Finite Group Databasebeta

← C2×#64330 #10596708

Subgroups — C2×#64330 #10596708

Order 576.

1914 conjugacy classes of subgroups; only the 536 normal subgroups are shown.

normal subgroups only

This group has 1914 conjugacy classes of subgroups — too many to draw. Only its 536 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×#638C211yesmaximal, characteristic, complemented
2C22×C3×#67C212yesmaximal, complemented
3#64330C2116yesmaximal, direct product factor
4C2×#3811C212yesmaximal, complemented
5C2×D3×#67C214yesmaximal, complemented
6C2×#3805C212yesmaximal, complemented
7C2×#3804C214yesmaximal, complemented
8C2×#1004—32nomaximal
9C24×C32C2211yescharacteristic, complemented
10C23×C3⋊D3C2211yescharacteristic, complemented
11C23×C3×D3C2212yescomplemented
12C22×D32C2212yescomplemented
13C2×#638C2214yescomplemented
14C2×#638C2218yescomplemented
15C22×C3⋊Dic3C2211yescharacteristic, complemented
16C2×C3×#67C2214yescomplemented
17C2×C3×#67C2214yescomplemented
18C2×C3×#67C22116yescomplemented
19C22×C3×Dic3C2212yescomplemented
20#3811C22116yescomplemented
21D3×#67C22132yescomplemented
22C2×#627C2212yescomplemented
23C2×#626C2214yescomplemented
24C2×#625C2212yescomplemented
25C2×#624C2212yescomplemented
26#3805C22116yescomplemented
27#3804C22132yescomplemented
28C2×D3×Dic3C2214yescomplemented
29C22×C3×D4—32no—
30C22×#67—32no—
31C22×#67—34no—
32C22×#67—61no—
33C22×#67D312yescomplemented
34#1004—332no—
35C2×#237—34no—
36C2×#237—34no—
37C2×D3×D4—34no—
38C2×D3×D4—34no—
39C2×#235—34no—
40C23×C32—42no—
41C23×C32C2312yes—
42C23×C32C2311yescharacteristic
43C23×C32C2314yescomplemented
44C22×C3⋊D3—42no—
45C22×C3⋊D3C2312yes—
46C22×C3⋊D3C2314yescomplemented
47C22×C3×D3—44no—
48C22×C3×D3C2314yes—
49C22×C3×D3C2318yescomplemented
50C2×D32—44no—
51C2×D32C2318yescomplemented
52#638C23116yescomplemented
53C2×C3⋊Dic3C2312yes—
54C2×C3⋊Dic3C2314yescomplemented
55C3×#67C23116yescomplemented
56C3×#67C23116yescomplemented
57C2×C3×Dic3C2314yes—
58C2×C3×Dic3C2318yescomplemented
59#627C2318yescomplemented
60#626C23116yescomplemented
61#625C2318yescomplemented
62#624C2318yescomplemented
63D3×Dic3C23116yescomplemented
64C2×H(5,𝔽2)—91noSylow
65C24×C3—21no—
66C24×C3—32no—
67C24×C3C2×D312yescomplemented
68C23×D3—32no—
69C23×D3—34no—
70C23×D3—61no—
71C23×D3C2×D312yescomplemented
72C2×C3×D4—34no—
73C2×C3×D4—34no—
74C2×C3×D4—316no—
75C22×C3×C4—32no—
76C2×#67—34no—
77C2×#67—34no—
78C2×#67—316no—
79C2×#67—34no—
80C2×#67—332no—
81C2×#67—34no—
82C2×#67—64no—
83C2×#67—68no—
84C2×#67C2×D314yescomplemented
85C2×#67C2×D314yescomplemented
86C2×#67C2×D3116yescomplemented
87C22×Dic3—32no—
88C22×Dic3—34no—
89C22×Dic3—61no—
