Finite Group Databasebeta

← #10594618

Subgroups — #10594618

Order 576.

903 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 903 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×#3802C212yesmaximal, complemented
2C2×D3×D12C212yesmaximal, complemented
3#64102C211yesmaximal, characteristic, complemented
4C3×#895C212yesmaximal, complemented
5#63922C214yesmaximal, complemented
6#63914C211yesmaximal, characteristic, complemented
7#63912C212yesmaximal, complemented
8C4×#625C211yesmaximal, characteristic, complemented
9#5609—32nomaximal
10C22×D32C2214yescomplemented
11C2×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C4×C3⋊D3C2212yescomplemented
13C2×C3×D12C2212yescomplemented
14C2×C3×D12C2214yescomplemented
15C2×C3×C4×D3C2212yescomplemented
16C2×#626C2214yescomplemented
17C2×#625C2211yescharacteristic
18C2×#625C2212yescomplemented
19C2×#625C2212yescomplemented
20#3802—28no—
21D3×D12—28no—
22C32×C4⋊C4C2211yescharacteristic, complemented
23#3752C2212yescomplemented
24C4×C3⋊Dic3C2211yescharacteristic, complemented
25C3×(C2×D3)⋊C4C2214yescomplemented
26#3724C2211yescharacteristic, complemented
27(C2×D3)⋊Dic3C2212yescomplemented
28C2×D3×D4—34no—
29C2×D3×D4—32no—
30C22×D12—32no—
31C3×#121—32no—
32#940—32no—
33#937—34no—
34C4×#67—32no—
35#895D312yescomplemented
36#893—32no—
37#893—61no—
38#884—34no—
39#882—34no—
40C22×C3⋊D3C2312yes—
41C22×C3×D3C2314yes—
42C22×C3×D3C2312yes—
43C2×D32—28no—
44C2×D32—28no—
45C2×D32—44no—
46C2×C32×C4C2312yes—
47C2×C32×C4C2311yescharacteristic
48C2×C3⋊Dic3C2311yescharacteristic
49C2×C3⋊Dic3C2311yescharacteristic
50C4⋊(C3⋊D3)D414yescomplemented
51C4×C3⋊D3—24no—
52C2×C3×Dic3C2312yes—
53C3×D12—28no—
54C3×D12—24no—
55C3×C4×D3—24no—
56#626—28no—
57#625—24no—
58#625—42no—
59#625D414yescomplemented
60D42—91noSylow
61C23×D3—34no—
62C23×D3—34no—
63C2×C3×D4—34no—
64C2×C3×D4—32no—
65C22×C3×C4—32no—
66C2×#67—32no—
67C2×#67—34no—
68C2×#67—34no—
69C2×#67—34no—
70D3×D4—68no—
71D3×D4—616no—
72C2×D12—32no—
73C2×D12—34no—
74C2×D12—61no—
75C2×D12—68no—
76C2×D12C2×D314yescomplemented
77C2×D12C2×D312yescomplemented
78C2×C4×D3—34no—
79C2×C4×D3—62no—
80C2×C4×D3C2×D312yescomplemented
81C3×C4⋊C4—21no—
82C3×C4⋊C4C2×D312yescomplemented
83C3×C22⋊C4—34no—
84#217—32no—
85(C2×D3)⋊C4—32no—
86(C2×D3)⋊C4—34no—
87(C2×D3)⋊C4—62no—
88(C2×D3)⋊C4C2×D314yescomplemented
89Dic3⋊C4—32no—
90C4×Dic3—32no—
91C4×Dic3—61no—
92C22×C32C2411yescharacteristic
93C2×C3⋊D3—22no—
94C2×C3⋊D3—22no—
95C2×C3⋊D3C2×D414yescomplemented
96C2×C3×D3—22no—
97C2×C3×D3—22no—
98C2×C3×D3—24no—
99C2×C3×D3—24no—
100C2×C3×D3—24no—
101C2×C3×D3C2×D414yescomplemented
102D32—416no—
103C32×C4—22no—
104C32×C4C2×D412yescomplemented
105C3⋊Dic3—21no—
106C3⋊Dic3C2×D412yescomplemented
107C3×Dic3—22no—
108C22×D4—92no—
109C22×D4—92no—
110#127—91no—
111#121—32no—
112#121—92no—
113#120—94no—
114C4×D4—91no—
115C4×D4—91no—
116C23×C3—32no—
117C23×C3—34no—
118C22×D3—34no—
119C22×D3—62no—
120C22×D3—68no—
121C22×D3—68no—
122C22×D3—68no—
123C22×D3—68no—
124C22×D3—124no—
125C22×D3—124no—
126C22×D3C22×D314yes—
127C22×D3C22×D312yes—
128C3×D4—68no—
129C3×D4—64no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4—32no—
133C2×C3×C4—64no—
134C2×C3×C4C22×D314yes—
135C2×C3×C4C22×D312yes—
136#67—38no—
137#67—68no—
138#67—68no—
139#67—124no—
140C2×Dic3—32no—
141C2×Dic3—32no—
142C2×Dic3—61no—
143C2×Dic3—61no—
144C2×Dic3C22×D312yes—
