Finite Group Databasebeta

← #10593960

Subgroups — #10593960

Order 576.

490 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 152 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
1#64078C211yesmaximal, characteristic, complemented
2C3×C4×D12C211yesmaximal, characteristic
3C3×C4×Dic6C211yesmaximal, characteristic, complemented
4#63896C214yesmaximal, complemented
5Dic3×D12C212yesmaximal
6D3×C4⋊Dic3C212yesmaximal, complemented
7#63869C212yesmaximal, complemented
8#63856C212yesmaximal, complemented
9#5579—31nomaximal
10#5543—31nomaximal
11C2×#634C2212yes—
12C2×C3×D12C2211yescharacteristic
13C2×C3×C4×D3C2212yes—
14C2×C3×Dic6C2211yescharacteristic
15C2×D3×Dic3C2214yes—
16C32×C42C2211yescharacteristic
17C4⋊(C3⋊Dic3)C2212yes—
18C4⋊(C3⋊Dic3)C2212yes—
19C3×(C2×D3)⋊C4C2212yes—
20C3×C4⋊Dic3C2211yescharacteristic
21C3×C4⋊Dic3C2211yescharacteristic
22C3×Dic3⋊C4C2212yes—
23C3×C4×Dic3C2212yes—
24#3724C2214yes—
25Dic3⋊Dic3C2214yes—
26(C2×D3)⋊Dic3C2214yes—
27C3×C4×Q8—31no—
28C3×C4×D4—31no—
29#946—32no—
30D4×Dic3—32no—
31C2×C4⋊Dic3—32no—
32#924—32no—
33#896—34no—
34D3×C4⋊C4—32no—
35#890—32no—
36#878—34no—
37C4×D12—32no—
38C4×D12D311yescharacteristic
39#870—32no—
40#869—31no—
41#869—31no—
42#869—61no—
43C4×Dic6D311yescharacteristic, complemented
44C22×C3×D3C2312yes—
45C2×C32×C4C2311yescharacteristic
46C2×C32×C4C2311yescharacteristic
47C2×C32×C4C2311yescharacteristic
48C2×C3⋊Dic3C2314yes—
49#634—42no—
50C2×C3×Dic3C2312yes—
51C2×C3×Dic3C2312yes—
52C2×C3×Dic3C2312yes—
53C3×D12Q814yescomplemented
54C3×C4×D3—24no—
55C3×Dic6—22no—
56D3×Dic3—44no—
57#554—91noSylow
58C2×C3×Q8—31no—
59C2×C3×D4—31no—
60C22×C3×C4—32no—
61C22×Dic3—34no—
62C2×D12C2×D311yescharacteristic
63C2×C4×D3—34no—
64C2×C4×D3C2×D312yes—
65C2×Dic6—32no—
66C2×Dic6—32no—
67C2×Dic6—62no—
68C2×Dic6C2×D311yescharacteristic
69C3×C4⋊C4—31no—
70C3×C4⋊C4—31no—
71C3×C4⋊C4—32no—
72C3×C22⋊C4—32no—
73C3×C42—21no—
74C3×C42—32no—
75C3×C42C2×D311yescharacteristic
76C3×C42C2×D311yescharacteristic
77#217—34no—
78(C2×D3)⋊C4—34no—
79(C2×D3)⋊C4C2×D312yes—
80C4⋊Dic3—32no—
81C4⋊Dic3—32no—
82C4⋊Dic3—32no—
83C4⋊Dic3—32no—
84C4⋊Dic3—34no—
85C4⋊Dic3—62no—
86C4⋊Dic3—62no—
87C4⋊Dic3C2×D311yescharacteristic
88C4⋊Dic3C2×D311yescharacteristic
89Dic3⋊C4—34no—
90Dic3⋊C4—34no—
91Dic3⋊C4—34no—
92Dic3⋊C4C2×D312yes—
93C4×Dic3C2×D312yes—
94C22×C32C2411yescharacteristic
95C2×C3×D3—22no—
96C2×C3×D3—22no—
97C2×C3×D3C2×Q814yes—
98C32×C4—21no—
99C32×C4C2×Q812yes—
100C32×C4#4112yes—
101C3⋊Dic3—24no—
102C3×Dic3—22no—
103C3×Dic3—22no—
104C3×Dic3—22no—
105C4⋊Q8—91no—
106#125—92no—
107C22⋊Q8—94no—
108C22⋊Q8—92no—
109C4×Q8—31no—
110C4×D4—31no—
111C4×D4—92no—
112C2×C4⋊C4—92no—
113C23×C3—32no—
114C22×D3C22×D312yes—
115C3×Q8—62no—
116C3×D4—34no—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—32no—
121C2×C3×C4—32no—
122C2×C3×C4—32no—
123C2×C3×C4—64no—
124C2×C3×C4C22×D311yescharacteristic
