Finite Group Databasebeta

← #11723639

Subgroups — #11723639

Order 864.

1316 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 1316 conjugacy classes of subgroups — too many to draw. Only its 132 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×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×C3×#638C211yesmaximal, characteristic, complemented
3C2×#6292C212yesmaximal, complemented
4C32×#217C211yesmaximal, characteristic, complemented
5#88804C212yesmaximal, complemented
6#64091—31nomaximal
7#63981—34nomaximal
8C23×C33C2211yescharacteristic
9C22×C3⋊(C3⋊D3)—22no—
10C22×C3⋊(C3⋊D3)—24no—
11C22×C3⋊(C3⋊D3)C2212yes—
12C22×C3×C3⋊D3C2211yescharacteristic, complemented
13C3×#638—22no—
14C2×C3×C3⋊Dic3C2211yescharacteristic, complemented
15C2×C32×Dic3C2212yescomplemented
16#6292—24no—
17C23×C3⋊D3—31no—
18C23×C3⋊D3—34no—
19C23×C3⋊D3—64no—
20C2×#638D311yescharacteristic, complemented
21C2×C4⋊(C3⋊D3)—32no—
22C2×C3×#67—34no—
23C2×#626—38no—
24C32×C22⋊C4—31no—
25#3752—32no—
26C3×#217D314yescomplemented
27#3722—38no—
28C22×C33—22no—
29C22×C33D412yescomplemented
30C22×C33C2311yescharacteristic
31C2×C3⋊(C3⋊D3)—28no—
32C2×C3⋊(C3⋊D3)—22no—
33C2×C3⋊(C3⋊D3)—22no—
34C2×C3⋊(C3⋊D3)D414yescomplemented
35C2×C3×C3⋊D3—22no—
36C2×C3×C3⋊D3—21no—
37C3×C3⋊Dic3—21no—
38C32×Dic3—22no—
39#937—91no—
40#882—94no—
41C23×C32—24no—
42C23×C32C2×D314yes—
43C23×C32C2×D311yescharacteristic
44C22×C3⋊D3—32no—
45C22×C3⋊D3—38no—
46C22×C3⋊D3—68no—
47C22×C3⋊D3—64no—
48C22×C3⋊D3—616no—
49C22×C3⋊D3—632no—
50C22×C3⋊D3—616no—
51C22×C3⋊D3—68no—
52C22×C3⋊D3—62no—
53C22×C3⋊D3C2×D311yescharacteristic, complemented
54C22×C3×D3—34no—
55C2×C32×C4—32no—
56#638—22no—
57C2×C3⋊Dic3C2×D311yescharacteristic, complemented
58C4⋊(C3⋊D3)—64no—
59C3×#67—68no—
60C2×C3×Dic3—34no—
61C2×C3×Dic3C2×D318yescomplemented
62#626—616no—
63C2×C33—22no—
64C2×C33C2×D411yescharacteristic
65C2×C33C2×D412yes—
66C3⋊(C3⋊D3)—24no—
67C3×C3⋊D3—41no—
68C23×D3—94no—
69C23×D3—91no—
70C23×D3—184no—
71C2×C3×D4—91no—
72C2×#67—34no—
73C2×#67—92no—
74C2×D12—98no—
75C3×C22⋊C4—34no—
76#217C3⋊D311yescharacteristic, complemented
77(C2×D3)⋊C4—98no—
78(C2×D3)⋊C4—92no—
79C22×C32—216no—
80C22×C32—28no—
81C22×C32—22no—
82C22×C32—28no—
83C22×C32—24no—
84C22×C32D1218yescomplemented
85C22×C32#6712yescomplemented
86C22×C32C22×D314yes—
87C22×C32C22×D311yescharacteristic
88C2×C3⋊D3—22no—
89C2×C3⋊D3—21no—
90C2×C3⋊D3—34no—
91C2×C3⋊D3—316no—
92C2×C3⋊D3—616no—
93C2×C3⋊D3—664no—
94C2×C3⋊D3—632no—
95C2×C3⋊D3—68no—
96C2×C3⋊D3—616no—
97C2×C3⋊D3—616no—
98C2×C3⋊D3—68no—
99C2×C3⋊D3—68no—
100C2×C3⋊D3—62no—
101C2×C3⋊D3—62no—
102C2×C3×D3—64no—
103C2×C3×D3—68no—
104C32×C4—62no—
105C3⋊Dic3—21no—
106C3×Dic3—28no—
107C3×Dic3—64no—
108#120—271noSylow
109C33#12011yescharacteristic, complemented, Sylow
110C23×C3—24no—
111C23×C3—91no—
112C23×C3D3214yes—
113C23×C3C2×C3⋊D311yescharacteristic
114C22×D3—34no—
115C22×D3—98no—
116C22×D3—92no—
117C22×D3—188no—
118C22×D3—1816no—
