Finite Group Databasebeta

← #10595434

Subgroups — #10595434

Order 576.

521 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 134 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#64155C211yesmaximal, characteristic, complemented
2C3×#946C212yesmaximal, complemented
3#63921C211yesmaximal, characteristic
4D3×(C2×D3)⋊C4C212yesmaximal
5#63899C212yesmaximal, complemented
6#63895C212yesmaximal, complemented
7#63893C212yesmaximal, complemented
8#63865C211yesmaximal, characteristic, complemented
9#63851C212yesmaximal, complemented
10#5634—32nomaximal
11C22×D32C2211yescharacteristic
12C2×C32×Q8C2211yescharacteristic, complemented
13C2×C4×C3⋊D3C2211yescharacteristic
14C2×C3×C4×D3C2212yes—
15C2×#627C2211yescharacteristic, complemented
16C2×#627C2212yescomplemented
17C2×#624C2211yescharacteristic
18C2×D3×Dic3C2212yes—
19#3752C2212yes—
20C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
21#3750C2212yescomplemented
22C3×(C2×D3)⋊C4C2212yes—
23C3×(C2×D3)⋊C4C2212yes—
24C3×C4⋊Dic3C2212yescomplemented
25C3×Dic3⋊C4C2212yescomplemented
26C3×Dic3⋊C4C2212yescomplemented
27#3724C2211yescharacteristic, complemented
28Dic3⋊Dic3C2212yescomplemented
29#3722C2212yes—
30(C2×D3)⋊Dic3C2212yes—
31(C2×D3)⋊Dic3C2212yes—
32C3×C22⋊Q8—32no—
33#946—32no—
34#946—61no—
35#946D312yescomplemented
36C2×(C2×D3)⋊C4—32no—
37#924—32no—
38#897—32no—
39#897—32no—
40#896—32no—
41#896—32no—
42#896—32no—
43#896—32no—
44D3×C22⋊C4—32no—
45D3×C22⋊C4—32no—
46#878—32no—
47#878—32no—
48C22×C3⋊D3C2311yescharacteristic
49C22×C3×D3C2312yes—
50C2×D32—41no—
51C2×D32—41no—
52C2×D32Q814yescomplemented
53C32×Q8—41no—
54C2×C32×C4C2311yescharacteristic
55C2×C32×C4C2312yes—
56C2×C3⋊Dic3C2311yescharacteristic
57C2×C3⋊Dic3C2312yes—
58C4×C3⋊D3—41no—
59C2×C3×Dic3C2312yes—
60C2×C3×Dic3C2312yes—
61C2×C3×Dic3C2312yes—
62C3×C4×D3—42no—
63#627—42no—
64#627—41no—
65#624—41no—
66D3×Dic3—42no—
67#543—91noSylow
68C23×D3—32no—
69C2×C3×Q8—21no—
70C2×C3×Q8C2×D312yescomplemented
71C22×C3×C4—32no—
72C22×Dic3—32no—
73C2×C4×D3—32no—
74C2×C4×D3—32no—
75C2×C4×D3—32no—
76C2×C4×D3—61no—
77C2×C4×D3C2×D312yes—
78C2×Dic6—32no—
79C2×Dic6—32no—
80C2×Dic6—32no—
81C3×C4⋊C4—32no—
82C3×C4⋊C4—32no—
83C3×C4⋊C4—32no—
84C3×C22⋊C4—32no—
85C3×C22⋊C4—32no—
86#217—32no—
87#217—32no—
88(C2×D3)⋊C4—32no—
89(C2×D3)⋊C4—32no—
90(C2×D3)⋊C4—32no—
91(C2×D3)⋊C4—32no—
92(C2×D3)⋊C4—32no—
93(C2×D3)⋊C4—32no—
94(C2×D3)⋊C4—62no—
95(C2×D3)⋊C4C2×D312yes—
96(C2×D3)⋊C4C2×D312yes—
97C4⋊Dic3—32no—
98C4⋊Dic3—32no—
99C4⋊Dic3—61no—
100C4⋊Dic3C2×D312yescomplemented
101Dic3⋊C4—32no—
102Dic3⋊C4—32no—
103Dic3⋊C4—32no—
104Dic3⋊C4—32no—
105Dic3⋊C4—62no—
106Dic3⋊C4C2×D312yescomplemented
107Dic3⋊C4C2×D312yescomplemented
108C22×C32C2411yescharacteristic
109C2×C3⋊D3—21no—
110C2×C3⋊D3—21no—
111C2×C3⋊D3C2×Q812yes—
112C2×C3×D3—22no—
113C2×C3×D3—22no—
114C2×C3×D3C2×Q814yes—
115D32—44no—
116C32×C4—21no—
117C32×C4—22no—
118C3⋊Dic3—22no—
119C3⋊Dic3—21no—
120C3×Dic3—22no—
121C3×Dic3—22no—
122C3×Dic3—22no—
123C22⋊Q8—32no—
124C22⋊Q8—92no—
125C22⋊Q8—92no—
126C22⋊Q8—91no—
127C22⋊Q8—92no—
128C22⋊Q8—92no—
129C22⋊Q8—91no—
130C2×C22⋊C4—91no—
131C2×C22⋊C4—92no—
132C23×C3—32no—
133C22×D3—32no—
134C22×D3—38no—
135C22×D3—61no—
136C22×D3—122no—
137C22×D3—122no—
138C22×D3C22×D312yes—
139C3×Q8—42no—
140C3×Q8—42no—
141C2×C3×C4—21no—
142C2×C3×C4—22no—
