Finite Group Databasebeta

← #10596512

Subgroups — #10596512

Order 576.

534 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 87 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#64306C211yesmaximal, characteristic, complemented
2C2×#3802C211yesmaximal, characteristic, complemented
3#64286C211yesmaximal, characteristic, complemented
4#64241C214yesmaximal, complemented
5#64231C212yesmaximal, complemented
6#64230C214yesmaximal, complemented
7#64221C212yesmaximal, complemented
8C22×D32C2212yes—
9C2×GO+(4,𝔽2)C2218yescomplemented
10C2×C4×C3⋊D3C2211yescharacteristic
11C3×#235—21no—
12C2×C3×D12—21no—
13#3811C2212yes—
14C2×#625—21no—
15#3805—21no—
16#3802—21no—
17#3802—21no—
18#3802C2212yes—
19#3802C2212yes—
20#3802C2211yescharacteristic, complemented
21C4×D32C2212yescomplemented
22#3798—21no—
23#3797C2211yescharacteristic, complemented
24#3796—21no—
25C22⋊#157C2212yes—
26C4⋊#157C2212yescomplemented
27C4×#157C2212yescomplemented
28D32⋊C4C2218yescomplemented
29D3×#41—61no—
30C2×D3×D4—61no—
31C22×C3⋊D3C2311yescharacteristic
32C22×C3×D3—22no—
33C2×D32—22no—
34C2×D32—44no—
35C2×D32C2312yes—
36C2×D32C2314yes—
37C2×#157C2314yes—
38GO+(4,𝔽2)—48no—
39C2×C32×C4D411yescharacteristic, complemented
40C2×C3⋊Dic3D411yescharacteristic, complemented
41C4×C3⋊D3D411yescharacteristic, complemented
42C4×C3⋊D3D411yescharacteristic, complemented
43C4×C3⋊D3C2311yescharacteristic
44C4×C3⋊D3C2311yescharacteristic
45C3×#67—22no—
46C3×D12—21no—
47C3×D12—22no—
48C3×D12—22no—
49C3×C4×D3—22no—
50C3×Dic6—21no—
51#627—21no—
52#626—22no—
53#625—21no—
54#625—22no—
55#625—22no—
56#624C2312yes—
57D3×Dic3—22no—
58#535—91nomaximal, Sylow
59C23×D3—62no—
60C3×#41—61no—
61C2×C3×D4—61no—
62C2×#67—62no—
63#239—61no—
64D3×Q8—61no—
65#237—61no—
66#237—62no—
67D3×D4—62no—
68D3×D4—62no—
69D3×D4—61no—
70D3×D4—62no—
71D3×D4—62no—
72D3×D4—62no—
73#235—21no—
74#235—62no—
75C2×D12—21no—
76C2×C4×D3—62no—
77C2×C4×D3—61no—
78C2×C4×D3—61no—
79C22×C32C2×D411yescharacteristic
80C2×C3⋊D3—22no—
81C2×C3⋊D3C2×D411yescharacteristic
82C2×C3⋊D3C2411yescharacteristic
83C2×C3×D3—22no—
84C2×C3×D3—24no—
85C2×C3×D3—44no—
86D32—24no—
87D32—28no—
88D32—44no—
89#157—24no—
90C32×C4C2×D411yescharacteristic
91C32×C4C2×D411yescharacteristic
92C3⋊Dic3C2×D411yescharacteristic
93C3⋊Dic3C2×D411yescharacteristic
94C3×Dic3—22no—
95C2×#41—91no—
96C22×D4—91no—
97#127—92no—
98#124—92no—
99#121—94no—
100#120—94no—
101#117—91no—
102C23×C3—62no—
103C22×D3—22no—
104C22×D3—62no—
105C22×D3—64no—
106C22×D3—61no—
107C22×D3—61no—
108C22×D3—64no—
109C22×D3—128no—
110C3×Q8—61no—
111C3×D4—62no—
112C3×D4—62no—
113C3×D4—61no—
114C3×D4—62no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—62no—
118#67—22no—
119#67—62no—
120#67—62no—
121#67—64no—
122#67—64no—
123C2×Dic3—61no—
124C2×Dic3—61no—
125C2×Dic3—62no—
126D12—22no—
127D12—22no—
128D12—21no—
129D12—62no—
130C4×D3—22no—
131C4×D3—62no—
