Finite Group Databasebeta

← #10594181

Subgroups — #10594181

Order 576.

714 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 192 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
1C22×#625C211yesmaximal, characteristic
2#64086C211yesmaximal, characteristic, complemented
3C3×D3×C22⋊C4C212yesmaximal, complemented
4D3×#217C212yesmaximal, complemented
5#63957C211yesmaximal, characteristic, complemented
6#63902C212yesmaximal, complemented
7C4×#625C212yesmaximal, complemented
8#63879C214yesmaximal, complemented
9#5486—32nomaximal
10C23×C3×D3C2212yes—
11C22×C3⋊Dic3C2211yescharacteristic
12C2×C3×C4×D3C2214yescomplemented
13C2×#625C418yescomplemented
14C2×#625C2212yes—
15C2×#625C2211yescharacteristic
16C2×#625C2211yescharacteristic
17C2×D3×Dic3C2214yescomplemented
18C32×C22⋊C4C2211yescharacteristic
19#3757C2211yescharacteristic
20#3750C2212yescomplemented
21C4×C3⋊Dic3C2212yescomplemented
22C3×#217C2212yes—
23C3×(C2×D3)⋊C4C2214yescomplemented
24#3724C2212yescomplemented
25(C2×D3)⋊Dic3C2214yescomplemented
26Dic32C2212yescomplemented
27C22×#67—32no—
28C2×C3×C22⋊C4—32no—
29C2×#217—32no—
30C4×#67—34no—
31C4×#67—34no—
32#67⋊C4—34no—
33#67⋊C4—34no—
34D3×C22⋊C4—32no—
35D3×C22⋊C4D312yescomplemented
36#877—32no—
37#877—32no—
38#877—61no—
39C23×C32C2311yescharacteristic
40C22×C3×D3—44no—
41C22×C3×D3C2×C418yescomplemented
42C22×C3×D3C2312yes—
43C22×C3×D3C2312yes—
44C2×C32×C4C2312yes—
45C2×C3⋊Dic3C2×C414yescomplemented
46C2×C3⋊Dic3C2312yes—
47C2×C3⋊Dic3C2311yescharacteristic
48C2×C3⋊Dic3C2311yescharacteristic
49C2×C3×Dic3C2314yes—
50C3×C4×D3—44no—
51#625C2×C418yescomplemented
52#625C2×C414yescomplemented
53#625C2×C414yescomplemented
54D3×Dic3—44no—
55#518—91noSylow
56C24×C3—32no—
57C23×D3C2×D312yes—
58C22×C3×C4—34no—
59C2×#67—316no—
60C2×#67—32no—
61C2×#67—32no—
62C2×#67—32no—
63C2×#67—32no—
64C22×Dic3—32no—
65C22×Dic3—34no—
66C22×Dic3—61no—
67C2×C4×D3—34no—
68C2×C4×D3C2×D314yescomplemented
69C3×C22⋊C4—21no—
70C3×C22⋊C4—32no—
71C3×C22⋊C4—34no—
72C3×C22⋊C4C2×D312yes—
73#217—32no—
74#217—34no—
75#217—61no—
76#217C2×D312yes—
77(C2×D3)⋊C4—34no—
78(C2×D3)⋊C4C2×D314yescomplemented
79Dic3⋊C4—34no—
80Dic3⋊C4—34no—
81Dic3⋊C4—62no—
82C4×Dic3—34no—
83C4×Dic3—34no—
84C4×Dic3—62no—
85C22×C32—22no—
86C22×C32C22×C412yes—
87C22×C32C2411yescharacteristic
88C2×C3×D3—24no—
89C2×C3×D3—24no—
90C2×C3×D3—48no—
91C2×C3×D3C22×C414yes—
92C2×C3×D3C22×C414yes—
93C32×C4—22no—
94C3⋊Dic3—22no—
95C3⋊Dic3C22×C412yes—
96C3⋊Dic3C22×C412yes—
97C3×Dic3—24no—
98C22×D4—91no—
99C4×D4—92no—
100C4×D4—94no—
101C4×D4—92no—
102#117—91no—
103#117—91no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—92no—
106C23×C3—21no—
107C23×C3—32no—
108C23×C3—32no—
109C23×C3—38no—
110C23×C3—124no—
111C23×C3C22×D312yes—
112C22×D3—44no—
113C22×D3C4×D318yescomplemented
114C22×D3C22×D312yes—
115C22×D3C22×D312yes—
116C2×C3×C4—22no—
117C2×C3×C4—34no—
118C2×C3×C4—124no—
119C2×C3×C4C22×D314yes—
120#67—38no—
121#67—38no—
