Finite Group Databasebeta

← #10595039

Subgroups — #10595039

Order 576.

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

normal subgroups only

This group has 566 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
1C2×#3752C211yesmaximal, characteristic, complemented
2C3×C4×#67C212yesmaximal, complemented
3#63982C211yesmaximal, characteristic, complemented
4#63981C212yesmaximal, complemented
5#63980C211yesmaximal, characteristic, complemented
6#63906C212yesmaximal, complemented
7#63868C212yesmaximal, complemented
8#63852C212yesmaximal, complemented
9#63850C212yesmaximal, complemented
10#5560—32nomaximal
11C23×C3⋊D3C2211yescharacteristic
12C22×C32×C4C2211yescharacteristic
13C22×C3⋊Dic3C2211yescharacteristic
14C2×C3×#67C2212yes—
15C2×C3×C4×D3C2212yescomplemented
16C2×#627C2211yescharacteristic, complemented
17C2×#626C2212yes—
18C2×#626C2212yes—
19C2×#625C2211yescharacteristic, complemented
20#3752C2212yes—
21#3752C2212yes—
22C3×#217C2212yes—
23C3×(C2×D3)⋊C4C2212yescomplemented
24C3×Dic3⋊C4C2212yescomplemented
25C3×C4×Dic3C2212yescomplemented
26#3724C2211yescharacteristic, complemented
27#3724C2211yescharacteristic, complemented
28Dic3⋊Dic3C2212yescomplemented
29#3722C2212yes—
30#3722C2212yes—
31(C2×D3)⋊Dic3C2212yescomplemented
32C3×C4×D4—32no—
33#939—32no—
34#937—32no—
35#930—32no—
36#929—32no—
37C4×#67D312yescomplemented
38C2×(C2×D3)⋊C4—32no—
39C2×(C2×D3)⋊C4—61no—
40#898—32no—
41#894—32no—
42#885—32no—
43#884—32no—
44#882—32no—
45#878—32no—
46#876—32no—
47#875—32no—
48C23×C32C2311yescharacteristic
49C22×C3⋊D3—22no—
50C22×C3⋊D3—41no—
51C22×C3⋊D3—41no—
52C22×C3⋊D3C2312yes—
53C22×C3×D3C2312yes—
54C2×C32×C4—41no—
55C2×C32×C4C2312yes—
56C2×C3⋊Dic3—41no—
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2311yescharacteristic
59C3×#67—24no—
60C2×C3×Dic3C2312yes—
61C2×C3×Dic3C2312yes—
62C2×C3×Dic3C2312yes—
63C3×C4×D3—42no—
64#627—41no—
65#626—42no—
66#626—42no—
67#625—41no—
68#538—91noSylow
69C23×D3—32no—
70C23×D3—61no—
71C2×C3×D4—32no—
72C22×C3×C4—21no—
73C22×C3×C4—32no—
74C22×C3×C4C2×D312yes—
75C2×#67—32no—
76C2×#67—32no—
77C2×#67—32no—
78C2×#67C2×D312yes—
79C22×Dic3—32no—
80C22×Dic3—61no—
81C2×D12—32no—
82C2×D12—32no—
83C2×C4×D3C2×D312yescomplemented
84C2×Dic6—32no—
85C3×C4⋊C4—32no—
86C3×C22⋊C4—32no—
87C3×C22⋊C4—32no—
88C3×C42—32no—
89#217—32no—
90#217C2×D312yes—
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—32no—
99(C2×D3)⋊C4—32no—
100(C2×D3)⋊C4—62no—
101(C2×D3)⋊C4—62no—
102(C2×D3)⋊C4C2×D312yescomplemented
103C4⋊Dic3—32no—
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4—32no—
107Dic3⋊C4C2×D312yescomplemented
108C4×Dic3C2×D312yescomplemented
109C22×C32—21no—
110C22×C32—21no—
111C22×C32#4112yes—
112C22×C32C2411yescharacteristic
113C2×C3⋊D3—22no—
114C2×C3⋊D3—22no—
115C2×C3⋊D3—22no—
116C2×C3⋊D3—44no—
117C2×C3×D3—22no—
118C2×C3×D3—22no—
119C2×C3×D3—22no—
120C32×C4—22no—
121C3⋊Dic3—21no—
122C3⋊Dic3—21no—
123C3×Dic3—22no—
124C3×Dic3—22no—
125C3×Dic3—22no—
126#126—92no—
127#124—92no—
128#123—92no—
129C22⋊Q8—91no—
130#121—92no—
131#121—91no—
132#120—92no—
133C4×D4—32no—
134C2×C22⋊C4—91no—
135C23×C3—21no—
136C23×C3—32no—
137C23×C3C22×D312yes—
138C22×D3—32no—
139C22×D3—32no—
140C22×D3—62no—
141C22×D3—64no—
142C22×D3—64no—
143C22×D3—122no—
144C22×D3—122no—
145C22×D3—122no—
146C22×D3—122no—
147C22×D3C22×D312yes—
148C3×D4—64no—
149C2×C3×C4—22no—
150C2×C3×C4—32no—
