Finite Group Databasebeta

← #77706

Subgroups — #77706

Order 384.

560 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C3×#397C211yesmaximal, characteristic, complemented
2#4990C212yesmaximal, complemented
3#4955C212yesmaximal
4#4955C211yesmaximal, characteristic
5#4951C212yesmaximal, complemented
6#4947C211yesmaximal, characteristic, complemented
7#4674C212yesmaximal, complemented
8#4673C212yesmaximal, complemented
9#4670C212yesmaximal, complemented
10#2736—31nomaximal, Sylow
11C24×D3C2211yescharacteristic
12C22×Dic6C2211yescharacteristic, complemented
13C2×C3×C4⋊C4C2211yescharacteristic, complemented
14C2×C3×C22⋊C4C2212yes—
15C2×C3×C22⋊C4C2211yescharacteristic
16C2×#217C2212yes—
17C2×#217C2211yescharacteristic
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×C4⋊Dic3C2212yescomplemented
25C2×C4×Dic3C2212yescomplemented
26C3×#95C2212yescomplemented
27C3×#95C2211yescharacteristic, complemented
28C3⋊#95C2211yescharacteristic, complemented
29C3⋊#95C2212yescomplemented
30C3⋊#95C2212yescomplemented
31C3⋊#95C2212yescomplemented
32#397—32no—
33#397D311yescharacteristic, complemented
34#393—31no—
35#393—32no—
36#388—32no—
37#388—32no—
38#388—32no—
39#379—31no—
40#379—32no—
41C24×C3C2311yescharacteristic
42C23×D3—22no—
43C23×D3—24no—
44C23×D3—41no—
45C23×D3—42no—
46C23×D3—41no—
47C23×D3C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C2×Dic6—41no—
56C2×Dic6—42no—
57C3×C4⋊C4—41no—
58C3×C22⋊C4—24no—
59C3×C22⋊C4—41no—
60#217—41no—
61#217—42no—
62(C2×D3)⋊C4—24no—
63(C2×D3)⋊C4—24no—
64(C2×D3)⋊C4—42no—
65(C2×D3)⋊C4—42no—
66(C2×D3)⋊C4—42no—
67(C2×D3)⋊C4—42no—
68C4⋊Dic3—42no—
69C4×Dic3—42no—
70C25—31no—
71C22×Q8—31no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4C2×D311yescharacteristic, complemented
74C2×C22⋊C4—32no—
75C2×C22⋊C4—32no—
76C2×C22⋊C4—31no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—32no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C22⋊C4—32no—
82C2×C22⋊C4C2×D312yes—
83C2×C22⋊C4C2×D311yescharacteristic
84C2×C42—32no—
85#95—32no—
86#95—31no—
87#95—32no—
88#95—32no—
89#95C2×D312yescomplemented
90#95C2×D311yescharacteristic, complemented
91C23×C3—22no—
92C23×C3—21no—
93C23×C3—22no—
94C23×C3—21no—
95C23×C3#4112yes—
96C23×C3C2411yescharacteristic
97C22×D3—22no—
98C22×D3—22no—
99C22×D3—22no—
100C22×D3—22no—
101C22×D3—22no—
102C22×D3—22no—
103C22×D3—44no—
104C22×D3—48no—
105C22×D3—48no—
106C22×D3—44no—
107C22×D3—44no—
108C22×D3#4114yes—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
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×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—22no—
128Dic6—44no—
129C24—32no—
130C24—64no—
131C24—62no—
132C24—121no—
133C24—122no—
134C24—121no—
135C24C22×D311yescharacteristic
136C2×Q8—121no—
137C2×Q8—122no—
138C22×C4—32no—
139C22×C4—32no—
140C22×C4—32no—
141C22×C4C22×D311yescharacteristic
142C22×C4C22×D312yes—
143C22×C4C22×D311yescharacteristic
144C22×C4C22×D312yes—
145C4⋊C4—41no—
146C4⋊C4—122no—
147C22⋊C4—24no—
148C22⋊C4—41no—
149C22⋊C4—64no—
150C22⋊C4—64no—
151C22⋊C4—122no—
152C22⋊C4—122no—
153C22⋊C4—122no—
154C22⋊C4—122no—
155C22⋊C4—121no—
156C22⋊C4—122no—
157C42—122no—
158C22×C3—22no—
159C22×C3—24no—
160C22×C3—42no—
161C22×C3—41no—
162C22×C3—41no—
163C22×C3C2×#4111yescharacteristic
164C22×C3C2×#4112yes—
165C22×C3H(5,𝔽2)11yescharacteristic
166C22×C3H(5,𝔽2)11yescharacteristic
167C22×C3H(5,𝔽2)12yes—
168C2×D3—24no—
169C2×D3—24no—
170C2×D3—24no—
171C2×D3—416no—
172C2×D3—42no—
173C2×D3—42no—
174C2×D3—42no—
175C2×D3—42no—
176C3×C4—42no—
177C3×C4—42no—
178C3×C4—41no—
179C3×C4—41no—
180Dic3—42no—
181Dic3—42no—
182Dic3—42no—
183C23—21no—
184C23—22no—
185C23—22no—
186C23—21no—
187C23—34no—
188C23—62no—
189C23—62no—
190C23—62no—
191C23—62no—
192C23—62no—
193C23—62no—
194C23—128no—
195C23—124no—
196C23—124no—
197C23—128no—
198C23—124no—
199C23#23712yes—
200C23C23×D311yescharacteristic
201Q8—124no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—62no—
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×C3—42no—
222C2×C3#53812yes—
223C2×C3#53811yescharacteristic
224C2×C3#55112yes—
225C2×C3#55111yescharacteristic
226C2×C3H(3,𝔽4)11yescharacteristic
227D3—44no—
228C22—24no—
229C22—22no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—64no—
234C22—64no—
235C22—64no—
236C22—1216no—
237C22—122no—
238C22—122no—
239C22—122no—
240C22—122no—
241C22C2×#23711yescharacteristic
242C22#100411yescharacteristic
243C22D3×#4112yes—
244C22#100912yes—
245C22#100911yescharacteristic
246C4—42no—
247C4—41no—
248C4—41no—
249C4—42no—
250C4—122no—
251C4—122no—
252C4—122no—
253C3#273611yescharacteristic, complemented, Sylow
254C2—42no—
255C2—124no—
256C2#561711yescharacteristic
257C2#565812yes—
258C2#565911yescharacteristic
259C2#568311yescharacteristic
260C2#570412yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
prime radicalC24×D3C22
hypercenter#397D3
hypocenterC3#2736
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C3×#397C2semidirect
2. #4990C2semidirect
5. #4951C2semidirect
6. #4947C2semidirect
7. #4674C2semidirect
8. #4673C2semidirect
9. #4670C2semidirect
12. C22×Dic6C22semidirect
13. C2×C3×C4⋊C4C22semidirect
24. C2×C4⋊Dic3C22semidirect
25. C2×C4×Dic3C22semidirect
26. C3×#95C22semidirect
27. C3×#95C22semidirect
28. C3⋊#95C22semidirect
29. C3⋊#95C22semidirect
30. C3⋊#95C22semidirect
31. C3⋊#95C22semidirect
33. #397D3semidirect
73. C2×C4⋊C4C2×D3semidirect
89. #95C2×D3semidirect
90. #95C2×D3semidirect
253. C3#2736semidirect

Maximal subgroups

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

subgroupindex
C3×#3972
#49902
#49552
#49512
#49472
#46742
#46732
#46702
#27363