Finite Group Databasebeta

← #11722953

Subgroups — #11722953

Order 864.

514 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 82 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×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#625C212yesmaximal, complemented
3C2×C3×#625C211yesmaximal, characteristic, complemented
4#88815C211yesmaximal, characteristic, complemented
5(C2×D3)⋊(C3⋊Dic3)C212yesmaximal, complemented
6#63980—31nomaximal
7#63970—32nomaximal
8C22×C32×D3C2212yes—
9C22×C32×D3C2211yescharacteristic
10C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C2×C3×C3⋊Dic3C2212yes—
12C2×C3×C3⋊Dic3C2211yescharacteristic
13D3×C3⋊Dic3—22no—
14C3×#625—24no—
15C3×#625—22no—
16C22×C3⋊Dic3—31no—
17C2×C3×#67—32no—
18C2×C3×#67—32no—
19C2×C3×#67—32no—
20C2×#625D312yescomplemented
21C2×#625D311yescharacteristic, complemented
22C2×D3×Dic3—32no—
23C2×D3×Dic3—61no—
24#3757—32no—
25#3724—31no—
26Dic3⋊Dic3—32no—
27(C2×D3)⋊Dic3—32no—
28(C2×D3)⋊Dic3—32no—
29(C2×D3)⋊Dic3—62no—
30C22×C33C2311yescharacteristic
31C2×C32×D3—21no—
32C2×C32×D3—21no—
33C2×C32×D3—22no—
34C2×C32×D3—22no—
35C2×C32×D3—22no—
36C2×C32×D3D412yescomplemented
37C3⋊(C3⋊Dic3)—21no—
38C3×C3⋊Dic3—22no—
39C3×C3⋊Dic3D412yescomplemented
40#939—92no—
41#938—91no—
42C23×C32—31no—
43C23×C32—32no—
44C22×C3×D3—21no—
45C22×C3×D3—22no—
46C22×C3×D3C2×D312yes—
47C22×C3×D3C2×D312yes—
48C22×C3×D3C2×D312yes—
49C2×C3⋊Dic3—31no—
50C2×C3⋊Dic3—32no—
51C2×C3⋊Dic3—61no—
52C2×C3⋊Dic3—62no—
53C2×C3⋊Dic3—62no—
54C2×C3⋊Dic3—121no—
55C2×C3⋊Dic3C2×D312yes—
56C2×C3⋊Dic3C2×D311yescharacteristic
57C3×#67—64no—
58C3×#67—64no—
59C3×#67—64no—
60C2×C3×Dic3—32no—
61C2×C3×Dic3—32no—
62C2×C3×Dic3—32no—
63C2×C3×Dic3—61no—
64C2×C3×Dic3—62no—
65#625—24no—
66#625—22no—
67D3×Dic3—64no—
68D3×Dic3—64no—
69C2×C33C2×D411yescharacteristic
70C2×C33C2×D411yescharacteristic
71C2×C33#4111yescharacteristic
72C32×D3—22no—
73C32×D3—42no—
74C2×C3×D4—91no—
75C2×C3×D4—92no—
76C2×#67—32no—
77C2×#67—32no—
78C2×#67—32no—
79C22×Dic3—92no—
80C22×Dic3—181no—
81C2×C4×D3—91no—
82#217—92no—
83#217—92no—
84#217—182no—
85(C2×D3)⋊C4—92no—
86C4⋊Dic3—91no—
87Dic3⋊C4—92no—
88C22×C32—21no—
89C22×C32—22no—
90C22×C32—32no—
91C22×C32—41no—
92C22×C32—61no—
93C22×C32—61no—
94C22×C32—62no—
95C22×C32—62no—
96C22×C32—62no—
97C22×C32C22×D312yes—
98C22×C32C22×D311yescharacteristic
99C2×C3×D3—22no—
100C2×C3×D3—22no—
101C2×C3×D3—22no—
102C2×C3×D3—22no—
103C2×C3×D3—24no—
104C2×C3×D3—22no—
105C2×C3×D3—22no—
106C2×C3×D3—22no—
107C2×C3×D3—22no—
108C2×C3×D3—22no—
109C2×C3×D3—22no—
110C2×C3×D3—22no—
111C2×C3×D3—42no—
112C2×C3×D3—42no—
113C2×C3×D3#6714yescomplemented
114C3⋊Dic3—22no—