122#67—38no—
123#67—38no—
124C2×Dic3—38no—
125C2×Dic3—32no—
126C2×Dic3—32no—
127C2×Dic3—34no—
128C2×Dic3—64no—
129C2×Dic3—61no—
130C2×Dic3—61no—
131C2×Dic3—62no—
132C2×Dic3—124no—
133C2×Dic3C22×D314yes—
134C4×D3—44no—
135C4×D3—124no—
136C2×C32—22no—
137C2×C32C23×C411yescharacteristic
138C2×C32H(5,𝔽2)12yes—
139C3×D3—24no—
140C3×D3—24no—
141C24—32no—
142C2×D4—91no—
143C2×D4—92no—
144C2×D4—91no—
145C2×D4—98no—
146C22×C4—34no—
147C22×C4—94no—
148C22×C4—91no—
149C4⋊C4—92no—
150C4⋊C4—92no—
151C22⋊C4—32no—
152C22⋊C4—34no—
153C22⋊C4—94no—
154C22⋊C4—91no—
155C22⋊C4D3211yescharacteristic
156C42—92no—
157C42—92no—
158C22×C3—24no—
159C22×C3—21no—
160C22×C3—22no—
161C22×C3—34no—
162C22×C3—34no—
163C22×C3—42no—
164C22×C3—64no—
165C22×C3—64no—
166C22×C3—128no—
167C22×C3C2×C4×D314yes—
168C22×C3C23×D312yes—
169C2×D3—24no—
170C2×D3—24no—
171C2×D3—48no—
172C2×D3C2×C4×D314yes—
173C2×D3C2×C4×D314yes—
174C3×C4—24no—
175C3×C4—42no—
176C3×C4—64no—
177Dic3—24no—
178Dic3—34no—
179Dic3—34no—
180Dic3—62no—
181Dic3—62no—
182Dic3—64no—
183Dic3—122no—
184C32#51811yescharacteristic, complemented, Sylow
185C23—32no—
186C23—32no—
187C23—38no—
188C23—124no—
189C23C2×D3211yescharacteristic
190D4—94no—
191D4—98no—
192D4—94no—
193C2×C4—34no—
194C2×C4—91no—
195C2×C4—91no—
196C2×C4—94no—
197C2×C4—92no—
198C2×C4—124no—
199C2×C4—364no—
200C2×C4C2×D3212yes—
201C2×C3—24no—
202C2×C3—21no—
203C2×C3—22no—
204C2×C3—42no—
205C2×C3—64no—
206C2×C3—64no—
207C2×C3C22×C4×D312yes—
208C2×C3#100414yes—
209D3—24no—
210D3—24no—
211C22—22no—
212C22—34no—
213C22—34no—
214C22—64no—
215C22—64no—
216C22—128no—
217C22C4×D3212yes—
218C22C22×D3211yescharacteristic
219C4—22no—
220C4—64no—
221C4—92no—
222C4—92no—
223C4—182no—
224C3—21no—
225C3#548612yescomplemented
226C2—22no—
227C2—64no—
228C2—64no—
229C2C2×C4×D3211yescharacteristic
230C2#6431312yes—
subgroupG/H
center, FrattiniC22C22×D32
commutatorC2×C32C23×C4
FittingC32×C22⋊C4C22
socle, cosocleC22×C32C24
prime radicalC22×#625C2
hypercenterC22⋊C4D32
hypocenterC32#518
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #64086C2semidirect
3. C3×D3×C22⋊C4C2semidirect
4. D3×#217C2semidirect
5. #63957C2semidirect
6. #63902C2semidirect
7. C4×#625C2semidirect
8. #63879C2semidirect
12. C2×C3×C4×D3C22semidirect
13. C2×#625C4semidirect
17. C2×D3×Dic3C22semidirect
20. #3750C22semidirect
21. C4×C3⋊Dic3C22semidirect
23. C3×(C2×D3)⋊C4C22semidirect
24. #3724C22semidirect
25. (C2×D3)⋊Dic3C22semidirect
26. Dic32C22semidirect
35. D3×C22⋊C4D3semidirect
41. C22×C3×D3C2×C4semidirect
45. C2×C3⋊Dic3C2×C4semidirect
51. #625C2×C4semidirect
52. #625C2×C4semidirect
53. #625C2×C4semidirect
68. C2×C4×D3C2×D3semidirect
78. (C2×D3)⋊C4C2×D3semidirect
113. C22×D3C4×D3semidirect
184. C32#518semidirect
225. C3#5486semidirect

Maximal subgroups

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

subgroupindex
C22×#6252
#640862
C3×D3×C22⋊C42
D3×#2172
#639572
#639022
C4×#6252
#638792
#54863