151C2×C3×C4—32no—
152C2×C3×C4—32no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—122no—
156C2×C3×C4C22×D312yes—
157C2×C3×C4C22×D312yes—
158#67—24no—
159#67—122no—
160#67—122no—
161#67—122no—
162C2×Dic3—32no—
163C2×Dic3—32no—
164C2×Dic3—61no—
165C2×Dic3—61no—
166C2×Dic3—122no—
167C2×Dic3—122no—
168C2×Dic3C22×D312yes—
169C2×Dic3C22×D312yes—
170C2×Dic3C22×D312yes—
171D12—122no—
172D12—122no—
173C4×D3—42no—
174Dic6—122no—
175C2×C32—22no—
176C2×C32C2×#4111yescharacteristic
177C2×C32H(5,𝔽2)11yescharacteristic
178C2×C32H(5,𝔽2)11yescharacteristic
179C3⋊D3—42no—
180C3×D3—42no—
181C24—91no—
182C2×Q8—91no—
183C2×D4—32no—
184C2×D4—92no—
185C2×D4—92no—
186C2×D4—91no—
187C22×C4—32no—
188C22×C4—91no—
189C22×C4D3211yescharacteristic
190C4⋊C4—32no—
191C4⋊C4—92no—
192C4⋊C4—91no—
193C4⋊C4—91no—
194C22⋊C4—32no—
195C22⋊C4—32no—
196C22⋊C4—92no—
197C22⋊C4—92no—
198C22⋊C4—92no—
199C22⋊C4—92no—
200C22⋊C4—92no—
201C42—32no—
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—21no—
207C22×C3—22no—
208C22×C3—62no—
209C22×C3—62no—
210C22×C3—62no—
211C22×C3#23514yes—
212C22×C3C23×D312yes—
213C2×D3—22no—
214C2×D3—22no—
215C2×D3—22no—
216C2×D3—64no—
217C2×D3—62no—
218C2×D3—62no—
219C2×D3—62no—
220C2×D3—62no—
221C2×D3—62no—
222C2×D3—62no—
223C2×D3—122no—
224C2×D3—122no—
225C2×D3—128no—
226C2×D3—128no—
227C3×C4—22no—
228C3×C4—22no—
229C3×C4—42no—
230C3×C4—62no—
231C3×C4—62no—
232C3×C4—62no—
233Dic3—22no—
234Dic3—22no—
235Dic3—22no—
236Dic3—62no—
237Dic3—62no—
238Dic3—121no—
239Dic3—121no—
240C32#53811yescharacteristic, complemented, Sylow
241C23—32no—
242C23—92no—
243C23—182no—
244C23—361no—
245C23—361no—
246C23C2×D3211yescharacteristic
247Q8—361no—
248D4—64no—
249D4—362no—
250D4—362no—
251D4—361no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—41no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—122no—
259C2×C4—361no—
260C2×C4C2×D3212yes—
261C2×C3—24no—
262C2×C3—24no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—122no—
267C2×C3C2×#23512yes—
268C2×C3#100412yes—
269C2×C3#100912yes—
270D3—42no—
271D3—124no—
272D3—122no—
273D3—122no—
274C22—21no—
275C22—21no—
276C22—62no—
277C22—62no—
278C22—62no—
279C22—182no—
280C22—182no—
281C22—182no—
282C22—364no—
283C22#379812yes—
284C22C22×D3211yescharacteristic
285C4—22no—
286C4—62no—
287C4—62no—
288C4—62no—
289C4—181no—
290C4—181no—
291C3—21no—
292C3#556012yescomplemented
293C2—22no—
294C2—122no—
295C2—362no—
296C2C2×#379811yescharacteristic
297C2#6430811yescharacteristic
298C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3752C2semidirect
2. C3×C4×#67C2semidirect
3. #63982C2semidirect
4. #63981C2semidirect
5. #63980C2semidirect
6. #63906C2semidirect
7. #63868C2semidirect
8. #63852C2semidirect
9. #63850C2semidirect
15. C2×C3×C4×D3C22semidirect
16. C2×#627C22semidirect
19. C2×#625C22semidirect
23. C3×(C2×D3)⋊C4C22semidirect
24. C3×Dic3⋊C4C22semidirect
25. C3×C4×Dic3C22semidirect
26. #3724C22semidirect
27. #3724C22semidirect
28. Dic3⋊Dic3C22semidirect
31. (C2×D3)⋊Dic3C22semidirect
37. C4×#67D3semidirect
83. C2×C4×D3C2×D3semidirect
102. (C2×D3)⋊C4C2×D3semidirect
107. Dic3⋊C4C2×D3semidirect
108. C4×Dic3C2×D3semidirect
240. C32#538semidirect
292. C3#5560semidirect

Maximal subgroups

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

subgroupindex
C2×#37522
C3×C4×#672
#639822
#639812
#639802
#639062
#638682
#638522
#638502
#55603