Finite Group Databasebeta

← #77262

Subgroups — #77262

Order 384.

568 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 141 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×#946C212yesmaximal, complemented
2C2×#894C212yesmaximal
3C3×#395C211yesmaximal, characteristic, complemented
4#4991C211yesmaximal, characteristic
5#4982C212yesmaximal, complemented
6#4975C211yesmaximal, characteristic, complemented
7#4675C212yesmaximal, complemented
8#4670C212yesmaximal, complemented
9#4668C212yesmaximal, complemented
10#2680—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×D12C2211yescharacteristic
13C22×C4×D3C2212yes—
14C2×C3×C4⋊C4C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic
16#946—28no—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×C4⋊Dic3C2212yes—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28#894—28no—
29C3×#95C2212yes—
30C3×#95C2211yescharacteristic
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2211yescharacteristic
34C2×#123—32no—
35C2×C22⋊Q8—32no—
36#398—32no—
37#397—32no—
38#395D311yescharacteristic, complemented
39#394—31no—
40#388—32no—
41#385—32no—
42#384—31no—
43C23×D3C2312yes—
44C2×C3×Q8—24no—
45C2×C3×Q8—41no—
46C22×C3×C4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C2×D12—24no—
55C2×D12—41no—
56C2×C4×D3—24no—
57C2×C4×D3—24no—
58C2×C4×D3—42no—
59C3×C4⋊C4—24no—
60C3×C4⋊C4—41no—
61(C2×D3)⋊C4—24no—
62(C2×D3)⋊C4—24no—
63(C2×D3)⋊C4—24no—
64(C2×D3)⋊C4—24no—
65(C2×D3)⋊C4D418yes—
66C4⋊Dic3—24no—
67C4⋊Dic3—41no—
68Dic3⋊C4—24no—
69Dic3⋊C4—24no—
70Dic3⋊C4—24no—
71C4×Dic3—22no—
72C22×Q8C2×D311yescharacteristic
73C22×D4—31no—
74C23×C4—32no—
75#123—68no—
76C22⋊Q8—68no—
77C2×C4⋊C4—32no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4C2×D312yes—
83C2×C4⋊C4C2×D311yescharacteristic
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C42—31no—
90#95—32no—
91#95—32no—
92#95—31no—
93#95C2×D312yes—
94#95C2×D311yescharacteristic
95C23×C3C2411yescharacteristic
96C22×D3—22no—
97C22×D3—22no—
98C22×D3—22no—
99C22×D3—22no—
100C22×D3—22no—
101C22×D3—22no—
102C22×D3C2×D414yes—
103C3×Q8—44no—
104C2×C3×C4—22no—
105C2×C3×C4—21no—
106C2×C3×C4—22no—
107C2×C3×C4—21no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4C2×D414yes—
116C2×C3×C4#4112yes—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3C2×D414yes—
126D12—44no—
127C4×D3—48no—
128C24—32no—
129C2×Q8—24no—
130C2×Q8—41no—
131C2×D4—64no—
132C2×D4—121no—
133C22×C4—32no—
134C22×C4—32no—
135C22×C4—32no—
136C22×C4—64no—
137C22×C4—64no—
138C22×C4—122no—
139C22×C4C22×D312yes—
140C22×C4C22×D311yescharacteristic
141C22×C4C22×D311yescharacteristic
142C22×C4C22×D312yes—
143C22×C4C22×D311yescharacteristic
144C4⋊C4—24no—
145C4⋊C4—41no—
146C4⋊C4—64no—
147C4⋊C4—64no—
148C4⋊C4—64no—
149C4⋊C4—64no—
150C4⋊C4—121no—
151C22⋊C4—38no—
152C22⋊C4—64no—
153C22⋊C4—64no—
154C22⋊C4—64no—
155C22⋊C4—64no—
156C42—62no—
157C22×C3C22×D412yes—
158C22×C3C2×#4111yescharacteristic
159C22×C3H(5,𝔽2)11yescharacteristic
160C22×C32−1+412yes—
161C22×C32−1+411yescharacteristic
162C2×D3—24no—
163C2×D3—24no—
164C2×D3—24no—
165C2×D3—42no—
166C2×D3—42no—
167C2×D3—42no—
168C2×D3—42no—
169C3×C4—22no—
170C3×C4—24no—
171C3×C4—42no—
172C3×C4—41no—
173C3×C4—41no—
174Dic3—24no—
175Dic3—42no—
176Dic3—42no—
177C23—34no—
178C23—62no—
179C23—62no—
180C23—62no—
181C23—62no—
182C23—62no—
183C23—62no—
184C23C23×D311yescharacteristic
185Q8—44no—
186D4—124no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—34no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—128no—
208C2×C4D3×D414yes—
209C2×C4#23912yes—
210C2×C3#53612yes—
211C2×C3#53911yescharacteristic
212C2×C3D4211yescharacteristic
213C2×C3#54712yes—
214C2×C3#56311yescharacteristic
215D3—44no—
216C22—64no—
217C22—64no—
218C22—64no—
219C22—122no—
220C22—122no—
221C22—122no—
222C22—122no—
223C22C2×D3×D412yes—
224C22#100411yescharacteristic
225C22C2×#23911yescharacteristic
226C22#100712yes—
227C22#101011yescharacteristic
228C4—22no—
229C4—24no—
230C4—41no—
231C4—41no—
232C4—42no—
233C4—64no—
234C4—122no—
235C4—122no—
236C3#268011yescharacteristic, complemented, Sylow
237C2—124no—
238C2#559011yescharacteristic
239C2#562812yes—
240C2#564511yescharacteristic
241C2#572612yes—
242C2#573511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC22×D12C22
hypercenter#395D3
hypocenterC3#2680
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#946C2semidirect
3. C3×#395C2semidirect
5. #4982C2semidirect
6. #4975C2semidirect
7. #4675C2semidirect
8. #4670C2semidirect
9. #4668C2semidirect
38. #395D3semidirect
236. C3#2680semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
C2×#8942
C3×#3952
#49912
#49822
#49752
#46752
#46702
#46682
#26803