no—
19#938—21no—
20#938C2212yes—
21#937—22no—
22C2×#213C2212yescomplemented
23C2×D24C2211yescharacteristic
24C2×D3×C8C2212yescomplemented
25#895C2212yescomplemented
26#894—22no—
27D3×C4⋊C4C2212yescomplemented
28C4×D12C2211yescharacteristic
29C4×D12C2211yescharacteristic
30C4×D12C2211yescharacteristic, complemented
31C3×#102C2212yescomplemented
32C3×#102C2212yescomplemented
33C3×C4×C8C2211yescharacteristic, complemented
34#832C2212yescomplemented
35#821C2211yescharacteristic
36#821C2211yescharacteristic, complemented
37(C2×D3)⋊C8C2212yescomplemented
38#818C2211yescharacteristic, complemented
39#809C2212yes—
40#807C2212yescomplemented
41C4⋊(C3⋊C8)C2211yescharacteristic, complemented
42#547—31no—
43D42—31no—
44#486D311yescharacteristic, complemented
45#480—32no—
46#474—31no—
47#466—32no—
48#459—31no—
49#447—32no—
50C4×D8—31no—
51C8×D4—31no—
52C2×#102—32no—
53C23×D3—22no—
54C2×C3×Q8—21no—
55C2×C3×D4—21no—
56C2×C3×D4—21no—
57C2×C3×D4C2312yes—
58C2×#67—22no—
59C2×#67—22no—
60#239—44no—
61D3×D4—24no—
62D3×D4—42no—
63D3×D4—44no—
64C2×D12—22no—
65C2×D12D411yescharacteristic
66C2×D12C2311yescharacteristic
67C2×D12C2311yescharacteristic
68C2×C4×D3—22no—
69C2×C4×D3—21no—
70C2×C4×D3—21no—
71C2×C4×D3C2312yes—
72C2×C3×C8C2311yescharacteristic
73C2×C3×C8C2311yescharacteristic
74C3×C4⋊C4—21no—
75C3×C4⋊C4C2312yes—
76C3×C42C2311yescharacteristic
77#217—22no—
78#213—42no—
79(C2×D3)⋊C4—21no—
80(C2×D3)⋊C4—21no—
81(C2×D3)⋊C4—22no—
82(C2×D3)⋊C4D412yes—
83C4⋊Dic3D411yescharacteristic
84C4⋊Dic3C2311yescharacteristic
85C4⋊Dic3C2311yescharacteristic
86Dic3⋊C4—22no—
87C2×C3⋊C8C2312yes—
88D24—22no—
89D3×C8—42no—
90C2×#41—61no—
91C22×D4—32no—
92C22×D4—61no—
93C2×D8—32no—
94C2×D8—31no—
95C22×C8—32no—
96C4⋊Q8C2×D311yescharacteristic, complemented
97#127C2×D311yescharacteristic
98#123—62no—
99C22⋊Q8—61no—
100#121—32no—
101#121—32no—
102#121—61no—
103#120—62no—
104C4×D4—31no—
105C4×D4—31no—
106C4×D4—31no—
107C2×C4⋊C4—32no—
108#107—31no—
109#107—32no—
110C4⋊C8—31no—
111#102—32no—
112#102—31no—
113#102—31no—
114#102—32no—
115#102C2×D312yescomplemented
116#102C2×D312yescomplemented
117C22⋊C8—32no—
118C4×C8C2×D311yescharacteristic, complemented
119C23×C3—22no—
120C22×D3—21no—
121C22×D3—21no—
122C22×D3—44no—
123C22×D3—44no—
124C22×D3—82no—
125C22×D3C2×D412yes—
126C3×Q8—42no—
127C3×D4—22no—
128C3×D4—22no—
129C3×D4—42no—
130C3×D4—81no—
131C2×C3×C4—22no—
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2411yescharacteristic
135#67—44no—
136#67—44no—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3C2×D412yes—
140D12—21no—
141D12—21no—
142D12—21no—
143D12—44no—
144D12D814yescomplemented
145D12#4112yescomplemented
146C4×D3—22no—
147C4×D3—22no—
148C4×D3—44no—
149C4×D3—42no—
150C4×D3—42no—
151C3×C8—21no—
152C3×C8—21no—
153C3⋊C8—22no—
154C24—62no—
155#41—124no—
156C2×Q8—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—31no—
160C2×D4—31no—
161C2×D4—31no—
162C2×D4—62no—
163C2×D4—62no—
164C2×D4—64no—
165C2×D4—62no—
166C2×D4—122no—
167C2×D4—124no—
168C2×D4C22×D312yes—
169C22×C4—32no—
170C22×C4—62no—
171C22×C4—61no—
172C22×C4—61no—
173D8—62no—
174D8—122no—
