Finite Group Databasebeta

← #81566

Subgroups — #81566

Order 384.

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

normal subgroups only

This group has 532 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#5725C211yesmaximal, characteristic, complemented
2D3×#127C211yesmaximal, characteristic, complemented
3C3×#486C211yesmaximal, characteristic, complemented
4#5088C211yesmaximal, characteristic, complemented
5#5073C211yesmaximal, characteristic, complemented
6#4786C214yesmaximal, complemented
7#4773C214yesmaximal, complemented
8#4695C211yesmaximal, characteristic, complemented
9#4685C211yesmaximal, characteristic, complemented
10#3205—31nomaximal, Sylow
11C2×#239—21no—
12C2×D3×D4—21no—
13C2×D3×D4—21no—
14C2×D3×D4C2212yes—
15C3×C4⋊Q8C2211yescharacteristic, complemented
16C3×#127C2211yescharacteristic
17#947—21no—
18#940—21no—
19#940—21no—
20#940C2212yes—
21C2×#213C2214yescomplemented
22C2×D24C2214yescomplemented
23#895—21no—
24#895C2212yes—
25#893—21no—
26#893C2212yescomplemented
27#874C2211yescharacteristic
28#874C2211yescharacteristic, complemented
29C42×D3C2211yescharacteristic
30C3×#102C2214yescomplemented
31C3×C4×C8C2211yescharacteristic, complemented
32(C2×D3)⋊C8C2212yes—
33Dic3⋊C8C2212yescomplemented
34#809C2214yescomplemented
35C4×C3⋊C8C2211yescharacteristic, complemented
36#532—31no—
37C2×#127—31no—
38#493—31no—
39#493—31no—
40#486—31no—
41#486D311yescharacteristic, complemented
42#459—34no—
43#447—34no—
44#431—31no—
45C23×D3—22no—
46C2×C3×Q8—21no—
47C2×C3×D4—21no—
48C2×C3×D4—21no—
49C2×C3×D4C2312yes—
50C2×#67—22no—
51C2×#67—22no—
52C2×#67—22no—
53#239—44no—
54D3×D4—44no—
55D3×D4—44no—
56D3×D4—44no—
57C2×D12—22no—
58C2×D12—21no—
59C2×D12—21no—
60C2×D12—21no—
61C2×D12—21no—
62C2×D12C2312yes—
63C2×D12C2312yes—
64C2×C4×D3—22no—
65C2×C4×D3D411yescharacteristic
66C2×C4×D3D411yescharacteristic
67C2×C4×D3C2311yescharacteristic
68C2×C3×C8C2312yes—
69C3×C4⋊C4—21no—
70C3×C4⋊C4C2312yes—
71C3×C42C2311yescharacteristic
72#213—44no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—22no—
75C4×Dic3D411yescharacteristic
76C4×Dic3D411yescharacteristic
77C4×Dic3C2311yescharacteristic
78C2×C3⋊C8C2312yes—
79D24—44no—
80C2×#41—61no—
81C22×D4—32no—
82C22×D4—61no—
83C22×D4—61no—
84C2×D8—34no—
85C2×D8—34no—
86C4⋊Q8C2×D311yescharacteristic, complemented
87#127—32no—
88#127—31no—
89#127—31no—
90#127—61no—
91#127—61no—
92#127C2×D311yescharacteristic
93#124—61no—
94#121—32no—
95#121—61no—
96C4×D4—32no—
97C4×D4—61no—
98C2×C42—31no—
99C4⋊C8—32no—
100#102—34no—
101#102C2×D314yescomplemented
102C22⋊C8—32no—
103C4×C8—31no—
104C4×C8C2×D311yescharacteristic, complemented
105C23×C3—22no—
106C22×D3—22no—
107C22×D3—22no—
108C22×D3—44no—
109C22×D3—44no—
110C22×D3—44no—
111C22×D3C2×D411yescharacteristic
112C3×Q8—42no—
113C3×D4—24no—
114C3×D4—42no—
115C3×D4—42no—
116C2×C3×C4—22no—
117C2×C3×C4C2×D411yescharacteristic
118C2×C3×C4C2×D411yescharacteristic
119C2×C3×C4C2411yescharacteristic
120#67—44no—
121#67—44no—
122#67—44no—
123C2×Dic3C2×D411yescharacteristic
124C2×Dic3C2×D411yescharacteristic
125C2×Dic3C2×D411yescharacteristic
126D12—24no—
127D12—24no—
128D12—44no—
129D12—42no—
130D12—42no—
131D12—42no—
132D12—42no—
