Finite Group Databasebeta

← C32×#882 #11723751

Subgroups — C32×#882 #11723751

Order 864.

916 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 222 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
1C23×C32×D3C211yesmaximal, characteristic, complemented
2C2×C32×#67C211yesmaximal, characteristic, complemented
3C2×C32×D12C212yesmaximal, complemented
4C33×C22⋊C4C211yesmaximal, characteristic, complemented
5C32×(C2×D3)⋊C4C212yesmaximal, complemented
6C32×#120—31nomaximal
7C3×#882C314yesmaximal, direct product factor
8C23×C33C2211yescharacteristic
9C22×C32×D3—22no—
10C22×C32×D3—24no—
11C22×C32×D3C2212yes—
12C22×C32×D3C2211yescharacteristic, complemented
13C2×C33×C4C2212yescomplemented
14C32×#67—22no—
15C2×C32×Dic3C2211yescharacteristic, complemented
16C32×D12—24no—
17C24×C32—31no—
18C23×C3×D3C2×C314yescomplemented
19C2×C32×D4—31no—
20C2×C32×D4—32no—
21C2×C3×#67C2×C314yescomplemented
22C2×C3×D12C2×C318yescomplemented
23C32×C22⋊C4—24no—
24C32×C22⋊C4—32no—
25C32×C22⋊C4D311yescharacteristic, complemented
26C32×C22⋊C4C2×C314yescomplemented
27C3×(C2×D3)⋊C4C2×C318yescomplemented
28C22×C33—22no—
29C22×C33D412yescomplemented
30C22×C33C2311yescharacteristic
31C2×C32×D3—28no—
32C2×C32×D3—22no—
33C2×C32×D3—22no—
34C2×C32×D3—22no—
35C2×C32×D3—21no—
36C2×C32×D3D414yescomplemented
37C33×C4—22no—
38C32×Dic3—21no—
39C3×#120—34no—
40#882C3211yescharacteristic, direct product factor
41C23×C32—24no—
42C23×C32—32no—
43C23×C32—31no—
44C23×C32—64no—
45C23×C32—62no—
46C23×C32C2×D311yescharacteristic
47C23×C32C22×C314yes—
48C22×C3×D3—28no—
49C22×C3×D3—216no—
50C22×C3×D3C22×C318yes—
51C22×C3×D3C22×C314yescomplemented
52C32×D4—64no—
53C32×D4—62no—
54C2×C32×C4—28no—
55C2×C32×C4—31no—
56C2×C32×C4C2×D312yescomplemented
57C2×C32×C4C22×C318yescomplemented
58C3×#67—28no—
59C2×C3×Dic3C22×C314yescomplemented
60C3×D12—216no—
61C2×C33—22no—
62C2×C33C2×D411yescharacteristic
63C2×C33C2×D412yes—
64C32×D3—24no—
65C32×D3—41no—
66C24×C3—34no—
67C23×D3C2×C3211yescharacteristic, complemented
68C2×C3×D4—34no—
69C2×C3×D4—38no—
70C2×#67C2×C3211yescharacteristic, complemented
71C2×D12C2×C3212yescomplemented
72C3×C22⋊C4—24no—
73C3×C22⋊C4—38no—
74C3×C22⋊C4C3×D314yescomplemented
75C3×C22⋊C4C2×C3211yescharacteristic, complemented
76(C2×D3)⋊C4C2×C3212yescomplemented
77C22×C32—22no—
78C22×C32—28no—
79C22×C32—24no—
80C22×C32—28no—
81C22×C32—34no—
82C22×C32—48no—
83C22×C32—68no—
84C22×C32—62no—
85C22×C32—62no—
86C22×C32—61no—
87C22×C32—62no—
88C22×C32D1212yescomplemented
89C22×C32C3×D418yescomplemented
90C22×C32C22×D311yescharacteristic
91C22×C32C23×C314yes—
92C2×C3×D3—232no—
93C2×C3×D3—28no—
94C2×C3×D3—28no—
95C2×C3×D3—28no—
96C2×C3×D3—24no—
97C2×C3×D3C3×D4116yescomplemented
98C32×C4—22no—
99C32×C4—28no—
100C32×C4—48no—
101C32×C4—61no—
102C3×Dic3—24no—
103#120—31noSylow
104C33#12011yescharacteristic, complemented, Sylow
105C23×C3—24no—
106C23×C3—38no—
107C23×C3—34no—
108C23×C3—616no—
109C23×C3—68no—
110C23×C3C2×C3×D314yes—
111C23×C3C22×C3211yescharacteristic
112C22×D3—22no—
113C22×D3—24no—
114C22×D3C22×C3212yes—
115C22×D3C22×C3211yescharacteristic, complemented
116C3×D4—616no—
117C3×D4—68no—
118C2×C3×C4—28no—
119C2×C3×C4—34no—
120C2×C3×C4C2×C3×D318yescomplemented
121C2×C3×C4C22×C3212yescomplemented
122#67—22no—
123C2×Dic3C22×C3211yescharacteristic, complemented
124D12—24no—
125C2×C32—22no—
126C2×C32—24no—
127C2×C32—28no—
128C2×C32—28no—
129C2×C32—48no—
