Finite Group Databasebeta

← #11721292

Subgroups — #11721292

Order 864.

444 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C22×D3×D9C211yesmaximal, characteristic, complemented
2C2×#6233C211yesmaximal, characteristic, complemented
3C2×C9×#67C211yesmaximal, characteristic, complemented
4C2×C3×#611C211yesmaximal, characteristic, complemented
5(C2×D3)⋊Dic9C211yesmaximal, characteristic, complemented
6#88445C211yesmaximal, characteristic, complemented
7(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
8#63983—31nomaximal
9#63499—31nomaximal
10C23×C3×C9C2211yescharacteristic, complemented
11C22×C3⋊D9C2211yescharacteristic
12C22×D3×C9C2211yescharacteristic
13C22×C3×D9C2211yescharacteristic
14C2×D3×D9—22no—
15C2×D3×D9—22no—
16C2×D3×D9—22no—
17#6233—22no—
18C2×C3⋊Dic9C2211yescharacteristic, complemented
19C9×#67—22no—
20C3×#611—22no—
21C2×C9×Dic3C2211yescharacteristic, complemented
22C2×C3×Dic9C2211yescharacteristic, complemented
23C22×D32—31no—
24C2×#638—31no—
25C2×C3×#67—31no—
26C2×C3×#67D311yescharacteristic
27C23×D9—31no—
28#3722—31no—
29(C2×D3)⋊Dic3—31no—
30(C2×D3)⋊Dic3—31no—
31C2×D4×C9—31no—
32C2×#611—31no—
33C2×#611—61no—
34C2×#611D311yescharacteristic, complemented
35#3676—31no—
36(C2×D9)⋊C4—31no—
37(C2×D9)⋊C4—31no—
38C22×C3×C9—21no—
39C22×C3×C9—21no—
40C22×C3×C9—21no—
41C22×C3×C9C2311yescharacteristic
42C2×C3⋊D9—21no—
43C2×C3⋊D9—21no—
44C2×C3⋊D9D412yescomplemented
45C2×D3×C9—21no—
46C2×D3×C9—21no—
47C2×D3×C9D412yescomplemented
48C2×C3×D9—21no—
49C2×C3×D9—21no—
50C2×C3×D9D412yescomplemented
51D3×D9—28no—
52C3⋊Dic9—21no—
53C9×Dic3—21no—
54C3×Dic9—21no—
55#937—91no—
56#937—91no—
57C23×C32C2×D311yescharacteristic
58C22×C3⋊D3—31no—
59C22×C3×D3—31no—
60C22×C3×D3C2×D311yescharacteristic
61C2×D32—62no—
62C2×D32—62no—
63C2×D32—62no—
64#638—62no—
65C2×C3⋊Dic3—31no—
66C3×#67—22no—
67C3×#67—62no—
68C2×C3×Dic3—31no—
69C2×C3×Dic3C2×D311yescharacteristic
70C23×C9—21no—
71C23×C9—31no—
72C23×C9C2×D311yescharacteristic, complemented
73C22×D9—31no—
74C22×D9—61no—
75C22×D9—62no—
76C22×D9—62no—
77C22×D9—62no—
78C22×D9C2×D311yescharacteristic
79D4×C9—62no—
80C2×C4×C9—31no—
81#611—22no—
82#611—62no—
83#611—64no—
84C2×Dic9—31no—
85C2×Dic9—61no—
86C2×Dic9C2×D311yescharacteristic, complemented
87C2×C3×C9—41no—
88C2×C3×C9C2×D411yescharacteristic
89C2×C3×C9C2×D411yescharacteristic
90C2×C3×C9C2×D411yescharacteristic
91C3⋊D9—22no—
92D3×C9—22no—
93C3×D9—22no—
94C23×D3—91no—
95C23×D3—91no—
96C2×C3×D4—31no—
97C2×C3×D4—91no—
98C2×#67—31no—
99C2×#67—91no—
100C2×#67—91no—
101C2×#67—181no—
102C2×#67D911yescharacteristic, complemented
103#217—91no—
104#217—91no—
105(C2×D3)⋊C4—91no—
106(C2×D3)⋊C4—91no—
107(C2×D3)⋊C4—91no—
108(C2×D3)⋊C4—91no—
109C22×C32—21no—
110C22×C32—21no—
111C22×C32—21no—
112C22×C32C22×D311yescharacteristic
113C2×C3⋊D3—32no—
114C2×C3⋊D3—61no—
115C2×C3⋊D3—61no—
116C2×C3×D3—21no—
117C2×C3×D3—21no—
118C2×C3×D3—32no—
119C2×C3×D3—61no—
120C2×C3×D3—61no—
121C2×C3×D3#6712yes—
122D32—68no—
123C3⋊Dic3—61no—
124C3×Dic3—21no—
125C3×Dic3—61no—
126C22×C9—22no—
127C22×C9—21no—
128C22×C9—21no—
129C22×C9—21no—
130C22×C9—21no—
131C22×C9—32no—
132C22×C9—41no—
133C22×C9—41no—
134C22×C9—61no—
135C22×C9—61no—
136C22×C9C22×D311yescharacteristic
137C2×D9—21no—
138C2×D9—21no—
139C2×D9—32no—
140C2×D9—62no—
141C2×D9—68no—
142C2×D9—61no—
143C2×D9—61no—
144C2×D9—121no—
145C2×D9—121no—
146C2×D9#6712yescomplemented
147C4×C9—61no—
148Dic9—21no—
149Dic9—62no—
150Dic9—61no—
151#120—271noSylow
152C3×C9#12011yescharacteristic, complemented, Sylow
153C23×C3—21no—
154C23×C3—31no—
155C23×C3—91no—
156C23×C3C2×D911yescharacteristic, complemented
157C23×C3D3211yescharacteristic
158C22×D3—31no—