125C2×C3×C4C22×D311yescharacteristic
126C2×C3×C4C22×D311yescharacteristic
127C2×C3×C4C22×D311yescharacteristic
128C2×C3×C4C22×D311yescharacteristic
129C2×C3×C4C22×D311yescharacteristic
130C2×Dic3—34no—
131C2×Dic3—34no—
132C2×Dic3—64no—
133C2×Dic3—124no—
134C2×Dic3C22×D312yes—
135C2×Dic3C22×D312yes—
136C2×Dic3C22×D312yes—
137D12Dic614yescomplemented
138C4×D3—24no—
139C4×D3—124no—
140Dic6—22no—
141Dic6—122no—
142Dic6—122no—
143Dic6—124no—
144C2×C32C22×Q811yescharacteristic
145C2×C32C2×#4111yescharacteristic
146C2×C32H(5,𝔽2)11yescharacteristic
147C3×D3—24no—
148C2×Q8—31no—
149C2×Q8—92no—
150C2×D4—31no—
151C22×C4—32no—
152C22×C4—94no—
153C4⋊C4—31no—
154C4⋊C4—31no—
155C4⋊C4—32no—
156C4⋊C4—94no—
157C4⋊C4—94no—
158C4⋊C4—92no—
159C4⋊C4—92no—
160C22⋊C4—32no—
161C22⋊C4—94no—
162C42—32no—
163C42D3211yescharacteristic
164C22×C3—21no—
165C22×C3—34no—
166C22×C3—62no—
167C22×C3—62no—
168C22×C3C23×D311yescharacteristic
169C22×C3C23×D311yescharacteristic
170C2×D3—22no—
171C2×D3—22no—
172C2×D3C2×Dic614yes—
173C3×C4—22no—
174C3×C4—21no—
175C3×C4—21no—
176C3×C4—22no—
177C3×C4—22no—
178C3×C4—62no—
179C3×C4—62no—
180C3×C4—62no—
181C3×C4C2×Dic612yes—
182C3×C4#23512yes—
183C3×C4#23712yes—
184C3×C4D3×Q812yes—
185Dic3—22no—
186Dic3—22no—
187Dic3—22no—
188Dic3—64no—
189Dic3—64no—
190Dic3—124no—
191C32#55411yescharacteristic, complemented, Sylow
192C23—32no—
193Q8—62no—
194Q8—362no—
195D4—34no—
196C2×C4—32no—
197C2×C4—32no—
198C2×C4—32no—
199C2×C4—64no—
200C2×C4—94no—
201C2×C4—364no—
202C2×C4C2×D3211yescharacteristic
203C2×C4C2×D3211yescharacteristic
204C2×C4C2×D3211yescharacteristic
205C2×C3—21no—
206C2×C3—21no—
207C2×C3—21no—
208C2×C3—64no—
209C2×C3C22×Dic611yescharacteristic
210C2×C3C2×#23511yescharacteristic
211C2×C3C2×#23711yescharacteristic
212C2×C3C2×D3×Q811yescharacteristic
213C2×C3#100911yescharacteristic
214C2×C3#100911yescharacteristic
215D3—24no—
216C22—34no—
217C22—62no—
218C22—62no—
219C22C22×D3211yescharacteristic
220C4—21no—
221C4—62no—
222C4—62no—
223C4—62no—
224C4—184no—
225C4D3×Dic612yes—
226C4#379512yes—
227C3—21no—
228C3#554311yescharacteristic, complemented
229C3#557911yescharacteristic, complemented
230C2—64no—
231C2C2×D3×Dic611yescharacteristic
232C2C2×#379511yescharacteristic
233C2#6430811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
prime radicalC2×C3×D12C22
hypercenterC42D32
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #64078C2semidirect
3. C3×C4×Dic6C2semidirect
4. #63896C2semidirect
6. D3×C4⋊Dic3C2semidirect
7. #63869C2semidirect
8. #63856C2semidirect
43. C4×Dic6D3semidirect
53. C3×D12Q8semidirect
137. D12Dic6semidirect
191. C32#554semidirect
228. C3#5543semidirect
229. C3#5579semidirect

Maximal subgroups

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

subgroupindex
#640782
C3×C4×D122
C3×C4×Dic62
#638962
Dic3×D122
D3×C4⋊Dic32
#638692
#638562
#55793
#55433