Finite Group Databasebeta

← #10594192

Subgroups — #10594192

Order 576.

706 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 136 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×#625C211yesmaximal, characteristic, complemented
2C2×#3805C211yesmaximal, characteristic, complemented
3#64086C211yesmaximal, characteristic, complemented
4C3×#882C212yesmaximal, complemented
5#63976C212yesmaximal, complemented
6#63915C212yesmaximal, complemented
7#63914C212yesmaximal, complemented
8#63885C214yesmaximal, complemented
9#5592—32nomaximal
10C23×C3×D3C2212yes—
11C22×C3⋊Dic3C2211yescharacteristic
12C2×C3×#67C2212yes—
13C2×C3×D12C2214yescomplemented
14C2×#627C2211yescharacteristic, complemented
15C2×#625—24no—
16C2×#625C2212yes—
17C2×#625C2212yes—
18C2×#625C2211yescharacteristic, complemented
19C2×#625C2214yescomplemented
20#3805—24no—
21C2×D3×Dic3C2212yescomplemented
22C32×C22⋊C4C2211yescharacteristic
23#3757C2211yescharacteristic
24#3750C2212yescomplemented
25C4×C3⋊Dic3C2212yescomplemented
26C3×(C2×D3)⋊C4C2214yescomplemented
27(C2×D3)⋊Dic3C2214yescomplemented
28C22×#67—32no—
29C2×#237—32no—
30C3×#120—32no—
31#953—32no—
32#940—34no—
33#939—34no—
34#937—32no—
35#885—34no—
36#884—34no—
37#882D312yescomplemented
38#877—32no—
39#877—61no—
40C23×C32C2311yescharacteristic
41C22×C3×D3—24no—
42C22×C3×D3—44no—
43C22×C3×D3C2314yes—
44C22×C3×D3C2312yes—
45C2×C32×C4C2312yes—
46C2×C3⋊Dic3D414yescomplemented
47C2×C3⋊Dic3C2312yes—
48C2×C3⋊Dic3C2311yescharacteristic
49C2×C3⋊Dic3C2311yescharacteristic
50C3×#67—24no—
51C2×C3×Dic3C2312yes—
52C3×D12—44no—
53#627—22no—
54#625—22no—
55#625—24no—
56#625—24no—
57#625—44no—
58D3×Dic3—24no—
59#535—91noSylow
60C24×C3—32no—
61C23×D3C2×D312yes—
62C2×C3×D4—32no—
63C2×C3×D4—34no—
64C2×#67—34no—
65C2×#67—34no—
66C2×#67—32no—
67C2×#67—34no—
68C2×#67—34no—
69C2×#67—68no—
70C2×#67C2×D312yes—
71C22×Dic3—32no—
72C22×Dic3—32no—
73C22×Dic3—61no—
74#237—68no—
75C2×D12C2×D314yescomplemented
76C2×C4×D3—32no—
77C2×Dic6—32no—
78C3×C22⋊C4—21no—
79C3×C22⋊C4—34no—
80C3×C22⋊C4C2×D312yes—
81#217—32no—
82#217—34no—
83#217—61no—
84(C2×D3)⋊C4—34no—
85(C2×D3)⋊C4C2×D314yescomplemented
86Dic3⋊C4—34no—
87Dic3⋊C4—62no—
88C4×Dic3—34no—
89C4×Dic3—62no—
90C22×C32—22no—
91C22×C32C2×D412yes—
92C22×C32C2411yescharacteristic
93C2×C3×D3—22no—
94C2×C3×D3—24no—
95C2×C3×D3—24no—
96C2×C3×D3—24no—
97C2×C3×D3—24no—
98C2×C3×D3—48no—
99C32×C4—22no—
100C3⋊Dic3—22no—
101C3⋊Dic3C2×D412yes—
102C3⋊Dic3C2×D412yes—
103C3×Dic3—22no—
104C2×#41—91no—
105C22×D4—91no—
106#127—92no—
107#124—92no—
108#121—94no—
109#120—32no—
110#120—92no—
111#117—91no—
112C23×C3—21no—
113C23×C3—34no—
114C23×C3—32no—
115C23×C3—64no—
116C23×C3—124no—
117C23×C3C22×D312yes—
118C22×D3—24no—
119C22×D3—44no—
120C22×D3C22×D314yes—
121C22×D3C22×D312yes—
122C3×D4—64no—
123C3×D4—124no—
124C2×C3×C4—22no—
125C2×C3×C4—32no—
126C2×C3×C4C22×D314yes—
127#67—24no—
