Finite Group Databasebeta

← #11721281

Subgroups — #11721281

Order 864.

316 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 69 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×C9×#67C211yesmaximal, characteristic, complemented
2C2×C3×#611C211yesmaximal, characteristic, complemented
3C2×#6194C211yesmaximal, characteristic, complemented
4C2×D3×Dic9C211yesmaximal, characteristic, complemented
5#88551C211yesmaximal, characteristic, complemented
6(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
7Dic9⋊Dic3C211yesmaximal, characteristic, complemented
8#63970—31nomaximal
9#63501—31nomaximal
10C23×C3×C9C2211yescharacteristic
11C22×D3×C9C2211yescharacteristic
12C22×C3×D9C2211yescharacteristic
13C2×C3⋊Dic9C2211yescharacteristic
14C2×C3⋊Dic9C2211yescharacteristic
15C9×#67—22no—
16C3×#611—22no—
17C2×C9×Dic3C2211yescharacteristic
18C2×C3×Dic9C2211yescharacteristic
19#6194—22no—
20D3×Dic9—22no—
21C2×C3×#67—31no—
22C2×C3×#67D311yescharacteristic
23C2×#625—31no—
24C2×D3×Dic3—31no—
25#3757—31no—
26Dic3⋊Dic3—31no—
27(C2×D3)⋊Dic3—31no—
28C2×D4×C9—31no—
29C2×#611—31no—
30C2×#611D311yescharacteristic, complemented
31C22×Dic9—31no—
32#3676—31no—
33#3676—61no—
34(C2×D9)⋊C4—31no—
35Dic9⋊C4—31no—
36C22×C3×C9—21no—
37C22×C3×C9—21no—
38C22×C3×C9—21no—
39C22×C3×C9C2311yescharacteristic
40C2×D3×C9—21no—
41C2×D3×C9—21no—
42C2×D3×C9D412yescomplemented
43C2×C3×D9—21no—
44C2×C3×D9—21no—
45C2×C3×D9—21no—
46C3⋊Dic9—21no—
47C3⋊Dic9—21no—
48C9×Dic3—21no—
49C3×Dic9D412yescomplemented
50#939—91no—
51#938—91no—
52C23×C32C2×D311yescharacteristic
53C22×C3×D3—31no—
54C22×C3×D3C2×D311yescharacteristic
55C2×C3⋊Dic3—31no—
56C2×C3⋊Dic3—31no—
57C3×#67—22no—
58C3×#67—62no—
59C2×C3×Dic3—31no—
60C2×C3×Dic3C2×D311yescharacteristic
61#625—62no—
62D3×Dic3—62no—
63C23×C9—21no—
64C23×C9—31no—
65C23×C9C2×D311yescharacteristic
66C22×D9C2×D311yescharacteristic
67D4×C9—62no—
68C2×C4×C9—31no—
69#611—22no—
70#611—62no—
71C2×Dic9—31no—
72C2×Dic9—31no—
73C2×Dic9—61no—
74C2×Dic9—61no—
75C2×Dic9—62no—
76C2×Dic9C2×D311yescharacteristic
77C2×C3×C9—41no—
78C2×C3×C9C2×D411yescharacteristic
79C2×C3×C9C2×D411yescharacteristic
80C2×C3×C9#4111yescharacteristic
81D3×C9—22no—
82C3×D9—41no—
83C2×C3×D4—31no—
84C2×C3×D4—91no—
85C2×#67—31no—
86C2×#67—91no—
87C2×#67—91no—
88C2×#67D911yescharacteristic, complemented
89C22×Dic3—91no—
90C2×C4×D3—91no—
91#217—91no—
92#217—91no—
93#217—91no—
94#217—181no—
95(C2×D3)⋊C4—91no—
96C4⋊Dic3—91no—
97Dic3⋊C4—91no—
98C22×C32—21no—
99C22×C32—21no—
100C22×C32—21no—
101C22×C32C22×D311yescharacteristic
102C2×C3×D3—21no—
103C2×C3×D3—21no—
104C2×C3×D3—61no—
105C2×C3×D3—61no—
106C2×C3×D3—61no—
107C2×C3×D3#6712yes—
108C3⋊Dic3—61no—
109C3⋊Dic3—61no—
110C3×Dic3—21no—
111C3×Dic3—32no—
112C22×C9—22no—
113C22×C9—21no—
114C22×C9—21no—
115C22×C9—21no—
116C22×C9—21no—
117C22×C9—32no—
118C22×C9—41no—
119C22×C9—41no—
120C22×C9—61no—
121C22×C9—61no—
122C22×C9C22×D311yescharacteristic
123C2×D9—21no—
124C2×D9—21no—
125C2×D9—21no—
126C4×C9—61no—
127Dic9—61no—
128Dic9—61no—
129Dic9—121no—
130Dic9—121no—
131Dic9#6712yescomplemented
132#121—271noSylow
133C3×C9#12111yescharacteristic, complemented, Sylow
134C23×C3—21no—
135C23×C3—31no—
136C23×C3—91no—
137C23×C3C2×D911yescharacteristic
