Finite Group Databasebeta

← #76823

Subgroups — #76823

Order 384.

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

normal subgroups only

This group has 544 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
1C2×#944C212yesmaximal, complemented
2C2×#878C212yesmaximal
3C3×#393C211yesmaximal, characteristic
4#4949C211yesmaximal, characteristic
5#4946C211yesmaximal, characteristic
6#4945C212yesmaximal
7#4657C212yesmaximal, complemented
8#4656C212yesmaximal, complemented
9#4652C212yesmaximal, complemented
10#2677—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C23×Dic3C2211yescharacteristic
13C22×Dic6C2212yes—
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C22⋊C4C2211yescharacteristic
16C2×#217C2212yes—
17C2×#217C2211yescharacteristic
18#944—28no—
19C2×C4⋊Dic3C2212yes—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×Dic3⋊C4C2212yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3C2212yes—
27#878—28no—
28C3×#95C2212yes—
29C3×#95C2211yescharacteristic
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C2×C4⋊Q8—32no—
35C2×C22⋊Q8—32no—
36#399—31no—
37#398—32no—
38#395—32no—
39#393D311yescharacteristic
40#388—32no—
41#387—32no—
42#385—31no—
43C24×C3C2311yescharacteristic
44C2×C3×Q8—24no—
45C2×C3×Q8—41no—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2312yes—
50C22×Dic3—24no—
51C22×Dic3—41no—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C2×Dic6—24no—
57C2×Dic6—24no—
58C2×Dic6—42no—
59C3×C22⋊C4—24no—
60C3×C22⋊C4—41no—
61#217—24no—
62#217Q814yescomplemented
63C4⋊Dic3—24no—
64C4⋊Dic3—41no—
65Dic3⋊C4—24no—
66Dic3⋊C4—24no—
67Dic3⋊C4—24no—
68Dic3⋊C4—24no—
69Dic3⋊C4D418yes—
70C4×Dic3—24no—
71C22×Q8—32no—
72C22×Q8C2×D311yescharacteristic
73C23×C4—31no—
74C4⋊Q8—68no—
75C22⋊Q8—68no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4—32no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—31no—
83C2×C22⋊C4—31no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4C2×D312yes—
86C2×C22⋊C4C2×D311yescharacteristic
87C2×C42—32no—
88#95—32no—
89#95—31no—
90#95—32no—
91#95—32no—
92#95C2×D312yes—
93#95C2×D311yescharacteristic
94C23×C3—22no—
95C23×C3—21no—
96C23×C3—21no—
97C23×C3—22no—
98C23×C3C2×Q812yes—
99C23×C3C2411yescharacteristic
100C3×Q8—44no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4C2×D414yes—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—44no—
121C2×Dic3C2×D414yes—
122C2×Dic3C2×D414yes—
123C2×Dic3C2×Q814yes—
124Dic6—48no—
125C24C22×D311yescharacteristic
126C2×Q8—24no—
127C2×Q8—41no—
128C2×Q8—64no—
129C2×Q8—64no—
130C2×Q8—122no—
131C22×C4—32no—
132C22×C4—32no—
133C22×C4—32no—
134C22×C4—32no—
135C22×C4—64no—
136C22×C4—121no—
137C22×C4C22×D311yescharacteristic
138C22×C4C22×D312yes—
139C22×C4C22×D311yescharacteristic
140C22×C4C22×D312yes—
141C4⋊C4—38no—
142C4⋊C4—64no—
143C4⋊C4—64no—
144C4⋊C4—64no—
145C4⋊C4—64no—
146C4⋊C4—64no—
147C4⋊C4—121no—
148C22⋊C4—24no—
149C22⋊C4—34no—
150C22⋊C4—41no—
151C22⋊C4—64no—
152C42—64no—
153C22×C3—22no—
154C22×C3—24no—
155C22×C3—42no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3C22×D412yes—
159C22×C3C22×Q811yescharacteristic
160C22×C3H(5,𝔽2)11yescharacteristic
161C22×C32−1+411yescharacteristic
162C22×C32−1+412yes—
163C3×C4—24no—
164C3×C4—42no—
165C3×C4—41no—
166C3×C4—41no—
167Dic3—24no—
168Dic3—24no—
169Dic3—24no—
170Dic3—42no—
171C23—22no—
172C23—21no—
173C23—22no—
174C23—21no—
175C23D3×Q812yes—
176C23C23×D311yescharacteristic
177Q8—44no—
178Q8—128no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—34no—
189C2×C4—34no—
190C2×C4—34no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—62no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—124no—
201C2×C4D3×D414yes—
202C2×C3—42no—
203C2×C3#53612yes—
204C2×C3#54411yescharacteristic
205C2×C3D4211yescharacteristic
206C2×C3D4×Q812yes—
207C2×C3#56311yescharacteristic
208C22—24no—
209C22—22no—
210C22—41no—
211C22—42no—
212C22—41no—
213C22C2×D3×D412yes—
214C22#100411yescharacteristic
215C22C2×D3×Q811yescharacteristic
216C22#101012yes—
217C22#101011yescharacteristic
218C4—24no—
219C4—41no—
220C4—42no—
221C4—41no—
222C4—64no—
223C4—64no—
224C4—64no—
225C4—122no—
226C3#267711yescharacteristic, complemented, Sylow
227C2—42no—
228C2#559011yescharacteristic
229C2#562311yescharacteristic
230C2#563212yes—
231C2#567211yescharacteristic
232C2#567512yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#393C2
prime radicalC24×C3C23
hypercenter#393D3
hypocenterC3#2677
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#944C2semidirect
7. #4657C2semidirect
8. #4656C2semidirect
9. #4652C2semidirect
62. #217Q8semidirect
226. C3#2677semidirect

Maximal subgroups

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

subgroupindex
C2×#9442
C2×#8782
C3×#3932
#49492
#49462
#49452
#46572
#46562
#46522
#26773