90C22×Dic3C2×D312yescomplemented
91#237—332no—
92#237—332no—
93D3×D4—332no—
94D3×D4—332no—
95#235—332no—
96C2×D12—34no—
97C2×C4×D3—34no—
98C2×C4×D3—34no—
99C2×Dic6—34no—
100C22×C32—22no—
101C22×C32—24no—
102C22×C32—44no—
103C22×C32C2414yes—
104C22×C32C2412yes—
105C22×C32C2411yescharacteristic
106C2×C3⋊D3—24no—
107C2×C3⋊D3—44no—
108C2×C3⋊D3C2414yes—
109C2×C3×D3—28no—
110C2×C3×D3—48no—
111C2×C3×D3C2418yes—
112D32—48no—
113C3⋊Dic3C2414yes—
114C3×Dic3C2418yes—
115H(5,𝔽2)—916no—
116C2×#41—94no—
117C2×#41—92no—
118C22×D4—32no—
119C22×D4—92no—
120C22×D4—94no—
121C22×D4—91no—
122C23×C3—21no—
123C23×C3—22no—
124C23×C3—24no—
125C23×C3—34no—
126C23×C3—38no—
127C23×C3—44no—
128C23×C3—44no—
129C23×C3—124no—
130C23×C3C22×D312yes—
131C23×C3C22×D312yes—
132C23×C3C22×D312yes—
133C23×C3C22×D318yescomplemented
134C22×D3—34no—
135C22×D3—38no—
136C22×D3—316no—
137C22×D3—44no—
138C22×D3—62no—
139C22×D3—64no—
140C22×D3—128no—
141C22×D3—124no—
142C22×D3—124no—
143C22×D3C22×D314yes—
144C22×D3C22×D318yescomplemented
145C3×D4—316no—
146C3×D4—316no—
147C2×C3×C4—34no—
148C2×C3×C4—38no—
149#67—316no—
150#67—316no—
151#67—316no—
152#67—316no—
153#67—616no—
154#67C22×D3116yescomplemented
155#67C22×D3116yescomplemented
156C2×Dic3—34no—
157C2×Dic3—38no—
158C2×Dic3—316no—
159C2×Dic3—62no—
160C2×Dic3—64no—
161C2×Dic3C22×D314yes—
162C2×Dic3C22×D318yescomplemented
163D12—316no—
164C4×D3—316no—
165C4×D3—316no—
166Dic6—316no—
167C2×C32—22no—
168C2×C32—24no—
169C2×C32H(5,𝔽2)12yescomplemented
170C2×C32C2511yescharacteristic
171C3⋊D3—24no—
172C3×D3—28no—
173C24—32no—
174C24—92no—
175C24—91no—
176C24D3211yescharacteristic, complemented
177#41—932no—
178#41—916no—
179C2×Q8—92no—
180C2×D4—34no—
181C2×D4—34no—
182C2×D4—316no—
183C2×D4—94no—
184C2×D4—916no—
185C2×D4—932no—
186C2×D4—92no—
187C2×D4—94no—
188C2×D4—98no—
189C22×C4—32no—
190C22×C4—92no—
191C22×C4—94no—
192C22×C4—91no—
193C22×C3—24no—
194C22×C3—24no—
195C22×C3—24no—
196C22×C3—24no—
197C22×C3—22no—
198C22×C3—24no—
199C22×C3—21no—
200C22×C3—38no—
201C22×C3—44no—
202C22×C3—48no—
203C22×C3—48no—
204C22×C3—68no—
205C22×C3—128no—
206C22×C3C23×D314yes—
207C22×C3C23×D314yes—
208C22×C3C23×D314yes—
209C22×C3C23×D312yes—
210C2×D3—28no—
211C2×D3—38no—
212C2×D3—48no—
213C2×D3—64no—
214C2×D3—68no—
215C2×D3—124no—
216C2×D3—1216no—
217C2×D3—128no—
218C2×D3—128no—
219C2×D3C23×D318yes—
220C3×C4—38no—
221Dic3—38no—
222Dic3—64no—
223Dic3C23×D318yes—
224C32C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
225C23—34no—
226C23—38no—
227C23—42no—
228C23—98no—
229C23—92no—
230C23—94no—
231C23—124no—
232C23—362no—
233C23—364no—
234C23C2×D3212yes—
235C23C2×D3211yescharacteristic
