Finite Group Databasebeta

← #79603

Subgroups — #79603

Order 384.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#911C212yesmaximal, complemented
2#5555C211yesmaximal, characteristic, complemented
3D4×D12C211yesmaximal, characteristic, complemented
4C3×C4×D8C211yesmaximal, characteristic, complemented
5#5019C211yesmaximal, characteristic, complemented
6#5015C211yesmaximal, characteristic, complemented
7#4886C212yesmaximal, complemented
8#4781C212yesmaximal, complemented
9#4776C212yesmaximal, complemented
10#4693C211yesmaximal, characteristic, complemented
11#4688C211yesmaximal, characteristic, complemented
12#3272—31nomaximal, Sylow
13C2×#237C2212yescomplemented
14C2×D3×D4C2212yes—
15C22×D12—21no—
16C2×C3×D8C2211yescharacteristic
17C3×C4×D4C2211yescharacteristic
18C3×C4×D4C2211yescharacteristic, complemented
19C2×#214C2212yes—
20C2×#213C2212yescomplemented
21#930—21no—
22#930—21no—
23C2×C4⋊Dic3—21no—
24#911—28no—
25C2×#204C2212yescomplemented
26C2×#204C2212yescomplemented
27C2×ΓL(1,𝔽25)C2212yescomplemented
28#897C2212yescomplemented
29#895C2212yes—
30#886—22no—
31#882—22no—
32#874C2211yescharacteristic
33C4×D12C2211yescharacteristic
34#869C2211yescharacteristic, complemented
35C3×#107C2211yescharacteristic, complemented
36C3×#102C2211yescharacteristic
37C3×#102C2211yescharacteristic, complemented
38C3×C4×C8C2211yescharacteristic, complemented
39(C2×D3)⋊C8C2212yescomplemented
40#810C2212yescomplemented
41#809C2212yescomplemented
42C4⋊(C3⋊C8)C2211yescharacteristic, complemented
43C2×AGL(1,ℤ/8)—32no—
44#547—31no—
45D42—31no—
46#492—31no—
47#468—32no—
48#461—31no—
49#459—31no—
50#450—32no—
51#449—32no—
52C4×D8D311yescharacteristic, complemented
53#436—31no—
54C23×D3—22no—
55C2×C3×D4C2311yescharacteristic
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C22×C3×C4—21no—
59C2×#67—22no—
60C2×#67—22no—
61C22×Dic3—22no—
62#237—24no—
63#237—42no—
64D3×D4—24no—
65D3×D4—42no—
66C2×D12—22no—
67C2×D12—21no—
68C2×D12—21no—
69C2×D12—44no—
70C2×D12D411yescharacteristic
71C2×D12C2312yes—
72C2×C4×D3C2312yes—
73C2×Dic6C2312yes—
74C3×D8D414yescomplemented
75C2×C3×C8C2311yescharacteristic
76C2×C3×C8C2311yescharacteristic
77C3×C4⋊C4C2311yescharacteristic
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80C3×C22⋊C4—21no—
81C3×C42C2311yescharacteristic
82#214—24no—
83#213—24no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4D412yes—
87C4⋊Dic3—21no—
88C4⋊Dic3—21no—
89C4⋊Dic3—22no—
90C4⋊Dic3D411yescharacteristic
91C2×C3⋊C8C2312yes—
92#204—24no—
93#204—42no—
94ΓL(1,𝔽25)—24no—
95C2×#41—32no—
96C22×D4—32no—
97C22×D4—61no—
98AGL(1,ℤ/8)—68no—
99C2×SD16—32no—
100C2×SD16—32no—
101C2×SD16—32no—
102C2×D8—32no—
103C2×D8C2×D311yescharacteristic
104C2×M16—32no—
105C4⋊Q8—31no—
106#127—31no—
107#123—62no—
108C22⋊Q8—32no—
109#121—32no—
110#121—61no—
111#121—61no—
112#120—62no—
113C4×D4—31no—
114C4×D4C2×D311yescharacteristic
115C4×D4C2×D311yescharacteristic, complemented
116C2×C4⋊C4—61no—
117#107C2×D311yescharacteristic, complemented
118C4⋊C8—31no—
119#103—32no—
120#102—32no—
121#102C2×D311yescharacteristic
122#102C2×D311yescharacteristic, complemented
123C22⋊C8—32no—
124C4×C8C2×D311yescharacteristic, complemented
125C23×C3—21no—
126C23×C3—21no—
127C22×D3—22no—
128C22×D3—44no—
129C22×D3—44no—
130C22×D3—82no—
131C22×D3C2×D412yes—
132C3×D4—21no—
133C3×D4—21no—
134C3×D4C2×D412yes—
135C3×D4C2×D412yescomplemented
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—42no—
139C2×C3×C4—42no—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2411yescharacteristic
143#67—44no—
144#67—44no—
145C2×Dic3—22no—
146C2×Dic3—44no—
147C2×Dic3C2×D412yes—
148D12—22no—
149D12—22no—
150D12—22no—
151D12—44no—
152D12—42no—
153D12—81no—
154C4×D3—22no—
155C4×D3—22no—
156Dic6—22no—