128#67—64no—
129#67—68no—
130#67—68no—
131#67—124no—
132#67—124no—
133C2×Dic3—38no—
134C2×Dic3—32no—
135C2×Dic3—32no—
136C2×Dic3—34no—
137C2×Dic3—64no—
138C2×Dic3—61no—
139C2×Dic3—61no—
140C2×Dic3—62no—
141C2×Dic3—64no—
142C2×Dic3C22×D312yes—
143D12—44no—
144C4×D3—64no—
145Dic6—64no—
146C2×C32—22no—
147C2×C32C22×D411yescharacteristic
148C2×C32H(5,𝔽2)12yes—
149C3×D3—42no—
150C3×D3—44no—
151C24—32no—
152#41—184no—
153C2×Q8—91no—
154C2×D4—32no—
155C2×D4—34no—
156C2×D4—92no—
157C2×D4—92no—
158C2×D4—91no—
159C2×D4—94no—
160C2×D4—184no—
161C22×C4—92no—
162C22×C4—91no—
163C4⋊C4—92no—
164C22⋊C4—34no—
165C22⋊C4—94no—
166C22⋊C4—91no—
167C22⋊C4D3211yescharacteristic
168C42—92no—
169C22×C3—24no—
170C22×C3—21no—
171C22×C3—22no—
172C22×C3—42no—
173C22×C3—62no—
174C22×C3—64no—
175C22×C3—64no—
176C22×C3—64no—
177C22×C3—64no—
178C22×C3—128no—
179C22×C3D3×D414yes—
180C22×C3C23×D312yes—
181C2×D3—22no—
182C2×D3—24no—
183C2×D3—24no—
184C2×D3—24no—
185C2×D3—24no—
186C2×D3—48no—
187C3×C4—24no—
188C3×C4—42no—
189C3×C4—62no—
190Dic3—22no—
191Dic3—34no—
192Dic3—34no—
193Dic3—62no—
194Dic3—62no—
195Dic3—64no—
196Dic3—122no—
197C32#53511yescharacteristic, complemented, Sylow
198C23—34no—
199C23—32no—
200C23—64no—
201C23—124no—
202C23C2×D3211yescharacteristic
203Q8—182no—
204D4—64no—
205D4—124no—
206D4—184no—
207D4—184no—
208D4—182no—
209D4—364no—
210C2×C4—32no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—94no—
214C2×C4—92no—
215C2×C4—184no—
216C2×C4C2×D3212yes—
217C2×C3—24no—
218C2×C3—21no—
219C2×C3—22no—
220C2×C3—42no—
221C2×C3—124no—
222C2×C3—122no—
223C2×C3C2×D3×D412yes—
224C2×C3#100414yes—
225D3—44no—
226D3—42no—
227C22—22no—
228C22—62no—
229C22—64no—
230C22—64no—
231C22—64no—
232C22—64no—
233C22—128no—
234C22#380212yes—
235C22C22×D3211yescharacteristic
236C4—22no—
237C4—62no—
238C4—92no—
239C4—92no—
240C4—182no—
241C3—21no—
242C3#559212yescomplemented
243C2—22no—
244C2—124no—
245C2—122no—
246C2C2×#380211yescharacteristic
247C2#6431312yes—
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
hypercenterC22⋊C4D32
hypocenterC32#535
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#625C2semidirect
2. C2×#3805C2semidirect
3. #64086C2semidirect
4. C3×#882C2semidirect
5. #63976C2semidirect
6. #63915C2semidirect
7. #63914C2semidirect
8. #63885C2semidirect
13. C2×C3×D12C22semidirect
14. C2×#627C22semidirect
18. C2×#625C22semidirect
19. C2×#625C22semidirect
21. C2×D3×Dic3C22semidirect
24. #3750C22semidirect
25. C4×C3⋊Dic3C22semidirect
26. C3×(C2×D3)⋊C4C22semidirect
27. (C2×D3)⋊Dic3C22semidirect
37. #882D3semidirect
46. C2×C3⋊Dic3D4semidirect
75. C2×D12C2×D3semidirect
85. (C2×D3)⋊C4C2×D3semidirect
197. C32#535semidirect
242. C3#5592semidirect

Maximal subgroups

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

subgroupindex
C22×#6252
C2×#38052
#640862
C3×#8822
#639762
#639152
#639142
#638852
#55923