Finite Group Databasebeta

← #82928

Subgroups — #82928

Order 384.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#235C211yesmaximal, characteristic, complemented
2C3×#525C211yesmaximal, characteristic, complemented
3C2×#951C211yesmaximal, characteristic, complemented
4C2×#949C211yesmaximal, characteristic, complemented
5#5207C211yesmaximal, characteristic, complemented
6#5047C214yesmaximal, complemented
7#5037C214yesmaximal, complemented
8#5004C211yesmaximal, characteristic, complemented
9#5001C211yesmaximal, characteristic, complemented
10#3005—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C22×#67—21no—
14C2×#235—22no—
15C2×#235—24no—
16C2×#235—24no—
17C2×#235C2214yes—
18C22×D12C2211yescharacteristic
19C22×C4×D3—21no—
20C22×Dic6C2211yescharacteristic
21C3×#123—21no—
22C3×C22⋊Q8C2212yescomplemented
23C3×#121C2212yescomplemented
24C3×#120—21no—
25C3×C4×D4—22no—
26C3×#117C2211yescharacteristic
27#951—24no—
28#949—24no—
29C2×#216C2212yescomplemented
30C2×#215C2212yescomplemented
31C2×#214C2212yescomplemented
32C2×#213C2212yescomplemented
33C2×#208C2211yescharacteristic
34C2×#208C2211yescharacteristic
35#830C2214yescomplemented
36#810C2214yescomplemented
37#809C2214yescomplemented
38C22×#41—31no—
39C2×#137—31no—
40C2×AGL(1,ℤ/8)—31no—
41#525D311yescharacteristic, complemented
42#452—34no—
43#449—34no—
44#419—31no—
45#418—31no—
46#407—31no—
47C24×C3D411yescharacteristic
48C23×D3—21no—
49C3×#41—22no—
50C3×#41—22no—
51C2×C3×Q8C2311yescharacteristic
52C2×C3×D4—22no—
53C2×C3×D4—21no—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—22no—
56C22×C3×C4—22no—
57C22×C3×C4—21no—
58C22×C3×C4D412yes—
59C22×C3×C4D411yescharacteristic
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C2×#67—28no—
63C2×#67—24no—
64C22×Dic3—21no—
65#235—216no—
66#235—28no—
67#235—28no—
68C2×D12—22no—
69C2×D12—21no—
70C2×D12D414yescomplemented
71C2×D12C2312yes—
72C2×C4×D3—28no—
73C2×C4×D3—24no—
74C2×Dic6—22no—
75C2×Dic6—21no—
76C2×Dic6D414yescomplemented
77C2×Dic6C2312yes—
78C3×C4⋊C4—22no—
79C3×C4⋊C4C2312yes—
80C3×C22⋊C4—21no—
81C3×C22⋊C4—22no—
82C3×C22⋊C4—22no—
83C3×C42—21no—
84#216—24no—
85#215—24no—
86#214—24no—
87#213—24no—
88#208—22no—
89#208—22no—
90C2×C3⋊C8C2312yes—
91C2×C3⋊C8C2312yes—
92C2×#41—34no—
93C2×#41—64no—
94C2×#41—64no—
95C2×#41—62no—
96C2×#41C2×D311yescharacteristic
97C22×Q8—31no—
98C22×D4—31no—
99C22×D4—61no—
100C23×C4—61no—
101C23×C4C2×D311yescharacteristic
102#137—64no—
103AGL(1,ℤ/8)—64no—
104C2×Q16—32no—
105C2×SD16—32no—
106C2×SD16—32no—
107C2×D8—32no—
108C2×M16—31no—
109C2×M16—31no—
110#123—21no—
111C22⋊Q8C2×D312yescomplemented
112#121C2×D312yescomplemented
113#120—21no—
114C4×D4—22no—
115#117C2×D311yescharacteristic
116#103—34no—
117#102—34no—
118C22⋊C8—34no—
119C23×C3—22no—
120C23×C3—21no—
121C23×C3—42no—
122C23×C3C2×D412yes—
123C23×C3C2×D411yescharacteristic
124C22×D3—24no—
125C22×D3—22no—
126C22×D3—44no—
127C3×Q8—21no—
128C3×Q8—21no—
129C3×D4—21no—
130C3×D4—21no—
131C3×D4—44no—
132C3×D4—42no—
133C2×C3×C4—22no—
134C2×C3×C4—24no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—42no—
140C2×C3×C4C2×D412yes—
141C2×C3×C4C2×D412yes—
142C2×C3×C4C2×D412yes—
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2411yescharacteristic
145#67—216no—