138C23×C3D3211yescharacteristic
139C22×D3—31no—
140C22×D3C2×D911yescharacteristic
141C3×D4—62no—
142C3×D4—182no—
143C2×C3×C4—31no—
144C2×C3×C4—91no—
145#67—22no—
146#67—62no—
147#67—182no—
148#67—182no—
149C2×Dic3—31no—
150C2×Dic3—91no—
151C2×Dic3—91no—
152C2×Dic3—91no—
153C2×Dic3—91no—
154C2×Dic3—181no—
155C2×Dic3—181no—
156C2×Dic3—182no—
157C2×Dic3C2×D911yescharacteristic
158C4×D3—182no—
159C2×C32—41no—
160C2×C32D3×D411yescharacteristic
161C2×C32#23711yescharacteristic
162C2×C32C2×#6711yescharacteristic
163C3×D3—22no—
164C3×D3—121no—
165C2×C9—21no—
166C2×C9—21no—
167C2×C9—21no—
168C2×C9—42no—
169C2×C9—41no—
170C2×C9—62no—
171C2×C9D3×D411yescharacteristic
172C2×C9#23711yescharacteristic
173C2×C9C2×#6711yescharacteristic
174D9—41no—
175C2×D4—31no—
176C2×D4—91no—
177C2×D4—271no—
178C22×C4—271no—
179C4⋊C4—271no—
180C22⋊C4—271no—
181C22⋊C4—271no—
182C22×C3—22no—
183C22×C3—21no—
184C22×C3—21no—
185C22×C3—21no—
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3—32no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3—61no—
194C22×C3—61no—
195C22×C3—181no—
196C22×C3—181no—
197C22×C3—181no—
198C22×C3C22×D911yescharacteristic
199C22×C3C2×D3211yescharacteristic
200C2×D3—21no—
201C2×D3—21no—
202C2×D3—61no—
203C2×D3—61no—
204C2×D3—61no—
205C2×D3#61112yescomplemented
206C3×C4—61no—
207C3×C4—92no—
208Dic3—21no—
209Dic3—32no—
210Dic3—181no—
211Dic3—181no—
212Dic3—181no—
213Dic3—181no—
214Dic3—361no—
215Dic3—361no—
216C32#93911yescharacteristic
217C9—21no—
218C9#93811yescharacteristic, complemented
219C23—31no—
220C23—91no—
221C23D3×D911yescharacteristic
222D4—62no—
223D4—182no—
224D4—542no—
225C2×C4—31no—
226C2×C4—91no—
227C2×C4—271no—
228C2×C4—271no—
229C2×C4—542no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—42no—
234C2×C3—41no—
235C2×C3—41no—
236C2×C3—62no—
237C2×C3—361no—
238C2×C3D4×D911yescharacteristic
239C2×C3#369911yescharacteristic
240C2×C3C2×#61111yescharacteristic
241C2×C3#380511yescharacteristic
242C2×C3D3×#6711yescharacteristic
243C2×C3D3×#6711yescharacteristic
244D3—22no—
245D3—121no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—32no—
250C22—61no—
251C22—61no—
252C22—181no—
253C22—181no—
254C22—181no—
255C22C2×D3×D911yescharacteristic
256C4—61no—
257C4—92no—
258C4—541no—
259C4—541no—
260C3—21no—
261C3#6350111yescharacteristic, complemented
262C3#6397011yescharacteristic
263C2—41no—
264C2—62no—
265C2—361no—
266C2#8866311yescharacteristic
267C2D3×#61111yescharacteristic
268C2D9×#6711yescharacteristic
subgroupG/H
centerC22C2×D3×D9
commutator, cosocleC22×C3×C9C23
FrattiniC22×C3C2×D32
FittingC23×C3×C9C22
socleC22×C32C22×D3
hypercenterC23D3×D9
hypocenterC3×C9#121
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×C9×#67C2semidirect
2. C2×C3×#611C2semidirect
3. C2×#6194C2semidirect
4. C2×D3×Dic9C2semidirect
5. #88551C2semidirect
6. (C2×D9)⋊Dic3C2semidirect
7. Dic9⋊Dic3C2semidirect
30. C2×#611D3semidirect
42. C2×D3×C9D4semidirect
49. C3×Dic9D4semidirect
88. C2×#67D9semidirect
131. Dic9#67semidirect
133. C3×C9#121semidirect
205. C2×D3#611semidirect
218. C9#938semidirect
261. C3#63501semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#672
C2×C3×#6112
C2×#61942
C2×D3×Dic92
#885512
(C2×D9)⋊Dic32
Dic9⋊Dic32
#639703
#635013