Finite Group Databasebeta

← #77815

Subgroups — #77815

Order 384.

540 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 137 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×#892C212yesmaximal, complemented
2C3×#398C211yesmaximal, characteristic, complemented
3#4994C212yesmaximal, complemented
4#4991C211yesmaximal, characteristic
5#4990C212yesmaximal, complemented
6#4977C211yesmaximal, characteristic, complemented
7#4668C212yesmaximal, complemented
8#4665C212yesmaximal
9#4655C212yesmaximal, complemented
10#2609—31nomaximal, Sylow
11C22×D12C2211yescharacteristic
12C22×C4×D3C2212yes—
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×(C2×D3)⋊C4C2212yes—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2212yes—
24C2×C4×Dic3C2212yes—
25C2×C4×Dic3C2211yescharacteristic
26#892—28no—
27C3×#95C2212yes—
28C3×#95C2211yescharacteristic
29C3⋊#95C2211yescharacteristic
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×#117—32no—
34#398D311yescharacteristic, complemented
35#396—32no—
36#394—31no—
37#388—32no—
38#387—32no—
39#386—32no—
40#385—32no—
41#382—31no—
42C23×D3C2312yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2311yescharacteristic
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C2×D12—24no—
52C2×D12—41no—
53C2×C4×D3—24no—
54C2×C4×D3—24no—
55C2×C4×D3—24no—
56C3×C4⋊C4—24no—
57C3×C4⋊C4—41no—
58C3×C4⋊C4—41no—
59(C2×D3)⋊C4—24no—
60(C2×D3)⋊C4—24no—
61(C2×D3)⋊C4—24no—
62(C2×D3)⋊C4—24no—
63(C2×D3)⋊C4—42no—
64C4⋊Dic3—24no—
65C4⋊Dic3—41no—
66Dic3⋊C4—24no—
67C4×Dic3—24no—
68C4×Dic3D414yescomplemented
69C22×D4—31no—
70C23×C4—32no—
71#117—68no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—31no—
75C2×C4⋊C4C2×D312yes—
76C2×C4⋊C4C2×D311yescharacteristic
77C2×C4⋊C4C2×D311yescharacteristic
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C22⋊C4—32no—
82C2×C22⋊C4—32no—
83C2×C42—31no—
84C2×C42—32no—
85#95—32no—
86#95—32no—
87#95—31no—
88#95—32no—
89#95C2×D311yescharacteristic
90#95C2×D312yes—
91C23×C3C2411yescharacteristic
92C22×D3—22no—
93C22×D3—22no—
94C22×D3—22no—
95C22×D3—22no—
96C22×D3—22no—
97C22×D3—22no—
98C22×D3#4114yes—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—22no—
110C2×C3×C4C2×D412yes—
111C2×C3×C4#4114yes—
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×Dic3C2×D414yes—
121D12—44no—
122C4×D3—216no—
123C24—32no—
124C2×D4—64no—
125C2×D4—121no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4—32no—
129C22×C4—64no—
130C22×C4—64no—
131C22×C4—64no—
132C22×C4C22×D311yescharacteristic
133C22×C4C22×D311yescharacteristic
134C22×C4C22×D312yes—
135C22×C4C22×D311yescharacteristic
136C22×C4C22×D312yes—
137C4⋊C4—24no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—64no—
141C4⋊C4—64no—
142C4⋊C4—121no—
143C22⋊C4—64no—
144C22⋊C4—64no—
145C22⋊C4—64no—
146C22⋊C4—64no—
147C22⋊C4—122no—
148C42—34no—
149C42—64no—
150C22×C3C22×D411yescharacteristic
151C22×C3C2×#4112yes—
152C22×C3C2×#4112yes—
153C22×C3H(5,𝔽2)11yescharacteristic
154C22×C32−1+411yescharacteristic
155C2×D3—24no—
156C2×D3—24no—
157C2×D3—24no—
158C2×D3—42no—
159C2×D3—42no—
160C2×D3—42no—
161C2×D3—42no—
162C3×C4—24no—
163C3×C4—22no—
164C3×C4—42no—
165C3×C4—41no—
166C3×C4—41no—
167Dic3—24no—
168Dic3—42no—
169Dic3—42no—
170C23—34no—
171C23—62no—
172C23—62no—
173C23—62no—
174C23—62no—
175C23—62no—
176C23—62no—
177C23C23×D311yescharacteristic
178D4—124no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—34no—
191C2×C4—616no—
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×C4D3×D412yes—
201C2×C4#23914yes—
202C2×C3#52512yes—
203C2×C3#53711yescharacteristic
204C2×C3#55111yescharacteristic
205C2×C3#55212yes—
206C2×C3#55511yescharacteristic
207D3—44no—
208C22—64no—
209C22—64no—
210C22—64no—
211C22—122no—
212C22—122no—
213C22—122no—
214C22—122no—
215C22C2×D3×D411yescharacteristic
216C22#100411yescharacteristic
217C22C2×#23912yes—
218C22D3×#4112yes—
219C22#101011yescharacteristic
220C4—22no—
221C4—24no—
222C4—41no—
223C4—41no—
224C4—42no—
225C4—64no—
226C4—122no—
227C4—122no—
228C3#260911yescharacteristic, complemented, Sylow
229C2—124no—
230C2#559711yescharacteristic
231C2#562712yes—
232C2#565911yescharacteristic
233C2#569412yes—
234C2#573411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#398C2
prime radicalC22×D12C22
hypercenter#398D3
hypocenterC3#2609
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#892C2semidirect
2. C3×#398C2semidirect
3. #4994C2semidirect
5. #4990C2semidirect
6. #4977C2semidirect
7. #4668C2semidirect
9. #4655C2semidirect
34. #398D3semidirect
68. C4×Dic3D4semidirect
228. C3#2609semidirect

Maximal subgroups

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

subgroupindex
C2×#8922
C3×#3982
#49942
#49912
#49902
#49772
#46682
#46652
#46552
#26093