130C2×C32—64no—
131C2×C32—121no—
132C2×C32C2×D1211yescharacteristic
133C2×C32D3×D412yes—
134C2×C32C2×C3×D414yes—
135C2×C32C2×C3×D418yes—
136C3×D3—216no—
137C3×D3—44no—
138C24—31no—
139C2×D4—31no—
140C2×D4—32no—
141C22⋊C4—32no—
142C22⋊C4C32×D311yescharacteristic, complemented
143C22×C3—28no—
144C22×C3—22no—
145C22×C3—28no—
146C22×C3—24no—
147C22×C3—316no—
148C22×C3—48no—
149C22×C3—632no—
150C22×C3—68no—
151C22×C3—68no—
152C22×C3—64no—
153C22×C3—68no—
154C22×C3C3×D1218yescomplemented
155C22×C3C32×D412yescomplemented
156C22×C3C22×C3×D314yes—
157C22×C3C23×C3211yescharacteristic
158C2×D3—28no—
159C2×D3—22no—
160C2×D3—22no—
161C2×D3—22no—
162C2×D3—21no—
163C2×D3C32×D414yescomplemented
164C3×C4—28no—
165C3×C4—22no—
166C3×C4—48no—
167C3×C4—64no—
168Dic3—21no—
169C32—24no—
170C32#88211yescharacteristic, direct product factor
171C32C3×#12014yescomplemented
172C23—32no—
173C23—31no—
174C23—64no—
175C23—62no—
176C23C2×C32×D311yescharacteristic
177D4—64no—
178D4—62no—
179C2×C4—31no—
180C2×C4C2×C32×D312yescomplemented
181C2×C3—28no—
182C2×C3—22no—
183C2×C3—24no—
184C2×C3—28no—
185C2×C3—48no—
186C2×C3—616no—
187C2×C3—124no—
188C2×C3C2×C3×D1214yes—
189C2×C3C3×D3×D418yes—
190C2×C3C2×C32×D411yescharacteristic
191C2×C3C2×C32×D412yes—
192D3—24no—
193D3—41no—
194C22—22no—
195C22—34no—
196C22—68no—
197C22—62no—
198C22—62no—
199C22—61no—
200C22—62no—
201C22C32×D1212yescomplemented
202C22C22×C32×D311yescharacteristic
203C4—22no—
204C4—61no—
205C3—24no—
206C3C3×#88214yesdirect product factor
207C3C32×#12011yescharacteristic, complemented
208C2—22no—
209C2—64no—
210C2—121no—
211C2C2×C32×D1211yescharacteristic
212C2C32×D3×D412yes—
subgroupG/H
centerC22×C32C22×D3
commutator, cosocleC22×C3C23×C32
FrattiniC22C22×C32×D3
FittingC33×C22⋊C4C2
socleC22×C33C23
hypercenterC32×C22⋊C4D3
hypocenterC3C32×#120
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C32×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C32×D3C2semidirect
2. C2×C32×#67C2semidirect
3. C2×C32×D12C2semidirect
4. C33×C22⋊C4C2semidirect
5. C32×(C2×D3)⋊C4C2semidirect
7. C3×#882C3direct (N × H)
12. C22×C32×D3C22semidirect
13. C2×C33×C4C22semidirect
15. C2×C32×Dic3C22semidirect
18. C23×C3×D3C2×C3semidirect
21. C2×C3×#67C2×C3semidirect
22. C2×C3×D12C2×C3semidirect
25. C32×C22⋊C4D3semidirect
26. C32×C22⋊C4C2×C3semidirect
27. C3×(C2×D3)⋊C4C2×C3semidirect
29. C22×C33D4semidirect
36. C2×C32×D3D4semidirect
40. #882C32direct (N × H)
51. C22×C3×D3C22×C3semidirect
56. C2×C32×C4C2×D3semidirect
57. C2×C32×C4C22×C3semidirect
59. C2×C3×Dic3C22×C3semidirect
67. C23×D3C2×C32semidirect
70. C2×#67C2×C32semidirect
71. C2×D12C2×C32semidirect
74. C3×C22⋊C4C3×D3semidirect
75. C3×C22⋊C4C2×C32semidirect
76. (C2×D3)⋊C4C2×C32semidirect
88. C22×C32D12semidirect
89. C22×C32C3×D4semidirect
97. C2×C3×D3C3×D4semidirect
104. C33#120semidirect
115. C22×D3C22×C32semidirect
120. C2×C3×C4C2×C3×D3semidirect
121. C2×C3×C4C22×C32semidirect
123. C2×Dic3C22×C32semidirect
142. C22⋊C4C32×D3semidirect
154. C22×C3C3×D12semidirect
155. C22×C3C32×D4semidirect
163. C2×D3C32×D4semidirect
170. C32#882direct (N × H)
171. C32C3×#120semidirect
180. C2×C4C2×C32×D3semidirect
201. C22C32×D12semidirect
206. C3C3×#882direct (N × H)
207. C3C32×#120semidirect

Maximal subgroups

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

subgroupindex
C23×C32×D32
C2×C32×#672
C2×C32×D122
C33×C22⋊C42
C32×(C2×D3)⋊C42
C32×#1203
C3×#8823