Finite Group Databasebeta

← #83396

Subgroups — #83396

Order 384.

772 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 145 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#5964C211yesmaximal, characteristic, complemented
2C3×#535C211yesmaximal, characteristic, complemented
3#5805C212yesmaximal, complemented
4#5240C211yesmaximal, characteristic, complemented
5#5032C218yesmaximal, complemented
6#5006C212yesmaximal, complemented
7#3027—31nomaximal, Sylow
8C2×C3×#41C2211yescharacteristic
9C22×C3×D4C2211yescharacteristic, complemented
10#1010—22no—
11#1009—22no—
12D3×#41—24no—
13#1007—21no—
14#1004—21no—
15#1004C2214yes—
16C2×#235—22no—
17C2×#235C2212yes—
18C3×#127—21no—
19C3×#124—21no—
20C3×#121—22no—
21C3×#120C2214yescomplemented
22C3×#117C2211yescharacteristic, complemented
23#953C2218yescomplemented
24#940—22no—
25#938—24no—
26#936—22no—
27#926C2212yescomplemented
28#834C2214yescomplemented
29#806C2218yescomplemented
30#585—31no—
31#535—32no—
32#535D311yescharacteristic, complemented
33B(4,𝔽2)—38no—
34#410—31no—
35#410—32no—
36C24×C3C2312yes—
37C3×#41—22no—
38C3×#41—22no—
39C2×C3×Q8D411yescharacteristic, complemented
40C2×C3×D4—24no—
41C2×C3×D4—22no—
42C2×C3×D4—22no—
43C2×C3×D4—21no—
44C2×C3×D4—21no—
45C2×C3×D4D411yescharacteristic, complemented
46C2×C3×D4C2312yes—
47C22×C3×C4D412yescomplemented
48C22×C3×C4C2311yescharacteristic
49C2×#67—24no—
50C2×#67—28no—
51C2×#67C2314yes—
52#239—24no—
53D3×Q8—24no—
54#237—24no—
55#237—28no—
56D3×D4—24no—
57D3×D4—28no—
58#235—24no—
59#235—24no—
60#235—22no—
61#235—24no—
62#235—22no—
63#235—22no—
64#235—22no—
65C2×D12—22no—
66C2×D12D412yescomplemented
67C2×C4×D3—24no—
68C2×C4×D3D414yescomplemented
69C2×Dic6—22no—
70C2×Dic6D412yescomplemented
71C3×C4⋊C4—21no—
72C3×C22⋊C4—24no—
73C3×C22⋊C4C2312yes—
74C3×C42—21no—
75#217—28no—
76#217C2314yes—
77C4⋊Dic3—22no—
78C4×Dic3—22no—
792−1+4—62no—
802−1+4—61no—
81H(5,𝔽2)—34no—
82H(5,𝔽2)—62no—
83H(5,𝔽2)—61no—
84C2×#41—32no—
85C2×#41—64no—
86C2×#41—62no—
87C2×#41C2×D311yescharacteristic
88C22×D4C2×D311yescharacteristic, complemented
89#127—21no—
90#127—62no—
91#124—21no—
92#124—62no—
93#121—22no—
94#121—64no—
95#120—38no—
96#120C2×D314yescomplemented
97#117—32no—
98#117C2×D311yescharacteristic, complemented
99#99—34no—
100#99—38no—
101C23×C3—24no—
102C23×C3—22no—
103C23×C3—82no—
104C23×C3C2×D412yes—
105C23×C3C2411yescharacteristic
106C22×D3—24no—
107C22×D3C2×D414yes—
108C3×Q8—22no—
109C3×D4—24no—
110C3×D4—22no—
111C3×D4—41no—
112C3×D4—42no—
113C3×D4—41no—
114C3×D4—84no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4C2×D412yes—
119C2×C3×C4C2×D412yes—
120C2×C3×C4C2×D412yes—
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2×D411yescharacteristic
123#67—28no—
124#67—24no—
125#67—24no—
126#67—88no—
127C2×Dic3—24no—
128C2×Dic3—24no—
129C2×Dic3C2×D414yes—
130D12—24no—
131D12—22no—
132D12—22no—
133C4×D3—28no—
134C4×D3—24no—
135C4×D3—24no—
136Dic6—24no—
137Dic6—22no—
138Dic6—22no—
139C24C22×D312yes—
140#41—22no—
141#41—22no—
142#41—64no—
143#41—64no—
144#41—68no—
145#41—64no—
146#41—64no—
