Finite Group Databasebeta

← #11721271

Subgroups — #11721271

Order 864.

336 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 65 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×#6233C211yesmaximal, characteristic, complemented
2C2×C3×#611C211yesmaximal, characteristic, complemented
3C2×#6192C211yesmaximal, characteristic, complemented
4C2×Dic3×D9C211yesmaximal, characteristic, complemented
5C9×#217C211yesmaximal, characteristic, complemented
6#88445C211yesmaximal, characteristic, complemented
7#88442C211yesmaximal, characteristic, complemented
8#63971—31nomaximal
9#63446—31nomaximal
10C23×C3×C9C2211yescharacteristic
11C22×C3⋊D9C2211yescharacteristic
12C22×C3×D9C2211yescharacteristic
13#6233—22no—
14C2×C3⋊Dic9C2211yescharacteristic
15C3×#611—22no—
16C2×C9×Dic3C2211yescharacteristic
17C2×C9×Dic3C2211yescharacteristic
18C2×C3×Dic9C2211yescharacteristic
19#6192—22no—
20Dic3×D9—22no—
21C2×#638—31no—
22C2×C3×#67—31no—
23C2×#626—31no—
24C2×D3×Dic3—31no—
25C3×#217D311yescharacteristic
26#3724—31no—
27#3722—31no—
28C2×#611—31no—
29C2×#611—61no—
30C2×#611D311yescharacteristic, complemented
31C2×D36—31no—
32C2×C4×D9—31no—
33C9×C22⋊C4—31no—
34(C2×D9)⋊C4—31no—
35Dic9⋊C4—31no—
36C22×C3×C9—21no—
37C22×C3×C9—21no—
38C22×C3×C9—21no—
39C22×C3×C9C2311yescharacteristic
40C2×C3⋊D9—21no—
41C2×C3⋊D9—21no—
42C2×C3⋊D9—21no—
43C2×C3×D9—21no—
44C2×C3×D9—21no—
45C2×C3×D9D412yescomplemented
46C3⋊Dic9D412yescomplemented
47C9×Dic3—21no—
48C9×Dic3—21no—
49C3×Dic9—21no—
50#939—91no—
51#884—91no—
52C23×C32C2×D311yescharacteristic
53C22×C3⋊D3—31no—
54C22×C3×D3—31no—
55#638—62no—
56C2×C3⋊Dic3—31no—
57C3×#67—62no—
58C2×C3×Dic3—31no—
59C2×C3×Dic3C2×D311yescharacteristic
60C2×C3×Dic3C2×D311yescharacteristic
61#626—62no—
62D3×Dic3—62no—
63C23×C9—21no—
64C23×C9C2×D311yescharacteristic
65C22×D9—31no—
66C22×D9—61no—
67C22×D9C2×D311yescharacteristic
68C2×C4×C9—31no—
69C2×C4×C9—31no—
70#611—22no—
71#611—62no—
72#611—64no—
73C2×Dic9—31no—
74C2×Dic9—61no—
75C2×Dic9C2×D311yescharacteristic
76D36—62no—
77C4×D9—62no—
78C2×C3×C9—41no—
79C2×C3×C9C2×D411yescharacteristic
80C2×C3×C9C2×D411yescharacteristic
81C2×C3×C9#4111yescharacteristic
82C3⋊D9—41no—
83C3×D9—22no—
84C2×C3×D4—91no—
85C2×#67—31no—
86C2×#67—91no—
87C2×#67—91no—
88C2×#67—91no—
89C2×#67—181no—
90C22×Dic3—91no—
91C2×D12—91no—
92C2×C4×D3—91no—
93C3×C22⋊C4—31no—
94#217D911yescharacteristic, complemented
95(C2×D3)⋊C4—91no—
96(C2×D3)⋊C4—91no—
97Dic3⋊C4—91no—
98Dic3⋊C4—91no—
99C22×C32—21no—
100C22×C32—21no—
101C22×C32—21no—
102C22×C32C22×D311yescharacteristic
103C2×C3⋊D3—61no—
104C2×C3⋊D3—61no—
105C2×C3⋊D3—61no—
106C2×C3×D3—32no—
107C2×C3×D3—61no—
108C2×C3×D3—61no—
109C3⋊Dic3—32no—
110C3×Dic3—21no—
111C3×Dic3—21no—
112C3×Dic3—61no—
113C22×C9—22no—
114C22×C9—21no—
115C22×C9—21no—
116C22×C9—21no—
117C22×C9—21no—
118C22×C9—41no—
119C22×C9—41no—
120C22×C9C22×D311yescharacteristic
121C2×D9—21no—
122C2×D9—21no—
123C2×D9—62no—
124C2×D9—61no—
125C2×D9—61no—
126C2×D9—61no—
127C2×D9—121no—
128C2×D9—121no—
129C2×D9#6712yescomplemented
130C4×C9—61no—
131C4×C9—61no—
132Dic9—21no—
133Dic9—32no—
134Dic9—62no—
135#121—271noSylow
136C3×C9#12111yescharacteristic, complemented, Sylow
137C23×C3—21no—
138C23×C3—91no—
