Finite Group Databasebeta

← #76718

Subgroups — #76718

Order 384.

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

normal subgroups only

This group has 568 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×#935C212yesmaximal, complemented
2C3×#392C211yesmaximal, characteristic
3#5220C212yesmaximal, complemented
4#5220C212yesmaximal, complemented
5#4953C211yesmaximal, characteristic, complemented
6#4948C212yesmaximal, complemented
7#4946C211yesmaximal, characteristic, complemented
8#4942C212yesmaximal, complemented
9#4658C212yesmaximal, complemented
10#2693—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C22×C3×D4C2212yes—
13C23×Dic3C2212yes—
14C23×Dic3C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×#217C2211yescharacteristic
22#935—28no—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C3×#95C2211yescharacteristic
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2211yescharacteristic
33C2×#123—32no—
34#400—32no—
35#397—31no—
36#396—32no—
37#392D311yescharacteristic
38#386—32no—
39#386—32no—
40#385—31no—
41#380—32no—
42C24×C3C2311yescharacteristic
43C24×C3C2312yes—
44C2×C3×D4—24no—
45C2×C3×D4—24no—
46C2×C3×D4—24no—
47C2×C3×D4—41no—
48C2×C3×D4—42no—
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×Dic3—24no—
53C22×Dic3—24no—
54C22×Dic3—42no—
55C22×Dic3—42no—
56C22×Dic3—41no—
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2312yes—
60C22×Dic3C2312yes—
61C3×C22⋊C4—42no—
62C3×C22⋊C4—41no—
63#217—24no—
64#217—24no—
65#217—42no—
66#217—42no—
67#217D414yescomplemented
68C4⋊Dic3—41no—
69Dic3⋊C4—24no—
70Dic3⋊C4—24no—
71Dic3⋊C4—42no—
72C22×D4C2×D312yes—
73C22×D4C2×D311yescharacteristic
74C23×C4—32no—
75C23×C4—31no—
76#123—68no—
77C2×C4⋊C4—32no—
78C2×C4⋊C4—32no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—31no—
81C2×C22⋊C4—31no—
82C2×C22⋊C4—32no—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4C2×D312yes—
87C2×C22⋊C4C2×D311yescharacteristic
88#95—31no—
89#95—32no—
90#95—32no—
91#95—32no—
92#95—32no—
93#95C2×D311yescharacteristic
94C23×C3—22no—
95C23×C3—22no—
96C23×C3—22no—
97C23×C3—22no—
98C23×C3—22no—
99C23×C3—21no—
100C23×C3—22no—
101C23×C3—22no—
102C23×C3—21no—
103C23×C3—22no—
104C23×C3C2×D412yes—
105C23×C3#4114yes—
106C23×C3C2411yescharacteristic
107C3×D4—48no—
108C3×D4—44no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—21no—
114C2×C3×C4—22no—
115C2×C3×C4—21no—
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—22no—
126C2×Dic3—22no—
127C2×Dic3—48no—
128C2×Dic3—44no—
129C2×Dic3C2×D414yes—
130C24C22×D311yescharacteristic
131C24C22×D312yes—
132C2×D4—24no—
133C2×D4—24no—
134C2×D4—24no—
135C2×D4—41no—
136C2×D4—42no—
137C22×C4—32no—
138C22×C4—32no—
139C22×C4—32no—
140C22×C4—32no—
141C22×C4—64no—
142C22×C4—64no—
143C22×C4—122no—
144C22×C4—121no—
145C22×C4—122no—
146C22×C4C22×D311yescharacteristic
147C22×C4C22×D312yes—
148C22×C4C22×D311yescharacteristic
149C4⋊C4—64no—
150C4⋊C4—64no—
151C4⋊C4—122no—
152C4⋊C4—121no—
153C22⋊C4—34no—
154C22⋊C4—42no—
155C22⋊C4—41no—
156C22⋊C4—64no—
157C22⋊C4—64no—
158C22⋊C4—122no—
159C22⋊C4—122no—
160C22×C3—22no—
161C22×C3—24no—
162C22×C3—24no—
163C22×C3—24no—
164C22×C3—24no—
165C22×C3—42no—
166C22×C3—42no—
167C22×C3—42no—
168C22×C3—42no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—42no—
172C22×C3C22×D411yescharacteristic
173C22×C3C2×#4112yes—
174C22×C3H(5,𝔽2)11yescharacteristic
175C22×C3H(5,𝔽2)12yes—
176C22×C32−1+411yescharacteristic
177C3×C4—42no—
178C3×C4—41no—
179C3×C4—41no—
180Dic3—24no—
181Dic3—42no—
182Dic3—42no—
183Dic3—42no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—21no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—21no—
193C23—22no—
194C23D3×D412yes—
195C23#23714yes—
196C23C23×D311yescharacteristic
197D4—44no—
198D4—48no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—34no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—128no—
219C2×C4—124no—
220C2×C3—42no—
221C2×C3—44no—
222C2×C3#53711yescharacteristic
223C2×C3#53912yes—
224C2×C3#54612yes—
225C2×C3#55111yescharacteristic
226C2×C3#56011yescharacteristic
227C22—24no—
228C22—24no—
229C22—24no—
230C22—22no—
231C22—24no—
232C22—41no—
233C22—42no—
234C22—41no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22C2×D3×D411yescharacteristic
240C22C2×#23712yes—
241C22#100411yescharacteristic
242C22#100412yes—
243C22#101011yescharacteristic
244C4—41no—
245C4—41no—
246C4—42no—
247C4—64no—
248C4—122no—
249C4—122no—
250C4—122no—
251C3#269311yescharacteristic, complemented, Sylow
252C2—44no—
253C2—42no—
254C2#558711yescharacteristic
255C2#559112yes—
256C2#559711yescharacteristic
257C2#560212yes—
258C2#572011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC3×#392C2
hypercenter#392D3
hypocenterC3#2693
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#935C2semidirect
3. #5220C2semidirect
4. #5220C2semidirect
5. #4953C2semidirect
6. #4948C2semidirect
7. #4946C2semidirect
8. #4942C2semidirect
9. #4658C2semidirect
67. #217D4semidirect
251. C3#2693semidirect

Maximal subgroups

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

subgroupindex
C2×#9352
C3×#3922
#52202
#49532
#49482
#49462
#49422
#46582
#26933