Finite Group Databasebeta

← #76309

Subgroups — #76309

Order 384.

512 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 173 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×#935C211yesmaximal, characteristic
2C3×#390C211yesmaximal, characteristic
3#5220C211yesmaximal, characteristic
4#4976C212yesmaximal, complemented
5#4946C212yesmaximal, complemented
6#4945C212yesmaximal, complemented
7#4944C212yesmaximal, complemented
8#4942C212yesmaximal, complemented
9#4933C212yesmaximal, complemented
10#2376—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×Dic3C2211yescharacteristic
13C23×Dic3C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C42C2211yescharacteristic
18C2×#217C2212yes—
19C2×#217C2211yescharacteristic
20C2×#217C2212yes—
21C2×#217C2211yescharacteristic
22#935C418yescomplemented
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2212yes—
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C2×#123—31no—
34#390D311yescharacteristic
35#389—32no—
36#388—32no—
37#388—32no—
38#387—32no—
39#386—31no—
40#385—32no—
41#380—32no—
42C24×C3C2311yescharacteristic
43C24×C3C2311yescharacteristic
44C2×C3×D4—41no—
45C2×C3×D4—41no—
46C2×C3×D4C2×C414yes—
47C22×C3×C4C2312yes—
48C22×C3×C4C2312yes—
49C22×C3×C4C2311yescharacteristic
50C22×Dic3—41no—
51C22×Dic3—41no—
52C22×Dic3—41no—
53C22×Dic3—42no—
54C22×Dic3C2×C414yescomplemented
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C3×C4⋊C4—41no—
60C3×C22⋊C4—42no—
61C3×C22⋊C4—42no—
62C3×C42—41no—
63#217—42no—
64#217—41no—
65#217C2×C418yes—
66#217C2×C414yescomplemented
67Dic3⋊C4—42no—
68Dic3⋊C4—42no—
69Dic3⋊C4—42no—
70Dic3⋊C4C2×C418yescomplemented
71C4×Dic3—42no—
72C22×D4C2×D311yescharacteristic
73C23×C4—31no—
74C23×C4—31no—
75#123—38no—
76C2×C4⋊C4—32no—
77C2×C4⋊C4—32no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4C2×D311yescharacteristic
81C2×C22⋊C4—32no—
82C2×C22⋊C4—31no—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—31no—
85C2×C22⋊C4C2×D312yes—
86C2×C22⋊C4C2×D312yes—
87C2×C42—32no—
88C2×C42C2×D311yescharacteristic
89#95—32no—
90#95—32no—
91#95—32no—
92#95—32no—
93#95—32no—
94C23×C3—22no—
95C23×C3—21no—
96C23×C3—22no—
97C23×C3—21no—
98C23×C3—21no—
99C23×C3—21no—
100C23×C3—22no—
101C23×C3—22no—
102C23×C3C22×C412yes—
103C23×C3C22×C412yes—
104C23×C3C2411yescharacteristic
105C3×D4—44no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4C22×C412yes—
115C2×Dic3—22no—
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×Dic3—22no—
125C2×Dic3—44no—
126C2×Dic3—44no—
127C2×Dic3C22×C414yes—
128C2×Dic3C22×C414yes—
129C24C22×D311yescharacteristic
130C24C22×D311yescharacteristic
131C2×D4—41no—
132C2×D4—41no—
133C2×D4C4×D314yes—
134C22×C4—32no—
135C22×C4—32no—
136C22×C4—34no—
137C22×C4—32no—
138C22×C4—32no—
139C22×C4—121no—
140C22×C4—121no—
141C22×C4—122no—
142C22×C4—121no—
143C22×C4C22×D312yes—
144C22×C4C22×D312yes—
145C22×C4C22×D311yescharacteristic
146C4⋊C4—38no—
147C4⋊C4—41no—
148C4⋊C4—122no—
149C4⋊C4—122no—
150C4⋊C4—122no—
151C22⋊C4—38no—
152C22⋊C4—34no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—122no—
156C22⋊C4—121no—
157C42—41no—
158C42—122no—
159C22×C3—22no—
160C22×C3—22no—
161C22×C3—24no—
162C22×C3—24no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3—42no—
166C22×C3—42no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3C23×C411yescharacteristic
170C22×C3H(5,𝔽2)11yescharacteristic
171C22×C3H(5,𝔽2)11yescharacteristic
172C22×C3H(5,𝔽2)12yes—
173C22×C32−1+412yes—
174C3×C4—22no—
175C3×C4—42no—
176C3×C4—42no—
177Dic3—24no—
178Dic3—24no—
179Dic3—42no—
180Dic3—42no—
181C23—22no—
182C23—22no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—22no—
188C23—22no—
189C23C2×C4×D312yes—
190C23C2×C4×D312yes—
191C23C23×D311yescharacteristic
192D4—44no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—34no—
202C2×C4—34no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—124no—
214C2×C4—124no—
215C2×C4C2×C4×D312yes—
216C2×C3—42no—
217C2×C3—42no—
218C2×C3#51811yescharacteristic
219C2×C3#52012yes—
220C2×C3#56011yescharacteristic
221C2×C3#56212yes—
222C2×C3#56311yescharacteristic
223C22—22no—
224C22—22no—
225C22—24no—
226C22—24no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—42no—
233C22C22×C4×D311yescharacteristic
234C22#100411yescharacteristic
235C22#100412yes—
236C22#100411yescharacteristic
237C22#101012yes—
238C4—22no—
239C4—42no—
240C4—42no—
241C4—64no—
242C4—64no—
243C4—122no—
244C4—122no—
245C3#237611yescharacteristic, complemented, Sylow
246C2—42no—
247C2—42no—
248C2#548611yescharacteristic
249C2#554612yes—
250C2#562112yes—
251C2#567211yescharacteristic
252C2#572011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#390C2
socle, cosocleC23×C3C24
prime radicalC22×C3×D4C22
hypercenter#390D3
hypocenterC3#2376
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
4. #4976C2semidirect
5. #4946C2semidirect
6. #4945C2semidirect
7. #4944C2semidirect
8. #4942C2semidirect
9. #4933C2semidirect
22. #935C4semidirect
54. C22×Dic3C2×C4semidirect
66. #217C2×C4semidirect
70. Dic3⋊C4C2×C4semidirect
245. C3#2376semidirect

Maximal subgroups

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

subgroupindex
C2×#9352
C3×#3902
#52202
#49762
#49462
#49452
#49442
#49422
#49332
#23763