236C23C2×D3214yescomplemented
237Q8—98no—
238D4—316no—
239D4—316no—
240D4—916no—
241D4—98no—
242D4—916no—
243C2×C4—38no—
244C2×C4—34no—
245C2×C4—98no—
246C2×C4—916no—
247C2×C4—94no—
248C2×C4—92no—
249C2×C3—24no—
250C2×C3—24no—
251C2×C3—24no—
252C2×C3—24no—
253C2×C3—22no—
254C2×C3—21no—
255C2×C3—44no—
256C2×C3—68no—
257C2×C3#100414yescomplemented
258C2×C3C24×D312yes—
259D3—28no—
260D3—68no—
261D3—124no—
262C22—24no—
263C22—22no—
264C22—38no—
265C22—44no—
266C22—68no—
267C22—94no—
268C22—128no—
269C22—184no—
270C22—364no—
271C22—368no—
272C22C22×D3214yes—
273C22C22×D3212yes—
274C22C22×D3211yescharacteristic
275C4—38no—
276C4—94no—
277C3—21no—
278C3C2×#100412yescomplemented
279C2—24no—
280C2—22no—
281C2—68no—
282C2—184no—
283C2#6433012yesdirect product factor
284C2C23×D3211yescharacteristic
subgroupG/H
centerC22C22×D32
commutator, cosocleC2×C32C25
FrattiniC2C23×D32
FittingC24×C32C22
socleC22×C32C24
hypercenterC24D32
hypocenterC32C2×H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#638C2semidirect
2. C22×C3×#67C2semidirect
3. #64330C2direct (N × H)
4. C2×#3811C2semidirect
5. C2×D3×#67C2semidirect
6. C2×#3805C2semidirect
7. C2×#3804C2semidirect
9. C24×C32C22semidirect
10. C23×C3⋊D3C22semidirect
11. C23×C3×D3C22semidirect
12. C22×D32C22semidirect
13. C2×#638C22semidirect
14. C2×#638C22semidirect
15. C22×C3⋊Dic3C22semidirect
16. C2×C3×#67C22semidirect
17. C2×C3×#67C22semidirect
18. C2×C3×#67C22semidirect
19. C22×C3×Dic3C22semidirect
20. #3811C22semidirect
21. D3×#67C22semidirect
22. C2×#627C22semidirect
23. C2×#626C22semidirect
24. C2×#625C22semidirect
25. C2×#624C22semidirect
26. #3805C22semidirect
27. #3804C22semidirect
28. C2×D3×Dic3C22semidirect
33. C22×#67D3semidirect
43. C23×C32C23semidirect
46. C22×C3⋊D3C23semidirect
49. C22×C3×D3C23semidirect
51. C2×D32C23semidirect
52. #638C23semidirect
54. C2×C3⋊Dic3C23semidirect
55. C3×#67C23semidirect
56. C3×#67C23semidirect
58. C2×C3×Dic3C23semidirect
59. #627C23semidirect
60. #626C23semidirect
61. #625C23semidirect
62. #624C23semidirect
63. D3×Dic3C23semidirect
67. C24×C3C2×D3semidirect
71. C23×D3C2×D3semidirect
84. C2×#67C2×D3semidirect
85. C2×#67C2×D3semidirect
86. C2×#67C2×D3semidirect
90. C22×Dic3C2×D3semidirect
133. C23×C3C22×D3semidirect
144. C22×D3C22×D3semidirect
154. #67C22×D3semidirect
155. #67C22×D3semidirect
162. C2×Dic3C22×D3semidirect
169. C2×C32H(5,𝔽2)semidirect
176. C24D32semidirect
224. C32C2×H(5,𝔽2)semidirect
236. C23C2×D32semidirect
257. C2×C3#1004semidirect
278. C3C2×#1004semidirect
283. C2#64330direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#6382
C22×C3×#672
#643302
C2×#38112
C2×D3×#672
C2×#38052
C2×#38042
C2×#10043