145D12—28no—
146D12—24no—
147D12—38no—
148D12—64no—
149D12—68no—
150C4×D3—24no—
151C4×D3—68no—
152C4×D3—124no—
153C2×C32C22×D411yescharacteristic
154C2×C32C22×D411yescharacteristic
155C2×C32H(5,𝔽2)11yescharacteristic
156C3⋊D3—24no—
157C3×D3—24no—
158C3×D3—44no—
159C24—94no—
160C2×D4—34no—
161C2×D4—32no—
162C2×D4—94no—
163C2×D4—91no—
164C2×D4—92no—
165C2×D4—92no—
166C2×D4—91no—
167C2×D4—188no—
168C2×D4—188no—
169C22×C4—32no—
170C22×C4—92no—
171C4⋊C4—91no—
172C4⋊C4D3211yescharacteristic, complemented
173C22⋊C4—34no—
174C22⋊C4—92no—
175C22⋊C4—92no—
176C42—91no—
177C22×C3—21no—
178C22×C3—34no—
179C22×C3—62no—
180C22×C3—62no—
181C22×C3—64no—
182C22×C3—64no—
183C22×C3—64no—
184C22×C3C23×D312yes—
185C2×D3—22no—
186C2×D3—22no—
187C2×D3—24no—
188C2×D3—24no—
189C2×D3—24no—
190C2×D3—38no—
191C2×D3—64no—
192C2×D3—64no—
193C2×D3—64no—
194C2×D3—122no—
195C2×D3—122no—
196C2×D3—1216no—
197C2×D3—1216no—
198C2×D3D3×D414yescomplemented
199C3×C4—24no—
200C3×C4—22no—
201C3×C4—42no—
202C3×C4—62no—
203C3×C4D3×D414yescomplemented
204Dic3—22no—
205Dic3—34no—
206Dic3—62no—
207Dic3—62no—
208Dic3—121no—
209C32D4211yescharacteristic, complemented, Sylow
210C23—32no—
211C23—34no—
212C23—92no—
213C23—188no—
214C23—188no—
215C23—364no—
216D4—68no—
217D4—64no—
218D4—94no—
219D4—94no—
220D4—188no—
221D4—184no—
222D4—362no—
223C2×C4—32no—
224C2×C4—64no—
225C2×C4—91no—
226C2×C4—91no—
227C2×C4—184no—
228C2×C4C2×D3212yes—
229C2×C4C2×D3211yescharacteristic
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—64no—
234C2×C3—124no—
235C2×C3C2×D3×D412yes—
236C2×C3C2×D3×D412yes—
237C2×C3#100912yes—
238D3—24no—
239D3—44no—
240D3—68no—
241D3—124no—
242C22—34no—
243C22—62no—
244C22—62no—
245C22—64no—
246C22—64no—
247C22—64no—
248C22—94no—
249C22—182no—
250C22—182no—
251C22—3616no—
252C22C22×D3211yescharacteristic
253C4—22no—
254C4—62no—
255C4—92no—
256C4—181no—
257C4#380212yescomplemented
258C3—21no—
259C3#560912yescomplemented
260C2—64no—
261C2—124no—
262C2—184no—
263C2C2×#380211yescharacteristic
264C2D32×D411yescharacteristic
265C2#6432011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32D42
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3802C2semidirect
2. C2×D3×D12C2semidirect
3. #64102C2semidirect
4. C3×#895C2semidirect
5. #63922C2semidirect
6. #63914C2semidirect
7. #63912C2semidirect
8. C4×#625C2semidirect
10. C22×D32C22semidirect
12. C2×C4×C3⋊D3C22semidirect
13. C2×C3×D12C22semidirect
14. C2×C3×D12C22semidirect
15. C2×C3×C4×D3C22semidirect
16. C2×#626C22semidirect
18. C2×#625C22semidirect
19. C2×#625C22semidirect
22. C32×C4⋊C4C22semidirect
23. #3752C22semidirect
24. C4×C3⋊Dic3C22semidirect
25. C3×(C2×D3)⋊C4C22semidirect
26. #3724C22semidirect
27. (C2×D3)⋊Dic3C22semidirect
35. #895D3semidirect
50. C4⋊(C3⋊D3)D4semidirect
59. #625D4semidirect
76. C2×D12C2×D3semidirect
77. C2×D12C2×D3semidirect
80. C2×C4×D3C2×D3semidirect
82. C3×C4⋊C4C2×D3semidirect
88. (C2×D3)⋊C4C2×D3semidirect
95. C2×C3⋊D3C2×D4semidirect
101. C2×C3×D3C2×D4semidirect
104. C32×C4C2×D4semidirect
106. C3⋊Dic3C2×D4semidirect
172. C4⋊C4D32semidirect
198. C2×D3D3×D4semidirect
203. C3×C4D3×D4semidirect
209. C32D42semidirect
257. C4#3802semidirect
259. C3#5609semidirect

Maximal subgroups

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

subgroupindex
C2×#38022
C2×D3×D122
#641022
C3×#8952
#639222
#639142
#639122
C4×#6252
#56093