132C4×D3—61no—
133C4×D3—61no—
134C4×D3—61no—
135C4×D3—61no—
136C4×D3—61no—
137C4×D3—61no—
138C4×D3—61no—
139C4×D3—61no—
140C4×D3—62no—
141Dic6—21no—
142Dic6—62no—
143C2×C32—21no—
144C2×C32C22×D411yescharacteristic
145C3⋊D3—21no—
146C3⋊D3H(5,𝔽2)12yes—
147C3×D3—44no—
148C3×D3—42no—
149C24—92no—
150#41—61no—
151#41—181no—
152#41—181no—
153#41—181no—
154C2×Q8—91no—
155C2×D4—61no—
156C2×D4—92no—
157C2×D4—92no—
158C2×D4—92no—
159C2×D4—91no—
160C2×D4—98no—
161C2×D4—181no—
162C2×D4—181no—
163C2×D4—181no—
164C22×C4—92no—
165C22×C4—91no—
166C4⋊C4—92no—
167C22⋊C4—92no—
168C22⋊C4—98no—
169C42—92no—
170C22×C3—21no—
171C22×C3—21no—
172C22×C3—62no—
173C22×C3—64no—
174C22×C3—124no—
175C2×D3—22no—
176C2×D3—24no—
177C2×D3—44no—
178C2×D3—61no—
179C2×D3—61no—
180C2×D3—61no—
181C2×D3—61no—
182C2×D3—64no—
183C2×D3—128no—
184C2×D3—124no—
185C2×D3—122no—
186C2×D3—128no—
187C3×C4—21no—
188C3×C4—21no—
189C3×C4—21no—
190C3×C4—21no—
191C3×C4—62no—
192Dic3—22no—
193Dic3—61no—
194Dic3—61no—
195Dic3—61no—
196Dic3—61no—
197C32#53511yescharacteristic, complemented, Sylow
198C23—62no—
199C23—94no—
200C23—92no—
201C23—91no—
202C23—182no—
203C23—364no—
204Q8—61no—
205Q8—181no—
206D4—62no—
207D4—62no—
208D4—62no—
209D4—61no—
210D4—182no—
211D4—182no—
212D4—182no—
213D4—181no—
214D4—368no—
215C2×C4—62no—
216C2×C4—92no—
217C2×C4—91no—
218C2×C4—91no—
219C2×C4—91no—
220C2×C4—91no—
221C2×C4—91no—
222C2×C4—94no—
223C2×C4—182no—
224C2×C4GO+(4,𝔽2)11yescharacteristic, complemented
225C2×C3—22no—
226C2×C3—21no—
227C2×C3—21no—
228C2×C3—41no—
229C2×C3—124no—
230C2×C3—122no—
231D3—44no—
232D3—42no—
233D3—62no—
234D3—62no—
235D3—62no—
236D3—121no—
237C22—62no—
238C22—64no—
239C22—91no—
240C22—91no—
241C22—124no—
242C22—182no—
243C22—184no—
244C22—188no—
245C22—364no—
246C22C2×GO+(4,𝔽2)11yescharacteristic
247C4—62no—
248C4—91no—
249C4—91no—
250C4—184no—
251C4C2×GO+(4,𝔽2)11yescharacteristic
252C4C2×GO+(4,𝔽2)11yescharacteristic
253C3—21no—
254C3—21no—
255C2—21no—
256C2—92no—
257C2—124no—
258C2—122no—
259C2—181no—
260C2C22×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC2C22×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24
FittingC2×C32×C4D4
socleC2×C32C22×D4
hypercenterC2×C4GO+(4,𝔽2)
hypocenterC32#535
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #64306C2semidirect
2. C2×#3802C2semidirect
3. #64286C2semidirect
4. #64241C2semidirect
5. #64231C2semidirect
6. #64230C2semidirect
7. #64221C2semidirect
9. C2×GO+(4,𝔽2)C22semidirect
20. #3802C22semidirect
21. C4×D32C22semidirect
23. #3797C22semidirect
26. C4⋊#157C22semidirect
27. C4×#157C22semidirect
28. D32⋊C4C22semidirect
39. C2×C32×C4D4semidirect
40. C2×C3⋊Dic3D4semidirect
41. C4×C3⋊D3D4semidirect
42. C4×C3⋊D3D4semidirect
197. C32#535semidirect
224. C2×C4GO+(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
#643062
C2×#38022
#642862
#642412
#642312
#642302
#642212
#5359