Finite Group Databasebeta

← #10595087

Subgroups — #10595087

Order 576.

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

normal subgroups only

This group has 535 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#64138C211yesmaximal, characteristic, complemented
2C3×C4×#67C212yesmaximal, complemented
3#63983C211yesmaximal, characteristic
4#63969C212yesmaximal, complemented
5#63953C211yesmaximal, characteristic, complemented
6D3×(C2×D3)⋊C4C212yesmaximal, complemented
7#63877C212yesmaximal, complemented
8#63868C212yesmaximal, complemented
9#63851C212yesmaximal, complemented
10#5556—32nomaximal
11C22×D32C2211yescharacteristic
12C22×C32×C4C2211yescharacteristic, complemented
13C2×#638C2211yescharacteristic
14C2×C3×#67C2212yes—
15C2×C3×C4×D3C2212yes—
16C2×#627C2211yescharacteristic, complemented
17C2×D3×Dic3C2212yes—
18#3757C2211yescharacteristic, complemented
19#3752C2212yes—
20#3750C2212yescomplemented
21C3×#217C2212yescomplemented
22C3×(C2×D3)⋊C4C2212yes—
23C3×Dic3⋊C4C2212yescomplemented
24C3×C4×Dic3C2212yescomplemented
25#3724C2212yescomplemented
26Dic3⋊Dic3C2212yescomplemented
27#3722C2211yescharacteristic
28#3722C2212yes—
29(C2×D3)⋊Dic3C2212yes—
30(C2×D3)⋊Dic3C2212yes—
31Dic32C2211yescharacteristic, complemented
32C3×C4×D4—32no—
33#937—32no—
34#935—32no—
35#929—32no—
36#929—61no—
37C4×#67D312yescomplemented
38C2×(C2×D3)⋊C4—32no—
39#924—32no—
40#898—32no—
41#896—32no—
42#885—32no—
43#883—32no—
44D3×C22⋊C4—32no—
45#877—32no—
46#876—32no—
47#872—32no—
48C23×C32C2311yescharacteristic
49C22×C3⋊D3C2311yescharacteristic
50C22×C3×D3C2312yes—
51C2×D32—24no—
52C2×D32—41no—
53C2×C32×C4—41no—
54C2×C32×C4C2312yes—
55#638—41no—
56C2×C3⋊Dic3C2311yescharacteristic
57C2×C3⋊Dic3C2312yes—
58C3×#67—24no—
59C2×C3×Dic3C2312yes—
60C2×C3×Dic3C2312yes—
61C2×C3×Dic3C2312yes—
62C3×C4×D3—42no—
63#627—41no—
64D3×Dic3—24no—
65#551—91noSylow
66C23×D3—32no—
67C2×C3×D4—32no—
68C22×C3×C4—21no—
69C22×C3×C4—32no—
70C22×C3×C4C2×D312yescomplemented
71C2×#67—32no—
72C2×#67—61no—
73C2×#67C2×D312yes—
74C22×Dic3—32no—
75C2×C4×D3—32no—
76C2×C4×D3C2×D312yes—
77C2×Dic6—32no—
78C3×C4⋊C4—32no—
79C3×C22⋊C4—32no—
80C3×C22⋊C4—32no—
81C3×C42—32no—
82#217—32no—
83#217—32no—
84#217—32no—
85#217—61no—
86#217C2×D312yescomplemented
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—32no—
94(C2×D3)⋊C4—62no—
95(C2×D3)⋊C4C2×D312yes—
96C4⋊Dic3—32no—
97Dic3⋊C4—32no—
98Dic3⋊C4—32no—
99Dic3⋊C4—32no—
100Dic3⋊C4—32no—
101Dic3⋊C4—32no—
102Dic3⋊C4—62no—
103Dic3⋊C4C2×D312yescomplemented
104C4×Dic3—32no—
105C4×Dic3C2×D312yescomplemented
106C22×C32—21no—
107C22×C32—22no—
108C22×C32C2411yescharacteristic
109C2×C3⋊D3—22no—
110C2×C3⋊D3—21no—
111C2×C3×D3—22no—
112C2×C3×D3—22no—
113C2×C3×D3#4114yes—
114D32—44no—
115C32×C4—22no—
116C3⋊Dic3—22no—
117C3⋊Dic3—21no—
118C3×Dic3—22no—
119C3×Dic3—22no—
120C3×Dic3—22no—
121#126—92no—
122#124—91no—
123#123—92no—
124#123—91no—
125C22⋊Q8—92no—
126#120—91no—
127C4×D4—32no—
128#117—92no—
129C2×C22⋊C4—92no—
130C23×C3—21no—
131C23×C3—32no—
132C23×C3C22×D312yes—
133C22×D3—32no—
134C22×D3—61no—
135C22×D3—64no—
136C22×D3—64no—
137C22×D3—122no—
138C22×D3C22×D312yes—
139C3×D4—64no—
140C2×C3×C4—22no—