143C2×C3×C4—32no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—122no—
147C2×C3×C4C22×D312yes—
148C2×C3×C4C22×D312yes—
149C2×C3×C4C22×D312yes—
150C2×Dic3—32no—
151C2×Dic3—32no—
152C2×Dic3—32no—
153C2×Dic3—61no—
154C2×Dic3—62no—
155C2×Dic3—122no—
156C2×Dic3C22×D312yes—
157C2×Dic3C22×D312yes—
158C2×Dic3C22×D312yes—
159C4×D3—42no—
160C4×D3—122no—
161C4×D3—122no—
162C4×D3—122no—
163C4×D3—122no—
164Dic6—122no—
165Dic6—122no—
166Dic6—122no—
167C2×C32C22×Q811yescharacteristic
168C2×C32H(5,𝔽2)11yescharacteristic
169C2×C32H(5,𝔽2)11yescharacteristic
170C3⋊D3—22no—
171C3×D3—24no—
172C24—91no—
173C2×Q8—92no—
174C2×Q8—91no—
175C2×Q8D3211yescharacteristic, complemented
176C22×C4—32no—
177C22×C4—91no—
178C22×C4—92no—
179C22×C4—91no—
180C4⋊C4—32no—
181C4⋊C4—32no—
182C4⋊C4—32no—
183C4⋊C4—91no—
184C4⋊C4—92no—
185C4⋊C4—92no—
186C4⋊C4—91no—
187C22⋊C4—32no—
188C22⋊C4—32no—
189C22⋊C4—92no—
190C22⋊C4—92no—
191C22⋊C4—92no—
192C22⋊C4—92no—
193C22×C3—21no—
194C22×C3—34no—
195C22×C3—62no—
196C22×C3—62no—
197C22×C3C23×D312yes—
198C2×D3—22no—
199C2×D3—22no—
200C2×D3—34no—
201C2×D3—62no—
202C2×D3—62no—
203C2×D3—62no—
204C2×D3—121no—
205C2×D3—121no—
206C2×D3—128no—
207C2×D3D3×Q814yes—
208C3×C4—22no—
209C3×C4—22no—
210C3×C4—22no—
211C3×C4—22no—
212C3×C4—42no—
213C3×C4—62no—
214C3×C4—62no—
215C3×C4—62no—
216Dic3—22no—
217Dic3—22no—
218Dic3—22no—
219Dic3—62no—
220Dic3—62no—
221Dic3—62no—
222Dic3—121no—
223Dic3—122no—
224C32#54311yescharacteristic, complemented, Sylow
225C23—32no—
226C23—94no—
227C23—91no—
228C23—361no—
229C23—361no—
230Q8—41no—
231Q8—362no—
232Q8—361no—
233C2×C4—32no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—92no—
237C2×C4—91no—
238C2×C4—122no—
239C2×C4—361no—
240C2×C4—361no—
241C2×C4—362no—
242C2×C4C2×D3211yescharacteristic
243C2×C4C2×D3212yes—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—21no—
247C2×C3—64no—
248C2×C3#100412yes—
249C2×C3C2×D3×Q812yes—
250C2×C3#100912yes—
251D3—24no—
252D3—64no—
253D3—122no—
254C22—34no—
255C22—62no—
256C22—62no—
257C22—92no—
258C22—181no—
259C22—181no—
260C22—364no—
261C22C22×D3211yescharacteristic
262C4—22no—
263C4—21no—
264C4—62no—
265C4—62no—
266C4—62no—
267C4—182no—
268C4—181no—
269C3—21no—
270C3#563412yescomplemented
271C2—64no—
272C2—182no—
273C2D32×Q811yescharacteristic
274C2#6432011yescharacteristic
275C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
prime radicalC22×D32C22
hypercenterC2×Q8D32
hypocenterC32#543
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #64155C2semidirect
2. C3×#946C2semidirect
5. #63899C2semidirect
6. #63895C2semidirect
7. #63893C2semidirect
8. #63865C2semidirect
9. #63851C2semidirect
12. C2×C32×Q8C22semidirect
15. C2×#627C22semidirect
16. C2×#627C22semidirect
20. C4⋊(C3⋊Dic3)C22semidirect
21. #3750C22semidirect
24. C3×C4⋊Dic3C22semidirect
25. C3×Dic3⋊C4C22semidirect
26. C3×Dic3⋊C4C22semidirect
27. #3724C22semidirect
28. Dic3⋊Dic3C22semidirect
35. #946D3semidirect
52. C2×D32Q8semidirect
70. C2×C3×Q8C2×D3semidirect
100. C4⋊Dic3C2×D3semidirect
106. Dic3⋊C4C2×D3semidirect
107. Dic3⋊C4C2×D3semidirect
175. C2×Q8D32semidirect
224. C32#543semidirect
270. C3#5634semidirect

Maximal subgroups

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

subgroupindex
#641552
C3×#9462
#639212
D3×(C2×D3)⋊C42
#638992
#638952
#638932
#638652
#638512
#56343