Finite Group Databasebeta

← #10590957

Subgroups — #10590957

Order 576.

358 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3796C211yesmaximal, characteristic
2C2×#3743C211yesmaximal, characteristic, complemented
3#63900C211yesmaximal, characteristic, complemented
4C3×#821C211yesmaximal, characteristic, complemented
5C3×#816C211yesmaximal, characteristic, complemented
6#63565C211yesmaximal, characteristic, complemented
7#63562C211yesmaximal, characteristic, complemented
8#4803—31nomaximal
9#4771—31nomaximal
10C2×C4×C3⋊D3C2211yescharacteristic
11C2×C3×D12C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C2×#626—21no—
14C2×#624—21no—
15C2×D3×Dic3—21no—
16#3796—22no—
17#3796C414yescomplemented
18C2×C32×C8C2211yescharacteristic
19C2×C3⋊(C3⋊C8)C2211yescharacteristic
20#3743—22no—
21C3×C4⋊Dic3C2211yescharacteristic
22C3×C4⋊Dic3C2211yescharacteristic
23#3724—21no—
24C2×#239—31no—
25C2×#237—31no—
26C2×ΓL(1,𝔽25)—31no—
27C2×ΓL(1,𝔽25)—31no—
28C2×ΓL(1,𝔽25)—61no—
29D3×C4⋊C4—31no—
30D3×C4⋊C4—31no—
31C3×#103—31no—
32C3×#102—31no—
33#835—31no—
34#832—31no—
35#821D311yescharacteristic, complemented
36#816D311yescharacteristic, complemented
37#810—31no—
38#809—31no—
39C22×C3⋊D3D411yescharacteristic
40C22×C3×D3—21no—
41C2×C32×C4C2311yescharacteristic
42C2×C3⋊Dic3D411yescharacteristic
43C4×C3⋊D3C2×C412yes—
44C4×C3⋊D3D412yes—
45C2×C3×Dic3—21no—
46C2×C3×Dic3—21no—
47C2×C3×Dic3—21no—
48C3×D12—21no—
49C3×D12C2×C412yescomplemented
50C3×Dic6—21no—
51C3×Dic6C2×C412yescomplemented
52#626—24no—
53#624—42no—
54D3×Dic3—24no—
55C32×C8—21no—
56C3⋊(C3⋊C8)—21no—
57#417—91noSylow
58C2×C3×Q8—31no—
59C2×C3×D4—31no—
60C2×#67—61no—
61C22×Dic3—61no—
62#239—34no—
63#239—62no—
64#237—34no—
65#237—62no—
66C2×D12—61no—
67C2×D12C2×D311yescharacteristic
68C2×C4×D3—31no—
69C2×C4×D3—31no—
70C2×C4×D3—61no—
71C2×C4×D3—61no—
72C2×C4×D3—61no—
73C2×C4×D3—61no—
74C2×Dic6C2×D311yescharacteristic
75C2×C3×C8—21no—
76C2×C3×C8C2×D311yescharacteristic
77C2×C3×C8C2×D311yescharacteristic
78C3×C4⋊C4—31no—
79C3×C4⋊C4—31no—
80C4⋊Dic3C2×D311yescharacteristic
81C4⋊Dic3C2×D311yescharacteristic
82Dic3⋊C4—61no—
83Dic3⋊C4—61no—
84C2×C3⋊C8—31no—
85C2×C3⋊C8—31no—
86C2×C3⋊C8—61no—
87ΓL(1,𝔽25)—62no—
88ΓL(1,𝔽25)—62no—
89ΓL(1,𝔽25)—64no—
90C22×C32C2×D411yescharacteristic
91C2×C3⋊D3—21no—
92C2×C3⋊D3—21no—
93C2×C3⋊D3C22⋊C412yes—
94C2×C3×D3—22no—
95C2×C3×D3—41no—
96C2×C3×D3—41no—
97C32×C4C22×C411yescharacteristic
98C32×C4C2×D411yescharacteristic
99C3⋊Dic3C22⋊C412yes—
100C3×Dic3—22no—
101C3×Dic3—41no—
102C3×Dic3—41no—
103C2×#41—91no—
104C2×M16—91no—
105C2×C4⋊C4—91no—
106#103—31no—
107#103—91no—
108#102—31no—
109#102—91no—
110C23×C3—61no—
111C22×D3—21no—
112C22×D3—31no—
113C22×D3—31no—
114C22×D3—61no—
115C3×Q8—32no—
116C3×Q8—61no—
117C3×D4—32no—
118C3×D4—61no—
119C2×C3×C4—21no—
120C2×C3×C4—61no—
121C2×C3×C4—61no—
122C2×C3×C4—61no—
123C2×C3×C4C22×D311yescharacteristic
124C2×C3×C4C22×D311yescharacteristic
125#67—64no—
126C2×Dic3—21no—
127C2×Dic3—21no—
128C2×Dic3—21no—
129C2×Dic3—31no—
130C2×Dic3—31no—
131C2×Dic3—61no—
132C2×Dic3—64no—
133D12—21no—
134D12—64no—
135D12C4×D312yescomplemented
136C4×D3—32no—
