Finite Group Databasebeta

← #10594165

Subgroups — #10594165

Order 576.

561 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 124 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#64091C211yesmaximal, characteristic
2C3×#879C212yesmaximal, complemented
3#63975C211yesmaximal, characteristic, complemented
4#63974C212yesmaximal, complemented
5#63953C211yesmaximal, characteristic, complemented
6#63897C212yesmaximal, complemented
7#63895C212yesmaximal, complemented
8#63894C212yesmaximal, complemented
9#63836C212yesmaximal, complemented
10#5705—32nomaximal
11C23×C3⋊D3C2211yescharacteristic
12C2×#638C2211yescharacteristic
13C2×C4⋊(C3⋊D3)C2212yes—
14C2×#627C2211yescharacteristic, complemented
15C2×#624C2212yes—
16C2×#624C2212yes—
17C2×#624C2211yescharacteristic, complemented
18C32×C22⋊C4C2211yescharacteristic
19#3752C2212yes—
20C3×#217C2212yes—
21C3×#217C2212yes—
22C3×C4⋊Dic3C2212yescomplemented
23C3×Dic3⋊C4C2212yescomplemented
24C3×Dic3⋊C4C2212yescomplemented
25C3×C4×Dic3C2212yescomplemented
26Dic3⋊Dic3C2212yescomplemented
27#3722C2212yes—
28#3722C2212yes—
29#3722C2212yescomplemented
30#3722C2211yescharacteristic, complemented
31Dic32C2211yescharacteristic, complemented
32C3×#126—32no—
33#897—32no—
34#896—32no—
35#894—32no—
36#894—32no—
37#893—32no—
38#892—32no—
39#885—32no—
40#883—32no—
41#882—32no—
42#882—61no—
43D3×C22⋊C4—32no—
44D3×C22⋊C4—32no—
45#879D312yescomplemented
46C4×D12—32no—
47#872—32no—
48C23×C32C2311yescharacteristic
49C22×C3⋊D3—24no—
50C22×C3⋊D3—41no—
51C22×C3⋊D3C2311yescharacteristic
52C22×C3⋊D3C2312yes—
53C2×C32×C4C2312yes—
54#638—41no—
55C2×C3⋊Dic3C2311yescharacteristic
56C4⋊(C3⋊D3)—24no—
57C2×C3×Dic3C2312yes—
58C2×C3×Dic3C2312yes—
59C2×C3×Dic3C2312yes—
60C2×C3×Dic3C2312yes—
61#627—41no—
62#624—24no—
63#624—42no—
64#624—41no—
65#551—91noSylow
66C23×D3—32no—
67C23×D3—61no—
68C2×#67—32no—
69C2×#67—61no—
70C2×D12—32no—
71C2×D12—32no—
72C2×D12—62no—
73C2×C4×D3—32no—
74C2×C4×D3—32no—
75C2×C4×D3—32no—
76C2×C4×D3—32no—
77C2×C4×D3—32no—
78C2×Dic6—32no—
79C3×C4⋊C4—32no—
80C3×C4⋊C4—32no—
81C3×C4⋊C4—32no—
82C3×C22⋊C4—21no—
83C3×C22⋊C4—32no—
84C3×C22⋊C4—32no—
85C3×C22⋊C4C2×D312yes—
86C3×C42—32no—
87#217C2×D312yes—
88#217C2×D312yes—
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—32no—
95(C2×D3)⋊C4—32no—
96(C2×D3)⋊C4—32no—
97(C2×D3)⋊C4—32no—
98(C2×D3)⋊C4—62no—
99C4⋊Dic3—32no—
100C4⋊Dic3C2×D312yescomplemented
101Dic3⋊C4—32no—
102Dic3⋊C4C2×D312yescomplemented
103Dic3⋊C4C2×D312yescomplemented
104C4×Dic3—32no—
105C4×Dic3C2×D312yescomplemented
106C22×C32—21no—
107C22×C32—22no—
108C22×C32C2411yescharacteristic
109C2×C3⋊D3—22no—
110C2×C3⋊D3—22no—
111C2×C3⋊D3—22no—
112C2×C3⋊D3—21no—
113C2×C3⋊D3—44no—
114C2×C3⋊D3#4114yes—
115C32×C4—22no—
116C3⋊Dic3—21no—
117C3×Dic3—22no—
118C3×Dic3—22no—
119C3×Dic3—22no—
120C3×Dic3—22no—
121#126—32no—
122#124—91no—
123#123—91no—
124#123—92no—
125C22⋊Q8—92no—
126#120—91no—
127C4×D4—92no—
128#117—92no—
129C2×C22⋊C4—92no—
130C23×C3—21no—
131C23×C3C22×D312yes—
132C22×D3—32no—
133C22×D3—32no—
134C22×D3—32no—
135C22×D3—62no—
136C22×D3—61no—
137C22×D3—64no—
138C22×D3—64no—
139C22×D3—124no—
140C22×D3—122no—
141C22×D3—122no—
142C2×C3×C4—22no—