146C2×Dic3—24no—
147C2×Dic3—22no—
148D12—24no—
149D12—22no—
150D12C2×D414yes—
151C4×D3—216no—
152Dic6—24no—
153Dic6—22no—
154Dic6C2×D414yes—
155C3⋊C8—22no—
156C3⋊C8—22no—
157C24—61no—
158C24#6711yescharacteristic
159#41—22no—
160#41—22no—
161#41—68no—
162#41—68no—
163#41—616no—
164C2×Q8—34no—
165C2×Q8—32no—
166C2×Q8—62no—
167C2×Q8—61no—
168C2×Q8C22×D311yescharacteristic
169C2×D4—22no—
170C2×D4—21no—
171C2×D4—34no—
172C2×D4—32no—
173C2×D4—68no—
174C2×D4—64no—
175C2×D4—62no—
176C2×D4—61no—
177C2×D4C22×D311yescharacteristic
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—68no—
182C22×C4—61no—
183C22×C4—64no—
184C22×C4#6712yes—
185C22×C4#6711yescharacteristic
186C22×C4C22×D312yes—
187C22×C4C22×D311yescharacteristic
188Q16—64no—
189SD16—64no—
190SD16—64no—
191D8—64no—
192M16—62no—
193M16—62no—
194C2×C8—32no—
195C2×C8—32no—
196C4⋊C4—22no—
197C4⋊C4C22×D312yes—
198C22⋊C4—21no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C42—21no—
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—44no—
207C22×C3—41no—
208C22×C3—42no—
209C22×C3#12012yes—
210C22×C3C22×D411yescharacteristic
211C2×D3—24no—
212C2×D3—48no—
213C2×D3—44no—
214C3×C4—22no—
215C3×C4—41no—
216C3×C4—42no—
217C3×C4#12012yes—
218C3×C4C22×D411yescharacteristic
219C3×C4C22×D411yescharacteristic
220Dic3—24no—
221C23—22no—
222C23—21no—
223C23—42no—
224C23—64no—
225C23—62no—
226C23—124no—
227C23C2×#6712yes—
228C23C2×#6711yescharacteristic
229Q8—21no—
230Q8—21no—
231Q8—34no—
232Q8—64no—
233Q8—62no—
234D4—21no—
235D4—21no—
236D4—34no—
237D4—44no—
238D4—42no—
239D4—616no—
240D4—64no—
241D4—62no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—616no—
250C2×C4—64no—
251C2×C4—62no—
252C2×C4D3×D412yes—
253C2×C4D3×D412yes—
254C2×C4C2×#6712yes—
255C2×C4C2×#6711yescharacteristic
256C2×C4C23×D311yescharacteristic
257C8—62no—
258C8—62no—
259C2×C3—22no—
260C2×C3—42no—
261C2×C3—81no—
262C2×C3C2×#12011yescharacteristic
263C2×C3#57512yes—
264D3—44no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—44no—
270C22—41no—
271C22—42no—
272C22—64no—
273C22—128no—
274C22—124no—
275C22#93712yes—
276C22C22×#6711yescharacteristic
277C4—22no—
278C4—41no—
279C4—42no—
280C4—64no—
281C4#93712yes—
282C4C2×D3×D411yescharacteristic
283C4C2×D3×D411yescharacteristic
284C3#300511yescharacteristic, complemented, Sylow
285C2—22no—
286C2—42no—
287C2—81no—
288C2—124no—
289C2C2×#93711yescharacteristic
290C2#582212yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#525C2
socleC22×C3C22×D4
hypercenter#525D3
hypocenterC3#3005
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#235C2semidirect
2. C3×#525C2semidirect
3. C2×#951C2semidirect
4. C2×#949C2semidirect
5. #5207C2semidirect
6. #5047C2semidirect
7. #5037C2semidirect
8. #5004C2semidirect
9. #5001C2semidirect
22. C3×C22⋊Q8C22semidirect
23. C3×#121C22semidirect
29. C2×#216C22semidirect
30. C2×#215C22semidirect
31. C2×#214C22semidirect
32. C2×#213C22semidirect
35. #830C22semidirect
36. #810C22semidirect
37. #809C22semidirect
41. #525D3semidirect
70. C2×D12D4semidirect
76. C2×Dic6D4semidirect
111. C22⋊Q8C2×D3semidirect
112. #121C2×D3semidirect
284. C3#3005semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C3×#5252
C2×#9512
C2×#9492
#52072
#50472
#50372
#50042
#50012
#30053