Finite Group Databasebeta

← #76693

Subgroups — #76693

Order 384.

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

normal subgroups only

This group has 604 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×D4×Dic3C212yesmaximal, complemented
2C3×#392C211yesmaximal, characteristic
3#5222C212yesmaximal, complemented
4#5220C212yesmaximal, complemented
5#4953C211yesmaximal, characteristic, complemented
6#4950C212yesmaximal, complemented
7#4945C212yesmaximal, complemented
8#4943C211yesmaximal, characteristic, complemented
9#4655C212yesmaximal, complemented
10#2600—31nomaximal, Sylow
11C22×C3×D4C2212yes—
12C22×C3×D4C2211yescharacteristic
13C23×Dic3C2212yes—
14C23×Dic3C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2212yes—
22D4×Dic3—28no—
23C2×C4⋊Dic3C2212yes—
24C2×C4⋊Dic3C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3C2212yes—
27C3×#95C2211yescharacteristic
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2211yescharacteristic
33C2×C4×D4—32no—
34#397—31no—
35#394—32no—
36#392D311yescharacteristic
37#388—32no—
38#387—32no—
39#386—32no—
40#385—32no—
41#381—31no—
42C24×C3C2311yescharacteristic
43C24×C3C2312yes—
44C2×C3×D4—24no—
45C2×C3×D4—24no—
46C2×C3×D4—24no—
47C2×C3×D4—24no—
48C2×C3×D4—41no—
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×Dic3—24no—
53C22×Dic3—41no—
54C22×Dic3—41no—
55C22×Dic3—42no—
56C22×Dic3—42no—
57C22×Dic3D414yescomplemented
58C22×Dic3C2312yes—
59C22×Dic3C2312yes—
60C22×Dic3C2312yes—
61C22×Dic3C2312yes—
62C3×C22⋊C4—24no—
63C3×C22⋊C4—41no—
64#217—24no—
65#217—24no—
66#217—24no—
67#217—41no—
68#217—42no—
69C4⋊Dic3—24no—
70C4⋊Dic3—41no—
71Dic3⋊C4—24no—
72C4×Dic3—24no—
73C22×D4C2×D312yes—
74C22×D4C2×D311yescharacteristic
75C23×C4—31no—
76C23×C4—32no—
77C4×D4—68no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—31no—
81C2×C22⋊C4—32no—
82C2×C22⋊C4—32no—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—31no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4C2×D312yes—
87C2×C22⋊C4C2×D311yescharacteristic
88C2×C42—32no—
89#95—32no—
90#95—32no—
91#95—32no—
92#95—32no—
93#95—31no—
94#95C2×D311yescharacteristic
95C23×C3—22no—
96C23×C3—22no—
97C23×C3—22no—
98C23×C3—22no—
99C23×C3—22no—
100C23×C3—22no—
101C23×C3—21no—
102C23×C3—22no—
103C23×C3—21no—
104C23×C3—22no—
105C23×C3C2×D412yes—
106C23×C3#4114yes—
107C23×C3C2411yescharacteristic
108C3×D4—216no—
109C3×D4—44no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4#4114yes—
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×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—44no—
129C2×Dic3—48no—
130C2×Dic3C2×D414yes—
131C24C22×D311yescharacteristic
132C24C22×D312yes—
133C2×D4—24no—
134C2×D4—24no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—41no—
138C22×C4—32no—
139C22×C4—34no—
140C22×C4—32no—
141C22×C4—32no—
142C22×C4—32no—
143C22×C4—64no—
144C22×C4—121no—
145C22×C4—122no—
146C22×C4—121no—
147C22×C4—122no—
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D311yescharacteristic
150C22×C4C22×D312yes—
151C4⋊C4—64no—
152C4⋊C4—64no—
153C4⋊C4—121no—
154C22⋊C4—24no—
155C22⋊C4—41no—
156C22⋊C4—64no—
157C22⋊C4—64no—
158C22⋊C4—64no—
159C22⋊C4—121no—
160C22⋊C4—122no—
161C42—64no—
162C22×C3—24no—
163C22×C3—24no—
164C22×C3—22no—
165C22×C3—24no—
166C22×C3—24no—
167C22×C3—42no—
168C22×C3—42no—
169C22×C3—41no—
170C22×C3—42no—
171C22×C3—42no—
172C22×C3—42no—
173C22×C3—41no—
174C22×C3C22×D411yescharacteristic
175C22×C3C2×#4112yes—
176C22×C3C2×#4112yes—
177C22×C3H(5,𝔽2)11yescharacteristic
178C22×C3H(5,𝔽2)11yescharacteristic
179C3×C4—24no—
180C3×C4—41no—
181C3×C4—41no—
182Dic3—24no—
183Dic3—42no—
184Dic3—42no—
185Dic3—42no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—21no—
193C23—22no—
194C23—21no—
195C23—22no—
196C23D3×D412yes—
197C23#23714yes—
198C23C23×D311yescharacteristic
199D4—216no—
200D4—44no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—34no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—124no—
220C2×C4—128no—
221C2×C4#23714yes—
222C2×C3—42no—
223C2×C3—44no—
224C2×C3#52512yes—
225C2×C3#53411yescharacteristic
226C2×C3#55111yescharacteristic
227C2×C3#55312yes—
228C2×C3#55911yescharacteristic
229C22—24no—
230C22—24no—
231C22—24no—
232C22—22no—
233C22—24no—
234C22—42no—
235C22—41no—
236C22—42no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22C2×D3×D411yescharacteristic
242C22C2×#23712yes—
243C22C2×#23712yes—
244C22#100411yescharacteristic
245C22#100911yescharacteristic
246C4—24no—
247C4—41no—
248C4—41no—
249C4—64no—
250C4—122no—
251C4—122no—
252C4—122no—
253C3#260011yescharacteristic, complemented, Sylow
254C2—42no—
255C2—44no—
256C2#558612yes—
257C2#558711yescharacteristic
258C2#560511yescharacteristic
259C2#561812yes—
260C2#571311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC3×#392C2
hypercenter#392D3
hypocenterC3#2600
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic3C2semidirect
3. #5222C2semidirect
4. #5220C2semidirect
5. #4953C2semidirect
6. #4950C2semidirect
7. #4945C2semidirect
8. #4943C2semidirect
9. #4655C2semidirect
57. C22×Dic3D4semidirect
253. C3#2600semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
C3×#3922
#52222
#52202
#49532
#49502
#49452
#49432
#46552
#26003