119C22×D3—184no—
120C22×D3—1832no—
121C22×D3—1816no—
122C22×D3—188no—
123C22×D3—182no—
124C3×D4—182no—
125C2×C3×C4—38no—
126C2×C3×C4—91no—
127#67—68no—
128#67—184no—
129C2×Dic3—34no—
130C2×Dic3C2×C3⋊D312yescomplemented
131D12—1816no—
132C2×C32—216no—
133C2×C32—28no—
134C2×C32—22no—
135C2×C32—24no—
136C2×C32—28no—
137C2×C32C2×D1214yes—
138C2×C32D3×D418yes—
139C2×C32D3×D412yes—
140C2×C32C2×#6711yescharacteristic
141C3⋊D3—41no—
142C3⋊D3—632no—
143C3⋊D3—616no—
144C3⋊D3—64no—
145C3×D3—124no—
146C24—271no—
147C2×D4—91no—
148C2×D4—272no—
149C22⋊C4—31no—
150C22⋊C4—272no—
151C22×C3—216no—
152C22×C3—28no—
153C22×C3—22no—
154C22×C3—28no—
155C22×C3—24no—
156C22×C3—181no—
157C22×C3—182no—
158C22×C3#62618yescomplemented
159C22×C3C4⋊(C3⋊D3)12yescomplemented
160C22×C3C2×D3214yes—
161C22×C3C22×C3⋊D311yescharacteristic
162C2×D3—64no—
163C2×D3—68no—
164C2×D3—916no—
165C2×D3—94no—
166C2×D3—1816no—
167C2×D3—1864no—
168C2×D3—188no—
169C2×D3—1832no—
170C2×D3—1816no—
171C2×D3—1816no—
172C2×D3—188no—
173C2×D3—188no—
174C2×D3—182no—
175C2×D3—182no—
176C3×C4—68no—
177C3×C4—181no—
178Dic3—22no—
179Dic3—64no—
180C32—24no—
181C32#88214yescomplemented
182C32#93711yescharacteristic, complemented
183C23—91no—
184C23—272no—
185C23—544no—
186C23—542no—
187C23D3×C3⋊D311yescharacteristic
188D4—182no—
189D4—544no—
190C2×C4—32no—
191C2×C4—91no—
192C2×C3—216no—
193C2×C3—22no—
194C2×C3—28no—
195C2×C3—24no—
196C2×C3—28no—
197C2×C3—361no—
198C2×C3C2×#62614yes—
199C2×C3#381118yes—
200C2×C3C2×C4⋊(C3⋊D3)11yescharacteristic
201C2×C3D4×C3⋊D312yes—
202D3—124no—
203D3—1832no—
204D3—184no—
205D3—1816no—
206C22—22no—
207C22—181no—
208C22—182no—
209C22—274no—
210C22—548no—
211C22—542no—
212C22—542no—
213C22#629212yescomplemented
214C22C2×D3×C3⋊D311yescharacteristic
215C4—62no—
216C4—181no—
217C3—24no—
218C3#6398114yescomplemented
219C3#6409111yescharacteristic, complemented
220C2—22no—
221C2—361no—
222C2—544no—
223C2C2×#629211yescharacteristic
224C2#8903912yes—
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC23×C33C22
hypercenterC23D3×C3⋊D3
hypocenterC33#120
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊(C3⋊D3)C2semidirect
2. C2×C3×#638C2semidirect
3. C2×#6292C2semidirect
4. C32×#217C2semidirect
5. #88804C2semidirect
12. C22×C3×C3⋊D3C22semidirect
14. C2×C3×C3⋊Dic3C22semidirect
15. C2×C32×Dic3C22semidirect
20. C2×#638D3semidirect
26. C3×#217D3semidirect
29. C22×C33D4semidirect
34. C2×C3⋊(C3⋊D3)D4semidirect
53. C22×C3⋊D3C2×D3semidirect
57. C2×C3⋊Dic3C2×D3semidirect
61. C2×C3×Dic3C2×D3semidirect
76. #217C3⋊D3semidirect
84. C22×C32D12semidirect
85. C22×C32#67semidirect
109. C33#120semidirect
130. C2×Dic3C2×C3⋊D3semidirect
158. C22×C3#626semidirect
159. C22×C3C4⋊(C3⋊D3)semidirect
181. C32#882semidirect
182. C32#937semidirect
213. C22#6292semidirect
218. C3#63981semidirect
219. C3#64091semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊(C3⋊D3)2
C2×C3×#6382
C2×#62922
C32×#2172
#888042
#640913
#639813