Finite Group Databasebeta

← #11721287

Subgroups — #11721287

Order 864.

444 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 83 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×D3×C9C211yesmaximal, characteristic, complemented
2C2×#6233C211yesmaximal, characteristic, complemented
3C2×#6195C212yesmaximal, complemented
4C3×#3676C211yesmaximal, characteristic, complemented
5(C2×D3)⋊Dic9C212yesmaximal, complemented
6#63977—31nomaximal
7#63515—31nomaximal
8C23×C3×C9C2211yescharacteristic
9C22×C3⋊D9C2211yescharacteristic, complemented
10C22×D3×C9—22no—
11C22×D3×C9—24no—
12C22×D3×C9C2212yes—
13#6233—22no—
14C2×C3⋊Dic9C2211yescharacteristic, complemented
15C2×C3×Dic9C2212yescomplemented
16#6195—24no—
17C23×C3×D3D311yescharacteristic
18C2×#638—31no—
19C2×#626—32no—
20C24×C9—31no—
21C3×#217—31no—
22(C2×D3)⋊Dic3—32no—
23C2×#611—31no—
24C2×#611—32no—
25C2×#611—61no—
26#3676—32no—
27#3676D311yescharacteristic, complemented
28C22×C3×C9—22no—
29C22×C3×C9D412yescomplemented
30C22×C3×C9C2311yescharacteristic
31C2×C3⋊D9—21no—
32C2×C3⋊D9—22no—
33C2×D3×C9—28no—
34C2×D3×C9—22no—
35C2×D3×C9—22no—
36C2×D3×C9D414yescomplemented
37C3⋊Dic9—21no—
38C3×Dic9—22no—
39#953—91no—
40#882—91no—
41C23×C32C2×D311yescharacteristic
42C22×C3⋊D3—31no—
43C22×C3×D3—22no—
44C22×C3×D3—24no—
45C22×C3×D3C2×D312yes—
46#638—62no—
47C2×C3⋊Dic3—31no—
48C2×C3×Dic3—32no—
49#626—64no—
50C23×C9—21no—
51C23×C9—32no—
52C23×C9—64no—
53C23×C9—62no—
54C23×C9C2×D311yescharacteristic
55C22×D9—31no—
56C22×D9—61no—
57#611—64no—
58#611—64no—
59#611—62no—
60C2×Dic9—31no—
61C2×Dic9—61no—
62C2×Dic9C2×D312yescomplemented
63C2×C3×C9—22no—
64C2×C3×C9C2×D411yescharacteristic
65C2×C3×C9C2×D412yes—
66C3⋊D9—41no—
67D3×C9—24no—
68C24×C3—31no—
69C23×D3D911yescharacteristic, complemented
70C2×#67—92no—
71C2×#67—91no—
72C2×#67—91no—
73C2×#67—181no—
74C2×D12—92no—
75C3×C22⋊C4—91no—
76#217—31no—
77#217—92no—
78(C2×D3)⋊C4—92no—
79C22×C32—22no—
80C22×C32#6712yes—
81C22×C32C22×D311yescharacteristic
82C2×C3⋊D3—61no—
83C2×C3⋊D3—62no—
84C2×C3×D3—28no—
85C2×C3×D3—22no—
86C2×C3×D3—22no—
87C2×C3×D3#6714yes—
88C3⋊Dic3—61no—
89C3×Dic3—62no—
90C22×C9—22no—
91C22×C9—22no—
92C22×C9—21no—
93C22×C9—34no—
94C22×C9—42no—
95C22×C9—68no—
96C22×C9—62no—
97C22×C9—62no—
98C22×C9D1212yescomplemented
99C22×C9C22×D311yescharacteristic
100C2×D9—62no—
101C2×D9—61no—
102C2×D9—62no—
103C2×D9—122no—
104Dic9—22no—
105Dic9—62no—
106Dic9—61no—
107#120—271noSylow
108C3×C9#12011yescharacteristic, complemented, Sylow
109C23×C3—21no—
110C23×C3—32no—
111C23×C3—64no—
112C23×C3—62no—
113C23×C3C2×D911yescharacteristic
114C23×C3D3211yescharacteristic
115C22×D3—22no—
116C22×D3—24no—
117C22×D3—91no—
118C22×D3—91no—
119C22×D3—181no—
120C22×D3C2×D912yes—
121C2×C3×C4—92no—