133C4×D3—22no—
134C4×D3—22no—
135C4×D3—44no—
136C4×D3D814yescomplemented
137C3×C8—22no—
138C3⋊C8—22no—
139C24—62no—
140#41—124no—
141C2×Q8—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—32no—
145C2×D4—32no—
146C2×D4—62no—
147C2×D4—62no—
148C2×D4—62no—
149C2×D4—62no—
150C2×D4—61no—
151C2×D4—61no—
152C2×D4—61no—
153C2×D4—61no—
154C2×D4—124no—
155C2×D4—124no—
156C2×D4—124no—
157C2×D4C22×D312yes—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—62no—
162D8—124no—
163D8—124no—
164C2×C8—32no—
165C2×C8C22×D312yes—
166C4⋊C4—21no—
167C4⋊C4C22×D312yes—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C42—31no—
171C42—31no—
172C42—31no—
173C42C22×D311yescharacteristic
174C22×C3—42no—
175C22×C3—42no—
176C22×C3—42no—
177C22×C3C22×D411yescharacteristic
178C2×D3—21no—
179C2×D3—21no—
180C2×D3—42no—
181C2×D3—42no—
182C2×D3—42no—
183C2×D3—42no—
184C2×D3—42no—
185C2×D3—42no—
186C2×D3—88no—
187C2×D3C2×D812yes—
188C3×C4—42no—
189C3×C4C2×D812yes—
190C3×C4AGL(1,ℤ/8)12yes—
191C3×C4H(5,𝔽2)12yes—
192Dic3—21no—
193Dic3—21no—
194Dic3C2×D812yes—
195C23—22no—
196C23—31no—
197C23—62no—
198C23—62no—
199C23—124no—
200C23—124no—
201C23—124no—
202Q8—42no—
203D4—24no—
204D4—42no—
205D4—42no—
206D4—64no—
207D4—64no—
208D4—124no—
209D4—124no—
210D4—124no—
211D4—124no—
212D4—122no—
213D4—122no—
214D4—122no—
215D4—122no—
216C2×C4—22no—
217C2×C4—31no—
218C2×C4—34no—
219C2×C4—31no—
220C2×C4—31no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—124no—
224C2×C4D3×D411yescharacteristic
225C2×C4D3×D411yescharacteristic
226C2×C4C23×D311yescharacteristic
227C8—22no—
228C8—62no—
229C2×C3—82no—
230C2×C3#53511yescharacteristic
231C2×C3C22×D811yescharacteristic
232C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
233D3—22no—
234D3—82no—
235D3—82no—
236C22—32no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—61no—
241C22—61no—
242C22—122no—
243C22—122no—
244C22—122no—
245C22—122no—
246C22—122no—
247C22—122no—
248C22—248no—
249C22C2×D3×D411yescharacteristic
250C4—32no—
251C4—42no—
252C4—61no—
253C4—61no—
254C4D3×D812yes—
255C4#91412yes—
256C4#100912yes—
257C3#320511yescharacteristic, complemented, Sylow
258C2—62no—
259C2—82no—
260C2—242no—
261C2—242no—
262C2#567411yescharacteristic
263C2C2×D3×D811yescharacteristic
264C2C2×#91411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#486C2
socleC22×C3C22×D4
hypercenter#486D3
hypocenterC3#3205
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. #5725C2semidirect
2. D3×#127C2semidirect
3. C3×#486C2semidirect
4. #5088C2semidirect
5. #5073C2semidirect
6. #4786C2semidirect
7. #4773C2semidirect
8. #4695C2semidirect
9. #4685C2semidirect
15. C3×C4⋊Q8C22semidirect
21. C2×#213C22semidirect
22. C2×D24C22semidirect
26. #893C22semidirect
28. #874C22semidirect
30. C3×#102C22semidirect
31. C3×C4×C8C22semidirect
33. Dic3⋊C8C22semidirect
34. #809C22semidirect
35. C4×C3⋊C8C22semidirect
41. #486D3semidirect
86. C4⋊Q8C2×D3semidirect
101. #102C2×D3semidirect
104. C4×C8C2×D3semidirect
136. C4×D3D8semidirect
257. C3#3205semidirect

Maximal subgroups

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

subgroupindex
#57252
D3×#1272
C3×#4862
#50882
#50732
#47862
#47732
#46952
#46852
#32053