Finite Group Databasebeta

← #76464

Subgroups — #76464

Order 384.

1000 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 157 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×#937C212yesmaximal, complemented
2C2×#882C214yesmaximal, complemented
3C3×#390C211yesmaximal, characteristic, complemented
4#4991C212yesmaximal, complemented
5#4961C214yesmaximal, complemented
6#4939C212yesmaximal, complemented
7#2679—31nomaximal, Sylow
8C24×D3C2212yescomplemented
9C22×C3×D4C2211yescharacteristic
10C22×#67C2214yes—
11C22×D12C2212yes—
12C22×D12C2212yes—
13C2×C3×C4⋊C4C2211yescharacteristic, complemented
14C2×C3×C22⋊C4C2214yescomplemented
15C2×C3×C42C2211yescharacteristic, complemented
16C2×#217C2212yescomplemented
17#937—28no—
18C2×(C2×D3)⋊C4C2214yescomplemented
19C2×(C2×D3)⋊C4C2214yescomplemented
20C2×(C2×D3)⋊C4C2214yescomplemented
21C2×C4⋊Dic3C2212yescomplemented
22#882—216no—
23C3⋊#95C2212yescomplemented
24C2×#120—32no—
25C2×#120—34no—
26#394—32no—
27#394—34no—
28#390—32no—
29#390D311yescharacteristic, complemented
30C24×C3C2312yes—
31C23×D3—24no—
32C23×D3—28no—
33C23×D3—42no—
34C23×D3—42no—
35C23×D3—44no—
36C23×D3C2314yes—
37C2×C3×D4—41no—
38C2×C3×D4—41no—
39C2×C3×D4D414yescomplemented
40C22×C3×C4C2312yes—
41C22×C3×C4C2312yes—
42C22×C3×C4C2311yescharacteristic
43C2×#67—28no—
44C2×#67—28no—
45C2×#67—44no—
46C22×Dic3C2314yes—
47C2×D12—28no—
48C2×D12—42no—
49C2×D12—42no—
50C2×D12—42no—
51C2×D12D418yescomplemented
52C3×C4⋊C4—41no—
53C3×C22⋊C4—28no—
54C3×C42—41no—
55#217—24no—
56(C2×D3)⋊C4—28no—
57(C2×D3)⋊C4—28no—
58(C2×D3)⋊C4—28no—
59C4⋊Dic3—42no—
60C25—32no—
61C22×D4—34no—
62C22×D4—32no—
63C22×D4—32no—
64C22×D4C2×D311yescharacteristic
65#120—616no—
66#120—68no—
67C2×C4⋊C4—32no—
68C2×C4⋊C4C2×D311yescharacteristic, complemented
69C2×C22⋊C4—32no—
70C2×C22⋊C4—34no—
71C2×C22⋊C4—34no—
72C2×C22⋊C4—34no—
73C2×C22⋊C4C2×D314yescomplemented
74C2×C42C2×D311yescharacteristic, complemented
75#95—32no—
76C23×C3—24no—
77C23×C3—24no—
78C23×C3—22no—
79C23×C3—22no—
80C23×C3C2×D414yes—
81C23×C3C2411yescharacteristic
82C22×D3—24no—
83C22×D3—24no—
84C22×D3—24no—
85C22×D3—24no—
86C22×D3—24no—
87C22×D3—24no—
88C22×D3—48no—
89C22×D3—416no—
90C22×D3—416no—
91C22×D3—48no—
92C22×D3—48no—
93C22×D3C2×D418yes—
94C3×D4—44no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—24no—
100C2×C3×C4—22no—
101C2×C3×C4C2×D412yes—
102C2×C3×C4C2×D414yes—
103#67—416no—
104C2×Dic3—24no—
105C2×Dic3—24no—
106C2×Dic3—24no—
107D12—48no—
108D12—48no—
109C24—34no—
110C24—68no—
111C24—64no—
112C24—122no—
113C24—122no—
114C24—124no—
115C24C22×D312yes—
116C2×D4—38no—
117C2×D4—41no—
118C2×D4—41no—
119C2×D4—68no—
120C2×D4—68no—
121C2×D4—68no—
122C2×D4—122no—
123C2×D4—122no—