159C22×D3—91no—
160C22×D3—91no—
161C22×D3—181no—
162C22×D3—182no—
163C22×D3—182no—
164C22×D3—182no—
165C22×D3—182no—
166C22×D3—182no—
167C22×D3—182no—
168C22×D3C2×D911yescharacteristic
169C3×D4—62no—
170C3×D4—182no—
171C2×C3×C4—31no—
172C2×C3×C4—91no—
173#67—22no—
174#67—62no—
175#67—182no—
176#67—182no—
177#67—184no—
178C2×Dic3—31no—
179C2×Dic3—91no—
180C2×Dic3—91no—
181C2×Dic3—181no—
182C2×Dic3C2×D911yescharacteristic, complemented
183C2×C32—41no—
184C2×C32D3×D411yescharacteristic
185C2×C32D3×D411yescharacteristic
186C2×C32C2×#6711yescharacteristic
187C3⋊D3—62no—
188C3×D3—22no—
189C3×D3—62no—
190C2×C9—21no—
191C2×C9—21no—
192C2×C9—21no—
193C2×C9—42no—
194C2×C9—41no—
195C2×C9—62no—
196C2×C9D3×D411yescharacteristic
197C2×C9D3×D411yescharacteristic
198C2×C9C2×#6711yescharacteristic
199D9—22no—
200D9—62no—
201D9—122no—
202C24—271no—
203C2×D4—31no—
204C2×D4—91no—
205C2×D4—271no—
206C22⋊C4—271no—
207C22⋊C4—271no—
208C22⋊C4—271no—
209C22×C3—22no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—32no—
218C22×C3—41no—
219C22×C3—41no—
220C22×C3—61no—
221C22×C3—61no—
222C22×C3—92no—
223C22×C3—181no—
224C22×C3—181no—
225C22×C3C22×D911yescharacteristic
226C22×C3C2×D3211yescharacteristic
227C2×D3—21no—
228C2×D3—21no—
229C2×D3—32no—
230C2×D3—61no—
231C2×D3—61no—
232C2×D3—92no—
233C2×D3—92no—
234C2×D3—182no—
235C2×D3—188no—
236C2×D3—188no—
237C2×D3—181no—
238C2×D3—181no—
239C2×D3—181no—
240C2×D3—181no—
241C2×D3—361no—
242C2×D3—361no—
243C2×D3#61112yescomplemented
244C3×C4—61no—
245C3×C4—181no—
246Dic3—21no—
247Dic3—61no—
248Dic3—182no—
249Dic3—181no—
250Dic3—181no—
251C32#93711yescharacteristic
252C9—21no—
253C9#93711yescharacteristic, complemented
254C23—31no—
255C23—91no—
256C23—271no—
257C23—542no—
258C23—542no—
259C23—542no—
260C23D3×D911yescharacteristic, complemented
261D4—62no—
262D4—182no—
263D4—542no—
264C2×C4—31no—
265C2×C4—91no—
266C2×C4—271no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—42no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—62no—
274C2×C3—182no—
275C2×C3D4×D911yescharacteristic
276C2×C3D4×D911yescharacteristic
277C2×C3C2×#61111yescharacteristic
278C2×C3D3×#6711yescharacteristic
279C2×C3D3×#6711yescharacteristic
280C2×C3#381111yescharacteristic
281D3—22no—
282D3—62no—
283D3—182no—
284D3—182no—
285D3—362no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—32no—
290C22—61no—
291C22—61no—
292C22—92no—
293C22—181no—
294C22—181no—
295C22—272no—
296C22—548no—
297C22—541no—
298C22—541no—
299C22C2×D3×D911yescharacteristic
300C4—61no—
301C4—181no—
302C4—541no—
303C3—21no—
304C3#6349911yescharacteristic, complemented
305C3#6398311yescharacteristic
306C2—41no—
307C2—62no—
308C2—182no—
309C2—542no—
310C2D3×#61111yescharacteristic
311C2D9×#6711yescharacteristic
312C2#8866811yescharacteristic
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. C22×D3×D9C2semidirect
2. C2×#6233C2semidirect
3. C2×C9×#67C2semidirect
4. C2×C3×#611C2semidirect
5. (C2×D3)⋊Dic9C2semidirect
6. #88445C2semidirect
7. (C2×D9)⋊Dic3C2semidirect
10. C23×C3×C9C22semidirect
18. C2×C3⋊Dic9C22semidirect
21. C2×C9×Dic3C22semidirect
22. C2×C3×Dic9C22semidirect
34. C2×#611D3semidirect
44. C2×C3⋊D9D4semidirect
47. C2×D3×C9D4semidirect
50. C2×C3×D9D4semidirect
72. C23×C9C2×D3semidirect
86. C2×Dic9C2×D3semidirect
102. C2×#67D9semidirect
146. C2×D9#67semidirect
152. C3×C9#120semidirect
156. C23×C3C2×D9semidirect
182. C2×Dic3C2×D9semidirect
243. C2×D3#611semidirect
253. C9#937semidirect
260. C23D3×D9semidirect
304. C3#63499semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D92
C2×#62332
C2×C9×#672
C2×C3×#6112
(C2×D3)⋊Dic92
#884452
(C2×D9)⋊Dic32
#639833
#634993