122#67—184no—
123#67—184no—
124#67—182no—
125#67—182no—
126C2×Dic3—32no—
127C2×Dic3—91no—
128C2×Dic3—91no—
129C2×Dic3—181no—
130D12—184no—
131C2×C32—22no—
132C2×C32C2×#6711yescharacteristic
133C2×C32C2×#6712yes—
134C3⋊D3—121no—
135C3×D3—24no—
136C2×C9—22no—
137C2×C9—21no—
138C2×C9—22no—
139C2×C9—42no—
140C2×C9—64no—
141C2×C9C2×D1211yescharacteristic
142C2×C9D3×D412yes—
143D9—122no—
144D9—121no—
145C24—31no—
146C2×D4—272no—
147C2×D4—271no—
148C22⋊C4—91no—
149C22⋊C4—272no—
150C22×C3—22no—
151C22×C3—22no—
152C22×C3—22no—
153C22×C3—21no—
154C22×C3—34no—
155C22×C3—42no—
156C22×C3—68no—
157C22×C3—62no—
158C22×C3—62no—
159C22×C3#61112yescomplemented
160C22×C3C22×D911yescharacteristic
161C22×C3#62612yes—
162C22×C3C2×D3211yescharacteristic
163C2×D3—28no—
164C2×D3—22no—
165C2×D3—22no—
166C2×D3—182no—
167C2×D3—182no—
168C2×D3—181no—
169C2×D3—181no—
170C2×D3—182no—
171C2×D3—362no—
172C2×D3#61114yescomplemented
173C3×C4—182no—
174Dic3—62no—
175Dic3—182no—
176Dic3—181no—
177Dic3—181no—
178C32#95311yescharacteristic
179C9—21no—
180C9#88211yescharacteristic, complemented
181C23—32no—
182C23—64no—
183C23—62no—
184C23—271no—
185C23D3×D911yescharacteristic
186D4—544no—
187D4—542no—
188C2×C4—92no—
189C2×C4—271no—
190C2×C3—22no—
191C2×C3—22no—
192C2×C3—21no—
193C2×C3—22no—
194C2×C3—42no—
195C2×C3—64no—
196C2×C3C2×#61111yescharacteristic
197C2×C3C2×#61112yes—
198C2×C3C2×#62611yescharacteristic
199C2×C3D3×#6712yes—
200D3—24no—
201D3—362no—
202D3—361no—
203D3—361no—
204C22—22no—
205C22—34no—
206C22—68no—
207C22—62no—
208C22—62no—
209C22—541no—
210C22—542no—
211C22#619512yescomplemented
212C22C2×D3×D911yescharacteristic
213C4—182no—
214C4—541no—
215C3—21no—
216C3#6351511yescharacteristic, complemented
217C3#6397711yescharacteristic
218C2—22no—
219C2—64no—
220C2—1081no—
221C2C2×#619511yescharacteristic
222C2D3×#61112yes—
subgroupG/H
centerC22C2×D3×D9
commutator, cosocleC22×C3×C9C23
FrattiniC22×C3C2×D32
FittingC23×C3×C9C22
socleC22×C32C22×D3
hypercenterC23D3×D9
hypocenterC3×C9#120
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×D3×C9C2semidirect
2. C2×#6233C2semidirect
3. C2×#6195C2semidirect
4. C3×#3676C2semidirect
5. (C2×D3)⋊Dic9C2semidirect
9. C22×C3⋊D9C22semidirect
14. C2×C3⋊Dic9C22semidirect
15. C2×C3×Dic9C22semidirect
27. #3676D3semidirect
29. C22×C3×C9D4semidirect
36. C2×D3×C9D4semidirect
62. C2×Dic9C2×D3semidirect
69. C23×D3D9semidirect
98. C22×C9D12semidirect
108. C3×C9#120semidirect
159. C22×C3#611semidirect
172. C2×D3#611semidirect
180. C9#882semidirect
211. C22#6195semidirect
216. C3#63515semidirect

Maximal subgroups

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

subgroupindex
C23×D3×C92
C2×#62332
C2×#61952
C3×#36762
(C2×D3)⋊Dic92
#639773
#635153