175C2×C8—32no—
176C2×C8—122no—
177C2×C8C22×D311yescharacteristic
178C2×C8C22×D311yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—31no—
181C4⋊C4—31no—
182C4⋊C4—31no—
183C4⋊C4—62no—
184C4⋊C4C22×D312yes—
185C22⋊C4—32no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C22⋊C4—61no—
189C22⋊C4—61no—
190C42C22×D311yescharacteristic
191C22×C3—42no—
192C22×C3—42no—
193C22×C3—42no—
194C22×C3C22×D411yescharacteristic
195C2×D3—22no—
196C2×D3—22no—
197C2×D3—22no—
198C2×D3—41no—
199C2×D3—41no—
200C2×D3—41no—
201C2×D3—41no—
202C2×D3—41no—
203C2×D3—88no—
204C2×D3C2×D814yes—
205C3×C4—21no—
206C3×C4—42no—
207C3×C4C2×D812yes—
208C3×C4C2×#4111yescharacteristic
209C3×C4H(5,𝔽2)11yescharacteristic
210Dic3—22no—
211Dic3—41no—
212Dic3—41no—
213C23—22no—
214C23—32no—
215C23—61no—
216C23—61no—
217C23—124no—
218C23—124no—
219C23—242no—
220Q8—42no—
221D4—22no—
222D4—22no—
223D4—34no—
224D4—32no—
225D4—42no—
226D4—61no—
227D4—61no—
228D4—61no—
229D4—81no—
230D4—124no—
231D4—124no—
232D4—124no—
233C2×C4—22no—
234C2×C4—32no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—124no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4D3×D411yescharacteristic
243C2×C4D3×D411yescharacteristic
244C2×C4C23×D311yescharacteristic
245C8—21no—
246C8—21no—
247C8—62no—
248C2×C3—82no—
249C2×C3#54611yescharacteristic
250C2×C3C22×D811yescharacteristic
251C2×C3#57711yescharacteristic
252D3—24no—
253D3—42no—
254D3—81no—
255C22—34no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—62no—
260C22—62no—
261C22—62no—
262C22—121no—
263C22—121no—
264C22—121no—
265C22—121no—
266C22—121no—
267C22—248no—
268C22C2×D3×D411yescharacteristic
269C4—21no—
270C4—42no—
271C4—62no—
272C4—121no—
273C4—121no—
274C4D3×D812yes—
275C4D3×#4111yescharacteristic
276C4#100911yescharacteristic
277C3#329411yescharacteristic, complemented, Sylow
278C2—64no—
279C2—82no—
280C2—122no—
281C2—241no—
282C2#567611yescharacteristic
283C2C2×D3×D811yescharacteristic
284C2#576211yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#486C2
socleC22×C3C22×D4
hypercenter#486D3
hypocenterC3#3294
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #5726C2semidirect
2. #5717C2semidirect
3. C3×#486C2semidirect
4. #5087C2semidirect
5. #5072C2semidirect
6. #4775C2semidirect
7. #4774C2semidirect
8. #4773C2semidirect
9. D3×#102C2semidirect
10. C4×D24C2semidirect
11. C8×D12C2semidirect
16. C3×C4⋊Q8C22semidirect
22. C2×#213C22semidirect
24. C2×D3×C8C22semidirect
25. #895C22semidirect
27. D3×C4⋊C4C22semidirect
30. C4×D12C22semidirect
31. C3×#102C22semidirect
32. C3×#102C22semidirect
33. C3×C4×C8C22semidirect
34. #832C22semidirect
36. #821C22semidirect
37. (C2×D3)⋊C8C22semidirect
38. #818C22semidirect
40. #807C22semidirect
41. C4⋊(C3⋊C8)C22semidirect
44. #486D3semidirect
96. C4⋊Q8C2×D3semidirect
115. #102C2×D3semidirect
116. #102C2×D3semidirect
118. C4×C8C2×D3semidirect
144. D12D8semidirect
145. D12#41semidirect
277. C3#3294semidirect

Maximal subgroups

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

subgroupindex
#57262
#57172
C3×#4862
#50872
#50722
#47752
#47742
#47732
D3×#1022
C4×D242
C8×D122
#32943