124C2×D4—122no—
125C2×D4—124no—
126C2×D4D1214yescomplemented
127C22×C4—34no—
128C22×C4C22×D312yes—
129C22×C4C22×D311yescharacteristic
130C22×C4C22×D312yes—
131C4⋊C4—41no—
132C4⋊C4—122no—
133C22⋊C4—28no—
134C22⋊C4—68no—
135C22⋊C4—64no—
136C22⋊C4—68no—
137C22⋊C4—68no—
138C42—41no—
139C22×C3—24no—
140C22×C3—28no—
141C22×C3—44no—
142C22×C3—42no—
143C22×C3—42no—
144C22×C3C22×D411yescharacteristic
145C22×C3C22×D412yes—
146C22×C3H(5,𝔽2)11yescharacteristic
147C22×C3H(5,𝔽2)11yescharacteristic
148C22×C3H(5,𝔽2)12yes—
149C2×D3—28no—
150C2×D3—28no—
151C2×D3—28no—
152C2×D3—432no—
153C2×D3—44no—
154C2×D3—44no—
155C2×D3—44no—
156C2×D3—44no—
157C3×C4—24no—
158C3×C4—22no—
159C3×C4—42no—
160Dic3—44no—
161C23—24no—
162C23—22no—
163C23—22no—
164C23—24no—
165C23—38no—
166C23—64no—
167C23—64no—
168C23—64no—
169C23—64no—
170C23—64no—
171C23—64no—
172C23—1216no—
173C23—128no—
174C23—128no—
175C23—1216no—
176C23—128no—
177C23C2×D1214yes—
178C23C23×D311yescharacteristic
179D4—44no—
180D4—128no—
181D4—128no—
182D4—1216no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—64no—
190C2×C4—64no—
191C2×C4—64no—
192C2×C4C2×D1212yes—
193C2×C4D3×D414yes—
194C2×C3—44no—
195C2×C3#53411yescharacteristic
196C2×C3#53412yes—
197C2×C3D4211yescharacteristic
198C2×C3D4212yes—
199C2×C3H(3,𝔽4)11yescharacteristic
200D3—48no—
201C22—24no—
202C22—28no—
203C22—42no—
204C22—42no—
205C22—44no—
206C22—68no—
207C22—68no—
208C22—68no—
209C22—1232no—
210C22—124no—
211C22—124no—
212C22—124no—
213C22—124no—
214C22C22×D1211yescharacteristic
215C22C2×D3×D412yes—
216C22#100411yescharacteristic
217C22#100411yescharacteristic
218C22#100912yes—
219C4—22no—
220C4—24no—
221C4—42no—
222C4—124no—
223C3#267911yescharacteristic, complemented, Sylow
224C2—44no—
225C2—128no—
226C2#549311yescharacteristic
227C2D4×D1212yes—
228C2#560512yes—
229C2#568311yescharacteristic
230C2#571711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
hypercenter#390D3
hypocenterC3#2679
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#937C2semidirect
2. C2×#882C2semidirect
3. C3×#390C2semidirect
4. #4991C2semidirect
5. #4961C2semidirect
6. #4939C2semidirect
8. C24×D3C22semidirect
13. C2×C3×C4⋊C4C22semidirect
14. C2×C3×C22⋊C4C22semidirect
15. C2×C3×C42C22semidirect
16. C2×#217C22semidirect
18. C2×(C2×D3)⋊C4C22semidirect
19. C2×(C2×D3)⋊C4C22semidirect
20. C2×(C2×D3)⋊C4C22semidirect
21. C2×C4⋊Dic3C22semidirect
23. C3⋊#95C22semidirect
29. #390D3semidirect
39. C2×C3×D4D4semidirect
51. C2×D12D4semidirect
68. C2×C4⋊C4C2×D3semidirect
73. C2×C22⋊C4C2×D3semidirect
74. C2×C42C2×D3semidirect
126. C2×D4D12semidirect
223. C3#2679semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×#8822
C3×#3902
#49912
#49612
#49392
#26793