Finite Group Databasebeta

← #10595436

Subgroups — #10595436

Order 576.

534 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 128 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#64156C211yesmaximal, characteristic, complemented
2C3×#946C212yesmaximal, complemented
3#63923C211yesmaximal, characteristic
4D3×(C2×D3)⋊C4C212yesmaximal, complemented
5#63904C212yesmaximal, complemented
6#63897C211yesmaximal, characteristic, complemented
7#63890C212yesmaximal, complemented
8#63878C212yesmaximal, complemented
9#63848C212yesmaximal, complemented
10#5637—32nomaximal
11C22×D32C2211yescharacteristic
12C2×C32×Q8C2211yescharacteristic, complemented
13C2×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×D3C2212yes—
15C2×#626C2212yes—
16C2×#624C2211yescharacteristic, complemented
17C2×D3×Dic3C2212yes—
18#3752C2212yes—
19#3752C2212yescomplemented
20C4×C3⋊Dic3C2211yescharacteristic, complemented
21C3×(C2×D3)⋊C4C2212yes—
22C3×(C2×D3)⋊C4C2212yes—
23C3×C4⋊Dic3C2212yescomplemented
24C3×Dic3⋊C4C2212yescomplemented
25C3×Dic3⋊C4C2212yescomplemented
26Dic3⋊Dic3C2212yescomplemented
27#3722C2211yescharacteristic
28#3722C2212yes—
29#3722C2211yescharacteristic, complemented
30(C2×D3)⋊Dic3C2212yes—
31Dic32C2212yescomplemented
32C3×C22⋊Q8—32no—
33#947—32no—
34#947—61no—
35#946D312yescomplemented
36C4×#67—32no—
37C2×(C2×D3)⋊C4—32no—
38#898—32no—
39#898—32no—
40#894—32no—
41#894—32no—
42#893—32no—
43#892—32no—
44#886—32no—
45#882—32no—
46D3×C22⋊C4—32no—
47#877—32no—
48C22×C3⋊D3C2311yescharacteristic
49C22×C3⋊D3C2311yescharacteristic
50C22×C3×D3C2312yes—
51C2×D32—24no—
52C2×D32—41no—
53C32×Q8—41no—
54C2×C32×C4C2311yescharacteristic
55C2×C32×C4C2312yes—
56C2×C3⋊Dic3C2312yes—
57C4⋊(C3⋊D3)—41no—
58C2×C3×Dic3C2312yes—
59C2×C3×Dic3C2312yes—
60C2×C3×Dic3C2312yes—
61C3×C4×D3—42no—
62#626—24no—
63#624—41no—
64D3×Dic3—24no—
65#551—91noSylow
66C23×D3—32no—
67C2×C3×Q8—21no—
68C2×C3×Q8C2×D312yescomplemented
69C22×C3×C4—32no—
70C2×#67—32no—
71C22×Dic3—32no—
72C2×D12—32no—
73C2×D12—32no—
74C2×D12—61no—
75C2×C4×D3—32no—
76C2×C4×D3—32no—
77C2×C4×D3C2×D312yes—
78C3×C4⋊C4—32no—
79C3×C4⋊C4—32no—
80C3×C4⋊C4—32no—
81C3×C22⋊C4—32no—
82C3×C22⋊C4—32no—
83#217—32no—
84(C2×D3)⋊C4—32no—
85(C2×D3)⋊C4—32no—
86(C2×D3)⋊C4—32no—
87(C2×D3)⋊C4—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—62no—
94(C2×D3)⋊C4—62no—
95(C2×D3)⋊C4C2×D312yes—
96(C2×D3)⋊C4C2×D312yes—
97C4⋊Dic3—32no—
98C4⋊Dic3C2×D312yescomplemented
99Dic3⋊C4—32no—
100Dic3⋊C4C2×D312yescomplemented
101Dic3⋊C4C2×D312yescomplemented
102C4×Dic3—32no—
103C4×Dic3—32no—
104C4×Dic3—32no—
105C4×Dic3—61no—
106C22×C32C2411yescharacteristic
107C2×C3⋊D3—22no—
108C2×C3⋊D3—22no—
109C2×C3⋊D3—21no—
110C2×C3⋊D3—21no—
111C2×C3×D3—22no—
112C2×C3×D3—22no—
113C2×C3×D3#4114yes—
114D32—44no—
115C32×C4—21no—
116C32×C4—22no—
117C3⋊Dic3—22no—
118C3×Dic3—22no—
119C3×Dic3—22no—
120C3×Dic3—22no—
121#126—92no—
122#124—91no—
123#123—92no—
124#123—91no—
125C22⋊Q8—32no—
126#120—91no—
127C4×D4—92no—
128#117—92no—
129C2×C22⋊C4—92no—
130C23×C3—32no—
131C22×D3—32no—
132C22×D3—32no—
133C22×D3—61no—
134C22×D3—61no—
135C22×D3—64no—