147#41—62no—
148#41—62no—
149#41—62no—
150#41—62no—
151#41—64no—
152C2×Q8—32no—
153C2×Q8—62no—
154C2×Q8—64no—
155C2×Q8#6711yescharacteristic, complemented
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—34no—
162C2×D4—32no—
163C2×D4—62no—
164C2×D4—68no—
165C2×D4—64no—
166C2×D4—64no—
167C2×D4—68no—
168C2×D4#6711yescharacteristic, complemented
169C2×D4C22×D312yes—
170C22×C4—34no—
171C22×C4—64no—
172C22×C4#6712yescomplemented
173C22×C4C22×D311yescharacteristic
174C4⋊C4—21no—
175C4⋊C4—62no—
176C22⋊C4—24no—
177C22⋊C4—34no—
178C22⋊C4—68no—
179C22⋊C4C22×D312yes—
180C42—21no—
181C42—62no—
182C22×C3—22no—
183C22×C3—41no—
184C22×C3—42no—
185C22×C3—48no—
186C22×C3—44no—
187C22×C3—42no—
188C22×C3C22×D412yes—
189C22×C3C22×D411yescharacteristic
190C2×D3—24no—
191C2×D3—44no—
192C2×D3—84no—
193C3×C4—22no—
194C3×C4—42no—
195C3×C4#12011yescharacteristic
196C3×C4#12012yes—
197C3×C4#12011yescharacteristic
198Dic3—24no—
199Dic3—44no—
200C23—24no—
201C23—22no—
202C23—34no—
203C23—64no—
204C23—82no—
205C23C2×#6712yes—
206C23C23×D311yescharacteristic
207Q8—22no—
208Q8—64no—
209Q8—62no—
210Q8—62no—
211D4—24no—
212D4—22no—
213D4—41no—
214D4—41no—
215D4—42no—
216D4—64no—
217D4—68no—
218D4—64no—
219D4—64no—
220D4—62no—
221D4—62no—
222D4—84no—
223D4—248no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—34no—
228C2×C4—68no—
229C2×C4—64no—
230C2×C4—64no—
231C2×C4—64no—
232C2×C4—64no—
233C2×C4D3×D412yes—
234C2×C4D3×D412yes—
235C2×C4C2×#6712yes—
236C2×C4C2×#6711yescharacteristic
237C2×C4C2×#6711yescharacteristic
238C2×C3—22no—
239C2×C3—21no—
240C2×C3—42no—
241C2×C3—82no—
242C2×C3C2×#12011yescharacteristic
243D3—44no—
244C22—22no—
245C22—41no—
246C22—42no—
247C22—44no—
248C22—48no—
249C22—42no—
250C22—64no—
251C22—124no—
252C22—244no—
253C22C2×D3×D412yes—
254C22C22×#6711yescharacteristic
255C4—22no—
256C4—42no—
257C4—64no—
258C4—124no—
259C4#93711yescharacteristic
260C4#93711yescharacteristic
261C4#93712yes—
262C3#302711yescharacteristic, complemented, Sylow
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—124no—
268C2C2×#93711yescharacteristic
subgroupG/H
centerC2C2×#937
commutator, cosocleC23×C3C24
FrattiniC23C23×D3
FittingC3×#535C2
socleC2×C3C2×#120
hypercenter#535D3
hypocenterC3#3027
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #5964C2semidirect
2. C3×#535C2semidirect
3. #5805C2semidirect
4. #5240C2semidirect
5. #5032C2semidirect
6. #5006C2semidirect
9. C22×C3×D4C22semidirect
21. C3×#120C22semidirect
22. C3×#117C22semidirect
23. #953C22semidirect
27. #926C22semidirect
28. #834C22semidirect
29. #806C22semidirect
32. #535D3semidirect
39. C2×C3×Q8D4semidirect
45. C2×C3×D4D4semidirect
47. C22×C3×C4D4semidirect
66. C2×D12D4semidirect
68. C2×C4×D3D4semidirect
70. C2×Dic6D4semidirect
88. C22×D4C2×D3semidirect
96. #120C2×D3semidirect
98. #117C2×D3semidirect
155. C2×Q8#67semidirect
168. C2×D4#67semidirect
172. C22×C4#67semidirect
262. C3#3027semidirect

Maximal subgroups

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

subgroupindex
#59642
C3×#5352
#58052
#52402
#50322
#50062
#30273