Finite Group Databasebeta

← #76674

Subgroups — #76674

Order 384.

660 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 149 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×#935C212yesmaximal, complemented
2C2×D4×Dic3C212yesmaximal, complemented
3C3×#392C211yesmaximal, characteristic
4#5222C212yesmaximal, complemented
5#4950C212yesmaximal, complemented
6#4949C211yesmaximal, characteristic, complemented
7#4946C211yesmaximal, characteristic, complemented
8#4945C212yesmaximal, complemented
9#4647C212yesmaximal, complemented
10#2455—31nomaximal, Sylow
11C22×C3×D4C2212yes—
12C22×C3×D4C2211yescharacteristic
13C23×Dic3C2211yescharacteristic
14C23×Dic3C2212yes—
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2211yescharacteristic
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×#217C2212yes—
22#935—28no—
23D4×Dic3—28no—
24C2×C4⋊Dic3C2212yes—
25C2×C4⋊Dic3C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2212yes—
30C3×#95C2211yescharacteristic
31C3⋊#95C2212yes—
32C3⋊#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C2×#123—32no—
35C2×C4×D4—32no—
36#399—31no—
37#394—32no—
38#392D311yescharacteristic
39#389—32no—
40#388—32no—
41#385—31no—
42#385—32no—
43C24×C3C2312yes—
44C24×C3C2311yescharacteristic
45C2×C3×D4—24no—
46C2×C3×D4—24no—
47C2×C3×D4—24no—
48C2×C3×D4—24no—
49C2×C3×D4—41no—
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×Dic3—24no—
54C22×Dic3—24no—
55C22×Dic3—24no—
56C22×Dic3—41no—
57C22×Dic3—42no—
58C22×Dic3C2312yes—
59C22×Dic3C2312yes—
60C22×Dic3C2312yes—
61C22×Dic3C2312yes—
62C3×C22⋊C4—24no—
63C3×C22⋊C4—41no—
64#217—24no—
65#217—22no—
66#217—24no—
67#217—24no—
68#217D418yescomplemented
69C4⋊Dic3—24no—
70C4⋊Dic3—41no—
71Dic3⋊C4—24no—
72Dic3⋊C4—24no—
73Dic3⋊C4—24no—
74C4×Dic3—24no—
75C22×D4C2×D312yes—
76C22×D4C2×D311yescharacteristic
77C23×C4—31no—
78C23×C4—32no—
79#123—68no—
80C4×D4—68no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—31no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4—31no—
91C2×C22⋊C4C2×D312yes—
92C2×C22⋊C4C2×D311yescharacteristic
93C2×C42—32no—
94#95—32no—
95#95—32no—
96#95—31no—
97#95C2×D311yescharacteristic
98C23×C3—22no—
99C23×C3—22no—
100C23×C3—22no—
101C23×C3—22no—
102C23×C3—22no—
103C23×C3—22no—
104C23×C3—21no—
105C23×C3—22no—
106C23×C3—21no—
107C23×C3—22no—
108C23×C3C2×D414yes—
109C23×C3#4112yes—
110C23×C3C2411yescharacteristic
111C3×D4—216no—
112C3×D4—44no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4#4114yes—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—44no—
131C2×Dic3—48no—
132C2×Dic3C2×D414yes—
133C2×Dic3C2×D414yes—
134C24C22×D312yes—
135C24C22×D311yescharacteristic
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—41no—
141C22×C4—32no—
142C22×C4—32no—
143C22×C4—32no—
144C22×C4—32no—
145C22×C4—64no—
146C22×C4—64no—
147C22×C4—64no—
148C22×C4—121no—
149C22×C4—122no—
150C22×C4C22×D311yescharacteristic
151C22×C4C22×D311yescharacteristic
152C22×C4C22×D312yes—
153C4⋊C4—64no—
154C4⋊C4—64no—
155C4⋊C4—64no—
156C4⋊C4—64no—
157C4⋊C4—121no—
158C22⋊C4—24no—
159C22⋊C4—38no—
160C22⋊C4—41no—
161C22⋊C4—64no—
162C22⋊C4—64no—
163C22⋊C4—64no—
164C22⋊C4—62no—
165C42—64no—
166C22×C3—24no—
167C22×C3—24no—
168C22×C3—22no—
169C22×C3—24no—
170C22×C3—24no—
171C22×C3—42no—
172C22×C3—42no—
173C22×C3—41no—
174C22×C3—42no—
175C22×C3—42no—
176C22×C3—42no—
177C22×C3—41no—
178C22×C3C22×D412yes—
179C22×C3C2×#4111yescharacteristic
180C22×C3C2×#4112yes—
181C22×C3H(5,𝔽2)11yescharacteristic
182C22×C32−1+411yescharacteristic
183C3×C4—24no—
184C3×C4—41no—
185C3×C4—41no—
186Dic3—24no—
187Dic3—24no—
188Dic3—42no—
189Dic3—42no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—21no—
197C23—22no—
198C23—21no—
199C23—22no—
200C23D3×D414yes—
201C23#23712yes—
202C23C23×D311yescharacteristic
203D4—216no—
204D4—44no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—34no—
212C2×C4—34no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—124no—
224C2×C4—128no—
225C2×C4#23714yes—
226C2×C3—44no—
227C2×C3—42no—
228C2×C3#52512yes—
229C2×C3#53911yescharacteristic
230C2×C3D4211yescharacteristic
231C2×C3#54712yes—
232C2×C3#55911yescharacteristic
233C22—24no—
234C22—24no—
235C22—24no—
236C22—22no—
237C22—24no—
238C22—42no—
239C22—41no—
240C22—42no—
241C22—41no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22C2×D3×D412yes—
246C22C2×#23712yes—
247C22C2×#23711yescharacteristic
248C22#100411yescharacteristic
249C22#101011yescharacteristic
250C4—24no—
251C4—41no—
252C4—41no—
253C4—64no—
254C4—64no—
255C4—122no—
256C4—122no—
257C3#245511yescharacteristic, complemented, Sylow
258C2—44no—
259C2—42no—
260C2#558612yes—
261C2#559011yescharacteristic
262C2#560211yescharacteristic
263C2#561112yes—
264C2#571311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC3×#392C2
hypercenter#392D3
hypocenterC3#2455
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#935C2semidirect
2. C2×D4×Dic3C2semidirect
4. #5222C2semidirect
5. #4950C2semidirect
6. #4949C2semidirect
7. #4946C2semidirect
8. #4945C2semidirect
9. #4647C2semidirect
68. #217D4semidirect
257. C3#2455semidirect

Maximal subgroups

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

subgroupindex
C2×#9352
C2×D4×Dic32
C3×#3922
#52222
#49502
#49492
#49462
#49452
#46472
#24553