143C2×C3×C4—32no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—32no—
147C2×C3×C4C22×D312yes—
148C2×C3×C4C22×D312yes—
149#67—122no—
150#67—122no—
151C2×Dic3—32no—
152C2×Dic3—61no—
153C2×Dic3C22×D312yes—
154C2×Dic3C22×D312yes—
155C2×Dic3C22×D312yes—
156C2×Dic3C22×D312yes—
157D12—64no—
158D12—64no—
159D12—124no—
160C4×D3—64no—
161C4×D3—64no—
162C4×D3—122no—
163C4×D3—122no—
164C4×D3—122no—
165Dic6—122no—
166C2×C32—41no—
167C2×C32C2×#4112yes—
168C2×C32H(5,𝔽2)11yescharacteristic
169C3⋊D3—24no—
170C3⋊D3—41no—
171C24—91no—
172C2×Q8—91no—
173C2×D4—91no—
174C2×D4—92no—
175C22×C4—92no—
176C22×C4—92no—
177C22×C4—91no—
178C4⋊C4—32no—
179C4⋊C4—32no—
180C4⋊C4—32no—
181C4⋊C4—92no—
182C22⋊C4—32no—
183C22⋊C4—32no—
184C22⋊C4—92no—
185C22⋊C4—92no—
186C22⋊C4—92no—
187C22⋊C4—91no—
188C22⋊C4—92no—
189C22⋊C4D3211yescharacteristic
190C42—32no—
191C42—91no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—22no—
195C22×C3—22no—
196C22×C3—21no—
197C22×C3—42no—
198C22×C3C23×D312yes—
199C2×D3—34no—
200C2×D3—34no—
201C2×D3—64no—
202C2×D3—62no—
203C2×D3—62no—
204C2×D3—62no—
205C2×D3—62no—
206C2×D3—62no—
207C2×D3—62no—
208C2×D3—62no—
209C2×D3—122no—
210C2×D3—122no—
211C2×D3—122no—
212C2×D3—121no—
213C2×D3—128no—
214C2×D3—128no—
215C3×C4—22no—
216C3×C4—22no—
217C3×C4—42no—
218C3×C4—62no—
219C3×C4—62no—
220C3×C4—62no—
221C3×C4—62no—
222Dic3—22no—
223Dic3—22no—
224Dic3—22no—
225Dic3—22no—
226Dic3—62no—
227Dic3—121no—
228C32#55111yescharacteristic, complemented, Sylow
229C23—92no—
230C23—91no—
231C23—184no—
232C23—361no—
233C23C2×D3211yescharacteristic
234Q8—361no—
235D4—184no—
236D4—361no—
237C2×C4—32no—
238C2×C4—32no—
239C2×C4—32no—
240C2×C4—32no—
241C2×C4—91no—
242C2×C4—184no—
243C2×C4—361no—
244C2×C4—362no—
245C2×C4C2×D3212yes—
246C2×C3—21no—
247C2×C3—22no—
248C2×C3—42no—
249C2×C3—42no—
250C2×C3D3×#4112yes—
251C2×C3D3×#4112yes—
252C2×C3#100912yes—
253D3—64no—
254D3—64no—
255D3—124no—
256D3—122no—
257D3—241no—
258C22—21no—
259C22—22no—
260C22—94no—
261C22—182no—
262C22—182no—
263C22—181no—
264C22—182no—
265C22—364no—
266C22C22×D3211yescharacteristic
267C4—22no—
268C4—62no—
269C4—62no—
270C4—62no—
271C4—62no—
272C4—181no—
273C3—21no—
274C3#570512yescomplemented
275C2—41no—
276C2—184no—
277C2—361no—
278C2#6430811yescharacteristic
279C2#6431212yes—
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radical#64091C2
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
2. C3×#879C2semidirect
3. #63975C2semidirect
4. #63974C2semidirect
5. #63953C2semidirect
6. #63897C2semidirect
7. #63895C2semidirect
8. #63894C2semidirect
9. #63836C2semidirect
14. C2×#627C22semidirect
17. C2×#624C22semidirect
22. C3×C4⋊Dic3C22semidirect
23. C3×Dic3⋊C4C22semidirect
24. C3×Dic3⋊C4C22semidirect
25. C3×C4×Dic3C22semidirect
26. Dic3⋊Dic3C22semidirect
29. #3722C22semidirect
30. #3722C22semidirect
31. Dic32C22semidirect
45. #879D3semidirect
100. C4⋊Dic3C2×D3semidirect
102. Dic3⋊C4C2×D3semidirect
103. Dic3⋊C4C2×D3semidirect
105. C4×Dic3C2×D3semidirect
228. C32#551semidirect
274. C3#5705semidirect

Maximal subgroups

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

subgroupindex
#640912
C3×#8792
#639752
#639742
#639532
#638972
#638952
#638942
#638362
#57053