136C22×D3—64no—
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—
150#67—64no—
151C2×Dic3—32no—
152C2×Dic3—32no—
153C2×Dic3—62no—
154C2×Dic3—64no—
155C2×Dic3C22×D312yes—
156C2×Dic3C22×D312yes—
157C2×Dic3C22×D312yes—
158D12—64no—
159D12—122no—
160D12—122no—
161C4×D3—42no—
162C4×D3—64no—
163C4×D3—122no—
164C2×C32C2×#4112yes—
165C2×C32H(5,𝔽2)11yescharacteristic
166C3⋊D3—41no—
167C3⋊D3—41no—
168C3×D3—24no—
169C24—91no—
170C2×Q8D3211yescharacteristic, complemented
171C2×D4—92no—
172C2×D4—91no—
173C22×C4—32no—
174C22×C4—91no—
175C22×C4—92no—
176C4⋊C4—32no—
177C4⋊C4—32no—
178C4⋊C4—32no—
179C4⋊C4—92no—
180C22⋊C4—32no—
181C22⋊C4—32no—
182C22⋊C4—92no—
183C22⋊C4—91no—
184C22⋊C4—91no—
185C22⋊C4—92no—
186C22⋊C4—92no—
187C22⋊C4—92no—
188C42—92no—
189C42—91no—
190C22×C3—21no—
191C22×C3—34no—
192C22×C3—62no—
193C22×C3—62no—
194C22×C3C23×D312yes—
195C2×D3—22no—
196C2×D3—22no—
197C2×D3—62no—
198C2×D3—62no—
199C2×D3—62no—
200C2×D3—62no—
201C2×D3—62no—
202C2×D3—62no—
203C2×D3—121no—
204C2×D3—121no—
205C2×D3—122no—
206C2×D3—122no—
207C2×D3—128no—
208C2×D3#23914yes—
209C3×C4—22no—
210C3×C4—22no—
211C3×C4—22no—
212C3×C4—22no—
213C3×C4—42no—
214C3×C4—62no—
215C3×C4—62no—
216C3×C4—62no—
217Dic3—22no—
218Dic3—22no—
219Dic3—22no—
220Dic3—62no—
221Dic3—62no—
222Dic3—122no—
223C32#55111yescharacteristic, complemented, Sylow
224C23—32no—
225C23—91no—
226C23—91no—
227C23—184no—
228C23—361no—
229Q8—41no—
230D4—184no—
231D4—361no—
232C2×C4—32no—
233C2×C4—32no—
234C2×C4—32no—
235C2×C4—92no—
236C2×C4—122no—
237C2×C4—184no—
238C2×C4—361no—
239C2×C4C2×D3211yescharacteristic
240C2×C4C2×D3212yes—
241C2×C3—22no—
242C2×C3—21no—
243C2×C3—64no—
244C2×C3#100412yes—
245C2×C3C2×#23912yes—
246C2×C3D3×#4112yes—
247D3—24no—
248D3—122no—
249D3—122no—
250D3—241no—
251D3—241no—
252C22—34no—
253C22—62no—
254C22—62no—
255C22—182no—
256C22—182no—
257C22—181no—
258C22—181no—
259C22—364no—
260C22C22×D3211yescharacteristic
261C4—22no—
262C4—21no—
263C4—62no—
264C4—62no—
265C4—62no—
266C4—182no—
267C3—21no—
268C3#563712yescomplemented
269C2—64no—
270C2—361no—
271C2—361no—
272C2D3×#23912yes—
273C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
prime radical#63923C2
hypercenterC2×Q8D32
hypocenterC32#551
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #64156C2semidirect
2. C3×#946C2semidirect
4. D3×(C2×D3)⋊C4C2semidirect
5. #63904C2semidirect
6. #63897C2semidirect
7. #63890C2semidirect
8. #63878C2semidirect
9. #63848C2semidirect
12. C2×C32×Q8C22semidirect
16. C2×#624C22semidirect
19. #3752C22semidirect
20. C4×C3⋊Dic3C22semidirect
23. C3×C4⋊Dic3C22semidirect
24. C3×Dic3⋊C4C22semidirect
25. C3×Dic3⋊C4C22semidirect
26. Dic3⋊Dic3C22semidirect
29. #3722C22semidirect
31. Dic32C22semidirect
35. #946D3semidirect
68. C2×C3×Q8C2×D3semidirect
98. C4⋊Dic3C2×D3semidirect
100. Dic3⋊C4C2×D3semidirect
101. Dic3⋊C4C2×D3semidirect
170. C2×Q8D32semidirect
223. C32#551semidirect
268. C3#5637semidirect

Maximal subgroups

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

subgroupindex
#641562
C3×#9462
#639232
D3×(C2×D3)⋊C42
#639042
#638972
#638902
#638782
#638482
#56373