Finite Group Databasebeta

← #78008

Subgroups — #78008

Order 384.

456 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 125 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×#898C212yesmaximal, complemented
2C3×#398C211yesmaximal, characteristic, complemented
3#4994C212yesmaximal, complemented
4#4994C212yesmaximal, complemented
5#4991C211yesmaximal, characteristic
6#4975C211yesmaximal, characteristic, complemented
7#4676C212yesmaximal, complemented
8#4672C212yesmaximal
9#4657C212yesmaximal, complemented
10#2823—31nomaximal, Sylow
11C22×D12C2211yescharacteristic
12C2×C3×C4⋊C4C2212yes—
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×(C2×D3)⋊C4C2212yes—
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×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28#898—28no—
29C3×#95C2212yes—
30C3×#95C2211yescharacteristic
31C3⋊#95C2211yescharacteristic
32C3⋊#95C2212yes—
33C2×#126—32no—
34#398—32no—
35#398D311yescharacteristic, complemented
36#396—32no—
37#396—32no—
38#396—32no—
39#394—31no—
40#394—32no—
41#384—31no—
42C23×D3C2312yes—
43C22×C3×C4C2311yescharacteristic
44C22×C3×C4C2312yes—
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C2×D12—42no—
52C2×D12—41no—
53C3×C4⋊C4—24no—
54C3×C4⋊C4—41no—
55C3×C4⋊C4—41no—
56(C2×D3)⋊C4—24no—
57(C2×D3)⋊C4—24no—
58(C2×D3)⋊C4—24no—
59(C2×D3)⋊C4—42no—
60(C2×D3)⋊C4—42no—
61(C2×D3)⋊C4—42no—
62C4⋊Dic3—24no—
63C4⋊Dic3—41no—
64Dic3⋊C4—24no—
65Dic3⋊C4—42no—
66Dic3⋊C4—42no—
67Dic3⋊C4—42no—
68C4×Dic3D414yescomplemented
69C22×D4—31no—
70#126—68no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—32no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—31no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4C2×D312yes—
78C2×C4⋊C4C2×D311yescharacteristic
79C2×C4⋊C4C2×D311yescharacteristic
80C2×C22⋊C4—32no—
81C2×C22⋊C4—32no—
82C2×C22⋊C4—32no—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C42—31no—
87#95—31no—
88#95—32no—
89#95C2×D312yes—
90#95C2×D311yescharacteristic
91C23×C3C2411yescharacteristic
92C22×D3—22no—
93C22×D3—22no—
94C22×D3—22no—
95C22×D3—22no—
96C22×D3—22no—
97C22×D3—22no—
98C22×D3—22no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
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×C4—22no—
111C2×C3×C4C2×D412yes—
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—
122C24—32no—
123C2×D4—121no—
124C2×D4—122no—
125C22×C4—32no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4C22×D311yescharacteristic
129C22×C4C22×D312yes—
130C22×C4C22×D312yes—
131C22×C4C22×D311yescharacteristic
132C22×C4C22×D311yescharacteristic
133C4⋊C4—24no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—64no—
137C4⋊C4—64no—
138C4⋊C4—122no—
139C4⋊C4—122no—
140C4⋊C4—122no—
141C4⋊C4—121no—
142C22⋊C4—64no—
143C22⋊C4—64no—
144C22⋊C4—64no—
145C22⋊C4—122no—
146C22⋊C4—122no—
147C22⋊C4—122no—
148C42—34no—
149C22×C3C22×D411yescharacteristic
150C22×C3H(5,𝔽2)11yescharacteristic
151C22×C3H(5,𝔽2)12yes—
152C22×C32−1+412yes—
153C22×C32−1+411yescharacteristic
154C2×D3—42no—
155C2×D3—42no—
156C2×D3—42no—
157C2×D3—42no—
158C2×D3—42no—
159C2×D3—42no—
160C2×D3—42no—
161C3×C4—22no—
162C3×C4—42no—
163C3×C4—42no—
164C3×C4—41no—
165C3×C4—41no—
166Dic3—24no—
167Dic3—42no—
168Dic3—42no—
169C23—62no—
170C23—62no—
171C23—62no—
172C23—62no—
173C23—62no—
174C23—62no—
175C23—62no—
176C23C23×D311yescharacteristic
177D4—124no—
178C2×C4—21no—
179C2×C4—21no—
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—22no—
189C2×C4—22no—
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×C4D3×D412yes—
200C2×C3#53711yescharacteristic
201C2×C3#53712yes—
202C2×C3#56011yescharacteristic
203C2×C3#56312yes—
204C2×C3#56311yescharacteristic
205D3—82no—
206C22—122no—
207C22—122no—
208C22—122no—
209C22—122no—
210C22—122no—
211C22—122no—
212C22—122no—
213C22C2×D3×D411yescharacteristic
214C22#100411yescharacteristic
215C22#100712yes—
216C22#100912yes—
217C22#101011yescharacteristic
218C4—22no—
219C4—41no—
220C4—41no—
221C4—42no—
222C4—42no—
223C4—64no—
224C4—122no—
225C4—122no—
226C3#282311yescharacteristic, complemented, Sylow
227C2—242no—
228C2#559711yescharacteristic
229C2#562912yes—
230C2#566611yescharacteristic
231C2#569912yes—
232C2#573511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#398C2
prime radicalC22×D12C22
hypercenter#398D3
hypocenterC3#2823
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#898C2semidirect
2. C3×#398C2semidirect
3. #4994C2semidirect
4. #4994C2semidirect
6. #4975C2semidirect
7. #4676C2semidirect
9. #4657C2semidirect
35. #398D3semidirect
68. C4×Dic3D4semidirect
226. C3#2823semidirect

Maximal subgroups

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

subgroupindex
C2×#8982
C3×#3982
#49942
#49912
#49752
#46762
#46722
#46572
#28233