Finite Group Databasebeta

← #78120

Subgroups — #78120

Order 384.

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

normal subgroups only

This group has 552 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×#399C211yesmaximal, characteristic, complemented
2#4994C212yesmaximal, complemented
3#4990C212yesmaximal, complemented
4#4990C212yesmaximal, complemented
5#4955C211yesmaximal, characteristic
6#4955C212yesmaximal
7#4953C211yesmaximal, characteristic, complemented
8#4944C212yesmaximal, complemented
9#4675C212yesmaximal, complemented
10#2735—31nomaximal, Sylow
11C24×D3C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic, complemented
13C2×C3×C4⋊C4C2212yescomplemented
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⋊Dic3C2211yescharacteristic, complemented
25C2×Dic3⋊C4C2212yescomplemented
26C2×C4×Dic3C2212yescomplemented
27C3×#95C2211yescharacteristic, complemented
28C3⋊#95C2212yescomplemented
29C3⋊#95C2212yescomplemented
30C3⋊#95C2212yescomplemented
31C3⋊#95C2211yescharacteristic, complemented
32#399D311yescharacteristic, complemented
33#397—31no—
34#397—32no—
35#396—32no—
36#388—32no—
37#388—32no—
38#388—32no—
39#379—32no—
40#379—31no—
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×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C22×Dic3C2312yes—
55C3×C4⋊C4—41no—
56C3×C4⋊C4—42no—
57C3×C22⋊C4—42no—
58C3×C22⋊C4—41no—
59#217—24no—
60#217—41no—
61(C2×D3)⋊C4—24no—
62(C2×D3)⋊C4—24no—
63(C2×D3)⋊C4—42no—
64(C2×D3)⋊C4—42no—
65(C2×D3)⋊C4—42no—
66(C2×D3)⋊C4—42no—
67C4⋊Dic3—41no—
68Dic3⋊C4—42no—
69C4×Dic3—42no—
70C25—31no—
71C2×C4⋊C4—32no—
72C2×C4⋊C4—31no—
73C2×C4⋊C4C2×D311yescharacteristic, complemented
74C2×C4⋊C4C2×D312yescomplemented
75C2×C22⋊C4—32no—
76C2×C22⋊C4—32no—
77C2×C22⋊C4—32no—
78C2×C22⋊C4—31no—
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C22⋊C4—32no—
82C2×C22⋊C4—32no—
83C2×C22⋊C4C2×D312yes—
84C2×C22⋊C4C2×D311yescharacteristic
85C2×C42—32no—
86#95—32no—
87#95—31no—
88#95—32no—
89#95—32no—
90#95C2×D311yescharacteristic, complemented
91C23×C3—22no—
92C23×C3—21no—
93C23×C3—21no—
94C23×C3—22no—
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—21no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
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—
128C24—32no—
129C24—64no—
130C24—62no—
131C24—122no—
132C24—121no—
133C24—121no—
134C24C22×D311yescharacteristic
135C22×C4—32no—
136C22×C4—32no—
137C22×C4—32no—
138C22×C4C22×D311yescharacteristic
139C22×C4C22×D312yes—
140C22×C4C22×D312yes—
141C22×C4C22×D311yescharacteristic
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—122no—
145C4⋊C4—121no—
146C22⋊C4—42no—
147C22⋊C4—41no—
148C22⋊C4—64no—
149C22⋊C4—64no—
150C22⋊C4—64no—
151C22⋊C4—122no—
152C22⋊C4—122no—
153C22⋊C4—122no—
154C22⋊C4—122no—
155C22⋊C4—121no—
156C42—122no—
157C22×C3—22no—
158C22×C3—24no—
159C22×C3—42no—
160C22×C3—41no—
161C22×C3—41no—
162C22×C3C2×#4111yescharacteristic
163C22×C3C2×#4112yes—
164C22×C3H(5,𝔽2)12yes—
165C22×C3H(5,𝔽2)11yescharacteristic
166C22×C3H(5,𝔽2)11yescharacteristic
167C2×D3—24no—
168C2×D3—24no—
169C2×D3—24no—
170C2×D3—416no—
171C2×D3—42no—
172C2×D3—42no—
173C2×D3—42no—
174C2×D3—42no—
175C3×C4—42no—
176C3×C4—41no—
177C3×C4—42no—
178C3×C4—41no—
179Dic3—42no—
180Dic3—42no—
181Dic3—42no—
182C23—22no—
183C23—21no—
184C23—22no—
185C23—21no—
186C23—34no—
187C23—62no—
188C23—62no—
189C23—62no—
190C23—62no—
191C23—62no—
192C23—62no—
193C23—128no—
194C23—124no—
195C23—124no—
196C23—128no—
197C23—124no—
198C23#23912yes—
199C23C23×D311yescharacteristic
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
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—62no—
219C2×C3—42no—
220C2×C3#53811yescharacteristic
221C2×C3#53812yes—
222C2×C3#55112yes—
223C2×C3#55111yescharacteristic
224C2×C3#56011yescharacteristic
225D3—44no—
226C22—24no—
227C22—22no—
228C22—42no—
229C22—41no—
230C22—41no—
231C22—64no—
232C22—64no—
233C22—64no—
234C22—1216no—
235C22—122no—
236C22—122no—
237C22—122no—
238C22—122no—
239C22#100412yes—
240C22#100411yescharacteristic
241C22C2×#23911yescharacteristic
242C22D3×#4112yes—
243C22#100911yescharacteristic
244C4—42no—
245C4—41no—
246C4—42no—
247C4—41no—
248C4—122no—
249C4—122no—
250C4—122no—
251C3#273511yescharacteristic, complemented, Sylow
252C2—42no—
253C2—124no—
254C2#563712yes—
255C2#564411yescharacteristic
256C2#565712yes—
257C2#565911yescharacteristic
258C2#572011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC24×D3C22
hypercenter#399D3
hypocenterC3#2735
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C3×#399C2semidirect
2. #4994C2semidirect
3. #4990C2semidirect
4. #4990C2semidirect
7. #4953C2semidirect
8. #4944C2semidirect
9. #4675C2semidirect
12. C2×C3×C4⋊C4C22semidirect
13. C2×C3×C4⋊C4C22semidirect
24. C2×C4⋊Dic3C22semidirect
25. C2×Dic3⋊C4C22semidirect
26. C2×C4×Dic3C22semidirect
27. C3×#95C22semidirect
28. C3⋊#95C22semidirect
29. C3⋊#95C22semidirect
30. C3⋊#95C22semidirect
31. C3⋊#95C22semidirect
32. #399D3semidirect
73. C2×C4⋊C4C2×D3semidirect
74. C2×C4⋊C4C2×D3semidirect
90. #95C2×D3semidirect
251. C3#2735semidirect

Maximal subgroups

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

subgroupindex
C3×#3992
#49942
#49902
#49552
#49532
#49442
#46752
#27353