Finite Group Databasebeta

← #76132

Subgroups — #76132

Order 384.

476 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 133 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×#389C211yesmaximal, characteristic, complemented
3#4994C212yesmaximal, complemented
4#4993C212yesmaximal, complemented
5#4990C212yesmaximal, complemented
6#4939C211yesmaximal, characteristic
7#4928C211yesmaximal, characteristic, complemented
8#4672C212yesmaximal
9#4647C212yesmaximal, complemented
10#2727—31nomaximal, Sylow
11C22×D12C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C42C2211yescharacteristic
16C2×(C2×D3)⋊C4C2212yes—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4⋊Dic3C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2212yes—
29#898—28no—
30C3×#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×#126—32no—
34#399—32no—
35#396—32no—
36#394—32no—
37#390—31no—
38#389—31no—
39#389—32no—
40#389D311yescharacteristic, complemented
41#388—32no—
42C23×D3C2312yes—
43C22×C3×C4C2312yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C22×Dic3C2312yes—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C2×D12—22no—
52C2×D12—41no—
53C2×D12—41no—
54C3×C4⋊C4—24no—
55C3×C4⋊C4—22no—
56C3×C4⋊C4—41no—
57C3×C42—41no—
58(C2×D3)⋊C4—24no—
59(C2×D3)⋊C4—24no—
60(C2×D3)⋊C4—24no—
61(C2×D3)⋊C4—24no—
62(C2×D3)⋊C4—42no—
63(C2×D3)⋊C4—42no—
64C4⋊Dic3—41no—
65C4⋊Dic3—41no—
66C4⋊Dic3D414yescomplemented
67Dic3⋊C4—24no—
68Dic3⋊C4—24no—
69Dic3⋊C4—42no—
70C4×Dic3—24no—
71C22×D4—31no—
72#126—68no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4—31no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4C2×D312yes—
80C2×C4⋊C4C2×D311yescharacteristic
81C2×C4⋊C4C2×D311yescharacteristic
82C2×C22⋊C4—32no—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C42—32no—
89C2×C42C2×D311yescharacteristic
90#95—32no—
91#95—32no—
92#95C2×D312yes—
93C23×C3C2411yescharacteristic
94C22×D3—22no—
95C22×D3—22no—
96C22×D3—22no—
97C22×D3—22no—
98C22×D3—22no—
99C22×D3—22no—
100C22×D3—22no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4C2×D412yes—
114C2×C3×C4#4112yes—
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—
126C24—32no—
127C2×D4—62no—
128C2×D4—121no—
129C2×D4—121no—
130C22×C4—32no—
131C22×C4—32no—
132C22×C4—32no—
133C22×C4C22×D312yes—
134C22×C4C22×D312yes—
135C22×C4C22×D311yescharacteristic
136C22×C4C22×D311yescharacteristic
137C22×C4C22×D311yescharacteristic
138C4⋊C4—22no—
139C4⋊C4—24no—
140C4⋊C4—34no—
141C4⋊C4—41no—
142C4⋊C4—64no—
143C4⋊C4—64no—
144C4⋊C4—122no—
145C4⋊C4—121no—
146C4⋊C4—121no—
147C22⋊C4—64no—
148C22⋊C4—64no—
149C22⋊C4—64no—
150C22⋊C4—64no—
151C22⋊C4—122no—
152C22⋊C4—122no—
153C42—41no—
154C42—64no—
155C22×C3C22×D411yescharacteristic
156C22×C3C2×#4111yescharacteristic
157C22×C3C2×#4111yescharacteristic
158C22×C3H(5,𝔽2)11yescharacteristic
159C22×C3H(5,𝔽2)11yescharacteristic
160C22×C32−1+411yescharacteristic
161C22×C32−1+411yescharacteristic
162C2×D3—42no—
163C2×D3—42no—
164C2×D3—42no—
165C2×D3—42no—
166C2×D3—42no—
167C2×D3—42no—
168C2×D3—42no—
169C3×C4—22no—
170C3×C4—22no—
171C3×C4—22no—
172C3×C4—42no—
173C3×C4—42no—
174Dic3—24no—
175Dic3—42no—
176Dic3—42no—
177C23—62no—
178C23—62no—
179C23—62no—
180C23—62no—
181C23—62no—
182C23—62no—
183C23—62no—
184C23C23×D311yescharacteristic
185D4—124no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
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#23512yes—
208C2×C4D3×D412yes—
209C2×C4#23912yes—
210C2×C3#53711yescharacteristic
211C2×C3#53811yescharacteristic
212C2×C3#53911yescharacteristic
213C2×C3#54711yescharacteristic
214C2×C3#55011yescharacteristic
215C2×C3#55611yescharacteristic
216C2×C3#56311yescharacteristic
217D3—82no—
218C22—122no—
219C22—122no—
220C22—122no—
221C22—122no—
222C22—122no—
223C22—122no—
224C22—122no—
225C22C2×#23511yescharacteristic
226C22C2×D3×D411yescharacteristic
227C22#100411yescharacteristic
228C22C2×#23911yescharacteristic
229C22#100711yescharacteristic
230C22#100911yescharacteristic
231C22#101011yescharacteristic
232C4—22no—
233C4—22no—
234C4—22no—
235C4—42no—
236C4—42no—
237C4—64no—
238C4—122no—
239C4—122no—
240C3#272711yescharacteristic, complemented, Sylow
241C2—242no—
242C2#551511yescharacteristic
243C2#555311yescharacteristic
244C2#558411yescharacteristic
245C2#559711yescharacteristic
246C2#564411yescharacteristic
247C2#569911yescharacteristic
248C2#572611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC22×D12C22
hypercenter#389D3
hypocenterC3#2727
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#898C2semidirect
2. C3×#389C2semidirect
3. #4994C2semidirect
4. #4993C2semidirect
5. #4990C2semidirect
7. #4928C2semidirect
9. #4647C2semidirect
40. #389D3semidirect
66. C4⋊Dic3D4semidirect
240. C3#2727semidirect

Maximal subgroups

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

subgroupindex
C2×#8982
C3×#3892
#49942
#49932
#49902
#49392
#49282
#46722
#46472
#27273