139C23×C3C2×D911yescharacteristic
140C23×C3D3211yescharacteristic
141C22×D3—31no—
142C22×D3—91no—
143C22×D3—91no—
144C22×D3—181no—
145C3×D4—182no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—91no—
149#67—62no—
150#67—182no—
151#67—182no—
152#67—182no—
153#67—184no—
154C2×Dic3—31no—
155C2×Dic3—91no—
156C2×Dic3—91no—
157C2×Dic3—181no—
158C2×Dic3—182no—
159C2×Dic3C2×D911yescharacteristic
160C2×Dic3C2×D911yescharacteristic
161D12—182no—
162C4×D3—182no—
163C2×C32—41no—
164C2×C32#23511yescharacteristic
165C2×C32D3×D411yescharacteristic
166C2×C32D3×D411yescharacteristic
167C3⋊D3—121no—
168C3×D3—62no—
169C2×C9—21no—
170C2×C9—21no—
171C2×C9—21no—
172C2×C9—42no—
173C2×C9—41no—
174C2×C9D3×D411yescharacteristic
175C2×C9#23711yescharacteristic
176C2×C9C2×#6711yescharacteristic
177D9—22no—
178D9—122no—
179D9—121no—
180C2×D4—91no—
181C2×D4—271no—
182C2×D4—271no—
183C22×C4—271no—
184C4⋊C4—271no—
185C22⋊C4—31no—
186C22⋊C4—271no—
187C22×C3—22no—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3—21no—
191C22×C3—21no—
192C22×C3—21no—
193C22×C3—21no—
194C22×C3—21no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—92no—
198C22×C3—181no—
199C22×C3—181no—
200C22×C3C22×D911yescharacteristic
201C22×C3C2×D3211yescharacteristic
202C2×D3—32no—
203C2×D3—61no—
204C2×D3—61no—
205C2×D3—182no—
206C2×D3—181no—
207C2×D3—181no—
208C2×D3—181no—
209C2×D3—181no—
210C2×D3—181no—
211C2×D3—181no—
212C2×D3—361no—
213C2×D3—361no—
214C3×C4—61no—
215C3×C4—61no—
216C3×C4—181no—
217Dic3—21no—
218Dic3—21no—
219Dic3—61no—
220Dic3—92no—
221Dic3—92no—
222Dic3—182no—
223C32#88411yescharacteristic
224C9—21no—
225C9#93911yescharacteristic, complemented
226C23—91no—
227C23—271no—
228C23D3×D911yescharacteristic
229D4—182no—
230D4—542no—
231D4—542no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—91no—
235C2×C4—271no—
236C2×C4—542no—
237C2×C3—21no—
238C2×C3—21no—
239C2×C3—21no—
240C2×C3—42no—
241C2×C3—41no—
242C2×C3—41no—
243C2×C3—182no—
244C2×C3#369711yescharacteristic
245C2×C3D4×D911yescharacteristic
246C2×C3D4×D911yescharacteristic
247C2×C3#380411yescharacteristic
248C2×C3D3×#6711yescharacteristic
249C2×C3#381111yescharacteristic
250D3—62no—
251D3—362no—
252D3—361no—
253D3—361no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—92no—
258C22—181no—
259C22—181no—
260C22—541no—
261C22—541no—
262C22—541no—
263C22C2×D3×D911yescharacteristic
264C4—61no—
265C4—61no—
266C4—181no—
267C4—272no—
268C3—21no—
269C3#6344611yescharacteristic, complemented
270C3#6397111yescharacteristic
271C2—41no—
272C2—182no—
273C2—1081no—
274C2#8865811yescharacteristic
275C2D9×#6711yescharacteristic
276C2#8866811yescharacteristic
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×#6233C2semidirect
2. C2×C3×#611C2semidirect
3. C2×#6192C2semidirect
4. C2×Dic3×D9C2semidirect
5. C9×#217C2semidirect
6. #88445C2semidirect
7. #88442C2semidirect
30. C2×#611D3semidirect
45. C2×C3×D9D4semidirect
46. C3⋊Dic9D4semidirect
94. #217D9semidirect
129. C2×D9#67semidirect
136. C3×C9#121semidirect
225. C9#939semidirect
269. C3#63446semidirect

Maximal subgroups

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

subgroupindex
C2×#62332
C2×C3×#6112
C2×#61922
C2×Dic3×D92
C9×#2172
#884452
#884422
#639713
#634463