Finite Group Databasebeta

← #74371

Subgroups — #74371

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×#896C212yesmaximal, complemented
2C2×#894C212yesmaximal
3C3×#384C211yesmaximal, characteristic, complemented
4#4992C212yesmaximal, complemented
5#4991C211yesmaximal, characteristic
6#4989C212yesmaximal, complemented
7#4983C212yesmaximal, complemented
8#4979C211yesmaximal, characteristic, complemented
9#4940C212yesmaximal, complemented
10#2680—31nomaximal, Sylow
11C22×D12C2211yescharacteristic
12C22×C4×D3C2212yes—
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2212yes—
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C42C2211yescharacteristic
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27#896—28no—
28#894—28no—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2211yescharacteristic
32C3⋊#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C2×#123—32no—
35C2×C22⋊Q8—32no—
36#398—32no—
37#397—32no—
38#395—31no—
39#394—31no—
40#388—32no—
41#385—32no—
42#384D311yescharacteristic, complemented
43C23×D3C2312yes—
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C2×D12—24no—
53C2×D12—41no—
54C2×C4×D3—24no—
55C2×C4×D3—24no—
56C2×C4×D3—42no—
57C2×Dic6—24no—
58C2×Dic6—41no—
59C3×C4⋊C4—24no—
60C3×C4⋊C4—24no—
61C3×C4⋊C4—41no—
62C3×C4⋊C4—41no—
63C3×C42—22no—
64(C2×D3)⋊C4—24no—
65(C2×D3)⋊C4—24no—
66(C2×D3)⋊C4—24no—
67(C2×D3)⋊C4—24no—
68(C2×D3)⋊C4D418yes—
69Dic3⋊C4—24no—
70Dic3⋊C4—24no—
71Dic3⋊C4—24no—
72C22×Q8—31no—
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⋊C4C2×D312yes—
81C2×C4⋊C4C2×D312yes—
82C2×C4⋊C4C2×D311yescharacteristic
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×C42C2×D311yescharacteristic
90#95—32no—
91#95—32no—
92#95—32no—
93#95—31no—
94#95—31no—
95C23×C3C2411yescharacteristic
96C22×D3—22no—
97C22×D3—22no—
98C22×D3—22no—
99C22×D3—22no—
100C22×D3—22no—
101C22×D3—22no—
102C22×D3C2×D414yes—
103C2×C3×C4—22no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—21no—
114C2×C3×C4C2×D414yes—
115C2×C3×C4#4112yes—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3C2×D414yes—
125D12—44no—
126C4×D3—48no—
127Dic6—44no—
128C24—32no—
129C2×Q8—64no—
130C2×Q8—121no—
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×D311yescharacteristic
140C22×C4C22×D312yes—
141C22×C4C22×D311yescharacteristic
142C22×C4C22×D311yescharacteristic
143C22×C4C22×D312yes—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—64no—
149C4⋊C4—64no—
150C4⋊C4—64no—
151C22⋊C4—38no—
152C22⋊C4—64no—
153C22⋊C4—64no—
154C22⋊C4—64no—
155C22⋊C4—64no—
156C42—22no—
157C22×C3C22×D412yes—
158C22×C3C2×#4111yescharacteristic
159C22×C3H(5,𝔽2)11yescharacteristic
160C22×C32−1+411yescharacteristic
161C22×C32−1+412yes—
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—41no—
172C3×C4—42no—
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—124no—
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—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
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×C4#23512yes—
209C2×C4D3×D414yes—
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×#23511yescharacteristic
224C22C2×D3×D412yes—
225C22#100411yescharacteristic
226C22#100711yescharacteristic
227C22#100712yes—
228C4—24no—
229C4—22no—
230C4—42no—
231C4—41no—
232C4—41no—
233C4—64no—
234C4—122no—
235C4—122no—
236C3#268011yescharacteristic, complemented, Sylow
237C2—124no—
238C2#550712yes—
239C2#551511yescharacteristic
240C2#571711yescharacteristic
241C2#572412yes—
242C2#573511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#384C2
prime radicalC22×D12C22
hypercenter#384D3
hypocenterC3#2680
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C2×#896C2semidirect
3. C3×#384C2semidirect
4. #4992C2semidirect
6. #4989C2semidirect
7. #4983C2semidirect
8. #4979C2semidirect
9. #4940C2semidirect
42. #384D3semidirect
236. C3#2680semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×#8942
C3×#3842
#49922
#49912
#49892
#49832
#49792
#49402
#26803