157Dic6—22no—
158C3×C8—21no—
159C3×C8C2×D412yescomplemented
160C3⋊C8—22no—
161C24—62no—
162#41—64no—
163#41—122no—
164C2×Q8—32no—
165C2×D4—31no—
166C2×D4—32no—
167C2×D4—62no—
168C2×D4—62no—
169C2×D4—64no—
170C2×D4—62no—
171C2×D4—61no—
172C2×D4—61no—
173C2×D4—124no—
174C2×D4—122no—
175C2×D4C22×D311yescharacteristic
176C2×D4C22×D311yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—32no—
180C22×C4—62no—
181SD16—64no—
182SD16—64no—
183SD16—122no—
184D8—64no—
185D8D1214yescomplemented
186M16—64no—
187C2×C8—32no—
188C2×C8C22×D311yescharacteristic
189C2×C8C22×D311yescharacteristic
190C4⋊C4—31no—
191C4⋊C4—62no—
192C4⋊C4—61no—
193C4⋊C4—61no—
194C4⋊C4C22×D311yescharacteristic
195C4⋊C4C22×D311yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—32no—
199C22⋊C4—62no—
200C22⋊C4—62no—
201C42C22×D311yescharacteristic
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—41no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3C22×D411yescharacteristic
209C2×D3—22no—
210C2×D3—22no—
211C2×D3—22no—
212C2×D3—42no—
213C2×D3—42no—
214C2×D3—42no—
215C2×D3—88no—
216C3×C4—21no—
217C3×C4—41no—
218C3×C4—41no—
219C3×C4AGL(1,ℤ/8)12yescomplemented
220C3×C4C22×D411yescharacteristic
221C3×C4H(5,𝔽2)11yescharacteristic
222Dic3—22no—
223Dic3—42no—
224C23—21no—
225C23—21no—
226C23—32no—
227C23—62no—
228C23—124no—
229C23—124no—
230C23—242no—
231Q8—62no—
232Q8—62no—
233D4—21no—
234D4—21no—
235D4—62no—
236D4—62no—
237D4—62no—
238D4—124no—
239D4—124no—
240D4—124no—
241D4—122no—
242D4—241no—
243D4C2×D1212yes—
244D4C2×D1212yescomplemented
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—32no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—124no—
254C2×C4D3×D411yescharacteristic
255C2×C4D3×D411yescharacteristic
256C2×C4C23×D311yescharacteristic
257C8—21no—
258C8—62no—
259C8C2×D1212yescomplemented
260C2×C3—42no—
261C2×C3—42no—
262C2×C3D4211yescharacteristic
263C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
264C2×C3#57711yescharacteristic
265D3—42no—
266D3—82no—
267C22—22no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—62no—
274C22—62no—
275C22—62no—
276C22—122no—
277C22—122no—
278C22—122no—
279C22—248no—
280C22C2×D3×D411yescharacteristic
281C4—21no—
282C4—41no—
283C4—41no—
284C4—62no—
285C4—122no—
286C4#91112yescomplemented
287C4C22×D1211yescharacteristic
288C4#100911yescharacteristic
289C3#327211yescharacteristic, complemented, Sylow
290C2—42no—
291C2—42no—
292C2—122no—
293C2—242no—
294C2D4×D1211yescharacteristic
295C2C2×#91111yescharacteristic
296C2#577011yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
hypercenterC4×D8D3
hypocenterC3#3272
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×#911C2semidirect
2. #5555C2semidirect
3. D4×D12C2semidirect
4. C3×C4×D8C2semidirect
5. #5019C2semidirect
6. #5015C2semidirect
7. #4886C2semidirect
8. #4781C2semidirect
9. #4776C2semidirect
10. #4693C2semidirect
11. #4688C2semidirect
13. C2×#237C22semidirect
18. C3×C4×D4C22semidirect
20. C2×#213C22semidirect
25. C2×#204C22semidirect
26. C2×#204C22semidirect
27. C2×ΓL(1,𝔽25)C22semidirect
28. #897C22semidirect
34. #869C22semidirect
35. C3×#107C22semidirect
37. C3×#102C22semidirect
38. C3×C4×C8C22semidirect
39. (C2×D3)⋊C8C22semidirect
40. #810C22semidirect
41. #809C22semidirect
42. C4⋊(C3⋊C8)C22semidirect
52. C4×D8D3semidirect
74. C3×D8D4semidirect
115. C4×D4C2×D3semidirect
117. #107C2×D3semidirect
122. #102C2×D3semidirect
124. C4×C8C2×D3semidirect
135. C3×D4C2×D4semidirect
159. C3×C8C2×D4semidirect
185. D8D12semidirect
219. C3×C4AGL(1,ℤ/8)semidirect
244. D4C2×D12semidirect
259. C8C2×D12semidirect
286. C4#911semidirect
289. C3#3272semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
#55552
D4×D122
C3×C4×D82
#50192
#50152
#48862
#47812
#47762
#46932
#46882
#32723