141C2×C3×C4—32no—
142C2×C3×C4—32no—
143C2×C3×C4—32no—
144C2×C3×C4—42no—
145C2×C3×C4—42no—
146C2×C3×C4—122no—
147C2×C3×C4C22×D312yes—
148C2×C3×C4C22×D312yes—
149#67—24no—
150#67—122no—
151#67—122no—
152C2×Dic3—32no—
153C2×Dic3—32no—
154C2×Dic3—32no—
155C2×Dic3—61no—
156C2×Dic3—62no—
157C2×Dic3—64no—
158C2×Dic3C22×D312yes—
159C2×Dic3C22×D312yes—
160C2×Dic3C22×D312yes—
161C4×D3—42no—
162C4×D3—64no—
163Dic6—122no—
164C2×C32—41no—
165C2×C32C2×#4112yes—
166C2×C32H(5,𝔽2)11yescharacteristic
167C3⋊D3—41no—
168C3×D3—24no—
169C24—91no—
170C2×Q8—91no—
171C2×D4—32no—
172C2×D4—91no—
173C22×C4—32no—
174C22×C4—92no—
175C22×C4D3211yescharacteristic, complemented
176C4⋊C4—32no—
177C4⋊C4—92no—
178C4⋊C4—92no—
179C4⋊C4—92no—
180C22⋊C4—32no—
181C22⋊C4—32no—
182C22⋊C4—92no—
183C22⋊C4—91no—
184C22⋊C4—92no—
185C22⋊C4—92no—
186C22⋊C4—91no—
187C22⋊C4—92no—
188C42—32no—
189C42—91no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—21no—
195C22×C3—34no—
196C22×C3—42no—
197C22×C3—62no—
198C22×C3—62no—
199C22×C3C23×D312yes—
200C2×D3—22no—
201C2×D3—22no—
202C2×D3—62no—
203C2×D3—62no—
204C2×D3—62no—
205C2×D3—121no—
206C2×D3—122no—
207C2×D3—128no—
208C2×D3#23514yes—
209C3×C4—22no—
210C3×C4—22no—
211C3×C4—42no—
212C3×C4—62no—
213C3×C4—62no—
214C3×C4—62no—
215Dic3—22no—
216Dic3—22no—
217Dic3—22no—
218Dic3—62no—
219Dic3—62no—
220Dic3—62no—
221Dic3—121no—
222Dic3—122no—
223C32#55111yescharacteristic, complemented, Sylow
224C23—32no—
225C23—91no—
226C23—184no—
227C23—361no—
228C23C2×D3211yescharacteristic
229Q8—361no—
230D4—64no—
231D4—361no—
232C2×C4—32no—
233C2×C4—32no—
234C2×C4—32no—
235C2×C4—41no—
236C2×C4—91no—
237C2×C4—92no—
238C2×C4—122no—
239C2×C4—184no—
240C2×C4C2×D3212yes—
241C2×C3—22no—
242C2×C3—21no—
243C2×C3—42no—
244C2×C3—42no—
245C2×C3—64no—
246C2×C3C2×#23512yes—
247C2×C3#100412yes—
248C2×C3D3×#4112yes—
249D3—24no—
250D3—122no—
251D3—241no—
252C22—22no—
253C22—21no—
254C22—34no—
255C22—62no—
256C22—62no—
257C22—182no—
258C22—181no—
259C22—364no—
260C22C22×D3211yescharacteristic
261C4—22no—
262C4—62no—
263C4—62no—
264C4—62no—
265C4—182no—
266C4—181no—
267C3—21no—
268C3#555612yescomplemented
269C2—41no—
270C2—64no—
271C2—361no—
272C2D3×#23512yes—
273C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radical#63983C2
hypercenterC22×C4D32
hypocenterC32#551
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #64138C2semidirect
2. C3×C4×#67C2semidirect
4. #63969C2semidirect
5. #63953C2semidirect
6. D3×(C2×D3)⋊C4C2semidirect
7. #63877C2semidirect
8. #63868C2semidirect
9. #63851C2semidirect
12. C22×C32×C4C22semidirect
16. C2×#627C22semidirect
18. #3757C22semidirect
20. #3750C22semidirect
21. C3×#217C22semidirect
23. C3×Dic3⋊C4C22semidirect
24. C3×C4×Dic3C22semidirect
25. #3724C22semidirect
26. Dic3⋊Dic3C22semidirect
31. Dic32C22semidirect
37. C4×#67D3semidirect
70. C22×C3×C4C2×D3semidirect
86. #217C2×D3semidirect
103. Dic3⋊C4C2×D3semidirect
105. C4×Dic3C2×D3semidirect
175. C22×C4D32semidirect
223. C32#551semidirect
268. C3#5556semidirect

Maximal subgroups

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

subgroupindex
#641382
C3×C4×#672
#639832
#639692
#639532
D3×(C2×D3)⋊C42
#638772
#638682
#638512
#55563