137C4×D3—32no—
138C4×D3—32no—
139C4×D3—32no—
140C4×D3—62no—
141C4×D3—62no—
142C4×D3—64no—
143C4×D3—122no—
144C4×D3—122no—
145Dic6—21no—
146Dic6C4×D312yescomplemented
147C3×C8—22no—
148C3×C8—21no—
149C3×C8—21no—
150C3⋊C8—62no—
151C3⋊C8—61no—
152C3⋊C8—61no—
153C2×C32C2×C22⋊C411yescharacteristic
154C2×C32AGL(1,ℤ/8)11yescharacteristic
155C2×C32#13711yescharacteristic
156C3⋊D3—22no—
157C3×D3—42no—
158#41—94no—
159#41—182no—
160C2×Q8—31no—
161C2×D4—31no—
162C2×D4—181no—
163C22×C4—91no—
164C22×C4—181no—
165C22×C4—181no—
166M16—182no—
167C2×C8—91no—
168C2×C8D3211yescharacteristic
169C4⋊C4—31no—
170C4⋊C4—31no—
171C4⋊C4—181no—
172C22×C3—21no—
173C22×C3—62no—
174C22×C3—121no—
175C22×C3—121no—
176C22×C3D3×D411yescharacteristic
177C22×C3D3×D411yescharacteristic
178C2×D3—22no—
179C2×D3—32no—
180C2×D3—32no—
181C2×D3—41no—
182C2×D3—41no—
183C2×D3—62no—
184C2×D3—61no—
185C2×D3—61no—
186C2×D3—61no—
187C2×D3—61no—
188C2×D3—121no—
189C2×D3—121no—
190C3×C4—21no—
191C3×C4—21no—
192C3×C4—62no—
193C3×C4—121no—
194C3×C4—121no—
195C3×C4C2×C4×D311yescharacteristic
196C3×C4C2×C4×D311yescharacteristic
197C3×C4D3×D411yescharacteristic
198C3×C4D3×D411yescharacteristic
199Dic3—22no—
200Dic3—32no—
201Dic3—32no—
202Dic3—41no—
203Dic3—41no—
204Dic3—62no—
205C32#41711yescharacteristic, complemented, Sylow
206C23—61no—
207C23—91no—
208Q8—32no—
209Q8—61no—
210D4—32no—
211D4—61no—
212D4—184no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—91no—
217C2×C4—92no—
218C2×C4—92no—
219C2×C4—184no—
220C2×C4—362no—
221C2×C4C2×D3211yescharacteristic
222C8—21no—
223C8—181no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—21no—
227C2×C3—122no—
228C2×C3D3×C22⋊C411yescharacteristic
229C2×C3D3×C22⋊C411yescharacteristic
230C2×C3#91111yescharacteristic
231C2×C3#91411yescharacteristic
232C2×C3#91511yescharacteristic
233C2×C3#91811yescharacteristic
234D3—42no—
235D3—62no—
236D3—62no—
237D3—122no—
238C22—62no—
239C22—92no—
240C22—121no—
241C22—121no—
242C22—181no—
243C22—181no—
244C22#380211yescharacteristic
245C4—62no—
246C4—92no—
247C4—121no—
248C4—121no—
249C4C4×D3211yescharacteristic
250C4#381111yescharacteristic
251C3—21no—
252C3#477111yescharacteristic, complemented
253C3#480311yescharacteristic, complemented
254C2—122no—
255C2—182no—
256C2#6379811yescharacteristic
257C2#6381111yescharacteristic
258C2#6392111yescharacteristic
subgroupG/H
centerC22#3802
commutatorC32×C4C22×C4
FrattiniC2×C4C2×D32
FittingC2×C32×C8C22
socleC22×C32C2×D4
prime radicalC2×#3796C2
cosocleC2×C32×C4C23
hypercenterC2×C8D32
hypocenterC32#417
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#3743C2semidirect
3. #63900C2semidirect
4. C3×#821C2semidirect
5. C3×#816C2semidirect
6. #63565C2semidirect
7. #63562C2semidirect
17. #3796C4semidirect
35. #821D3semidirect
36. #816D3semidirect
49. C3×D12C2×C4semidirect
51. C3×Dic6C2×C4semidirect
135. D12C4×D3semidirect
146. Dic6C4×D3semidirect
205. C32#417semidirect
252. C3#4771semidirect
253. C3#4803semidirect

Maximal subgroups

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

subgroupindex
C2×#37962
C2×#37432
#639002
C3×#8212
C3×#8162
#635652
#635622
#48033
#47713