115C3⋊Dic3—61no—
116C3⋊Dic3—62no—
117C3⋊Dic3—62no—
118C3⋊Dic3—122no—
119C3⋊Dic3—122no—
120C3⋊Dic3#6712yescomplemented
121C3×Dic3—34no—
122C3×Dic3—62no—
123C3×Dic3—62no—
124C3×Dic3—62no—
125C3×Dic3—122no—
126#121—271noSylow
127C33#12111yescharacteristic, complemented, Sylow
128C23×C3—32no—
129C23×C3—32no—
130C23×C3—32no—
131C23×C3—61no—
132C23×C3—62no—
133C22×D3D3212yes—
134C22×D3D3211yescharacteristic
135C3×D4—182no—
136C3×D4—184no—
137C2×C3×C4—91no—
138C2×C3×C4—92no—
139#67—64no—
140#67—64no—
141#67—64no—
142C2×Dic3—32no—
143C2×Dic3—32no—
144C2×Dic3—32no—
145C2×Dic3—61no—
146C2×Dic3—62no—
147C2×Dic3—92no—
148C2×Dic3—91no—
149C2×Dic3—181no—
150C2×Dic3—182no—
151C2×Dic3—184no—
152C2×Dic3—184no—
153C2×Dic3—361no—
154C4×D3—182no—
155C2×C32—22no—
156C2×C32—21no—
157C2×C32—21no—
158C2×C32—22no—
159C2×C32—22no—
160C2×C32—21no—
161C2×C32—41no—
162C2×C32—41no—
163C2×C32—41no—
164C2×C32—62no—
165C2×C32—122no—
166C2×C32D3×D411yescharacteristic
167C2×C32D3×D412yes—
168C2×C32#23711yescharacteristic
169C2×C32#23712yes—
170C2×C32C2×#6712yes—
171C2×C32C2×#6711yescharacteristic
172C3×D3—24no—
173C3×D3—24no—
174C3×D3—44no—
175C3×D3—42no—
176C3×D3—42no—
177C2×D4—91no—
178C2×D4—92no—
179C22×C4—271no—
180C4⋊C4—271no—
181C22⋊C4—272no—
182C22×C3—22no—
183C22×C3—21no—
184C22×C3—34no—
185C22×C3—41no—
186C22×C3—62no—
187C22×C3—62no—
188C22×C3—62no—
189C22×C3—64no—
190C22×C3—62no—
191C22×C3—62no—
192C22×C3—62no—
193C22×C3—62no—
194C22×C3—62no—
195C22×C3—62no—
196C22×C3—62no—
197C22×C3—62no—
198C22×C3—122no—
199C22×C3—122no—
200C22×C3C2×D3211yescharacteristic
201C22×C3C2×D3212yes—
202C2×D3—21no—
203C2×D3—21no—
204C2×D3—22no—
205C2×D3—22no—
206C2×D3—22no—
207C2×D3#62512yescomplemented
208C3×C4—92no—
209C3×C4—182no—
210Dic3—34no—
211Dic3—62no—
212Dic3—62no—
213Dic3—62no—
214Dic3—122no—
215Dic3—182no—
216Dic3—182no—
217Dic3—181no—
218Dic3—362no—
219Dic3—362no—
220C32—22no—
221C32—21no—
222C32—41no—
223C32#93811yescharacteristic, complemented
224C32#93912yescomplemented
225C23—31no—
226C23—32no—
227D4—182no—
228D4—184no—
229C2×C4—91no—
230C2×C4—92no—
231C2×C4—271no—
232C2×C4—542no—
233C2×C3—22no—
234C2×C3—21no—
235C2×C3—21no—
236C2×C3—22no—
237C2×C3—21no—
238C2×C3—22no—
239C2×C3—41no—
240C2×C3—41no—
241C2×C3—41no—
242C2×C3—64no—
243C2×C3—64no—
244C2×C3—124no—
245C2×C3—122no—
246C2×C3—122no—
247C2×C3#380512yes—
248C2×C3#380511yescharacteristic
249C2×C3C2×#62511yescharacteristic
250C2×C3D3×#6712yes—
251C2×C3D3×#6712yes—
252C2×C3#381111yescharacteristic
253D3—22no—
254D3—42no—
255C22—32no—
256C22—61no—
257C22—61no—
258C22—62no—
259C22—62no—
260C22—62no—
261C22D3311yescharacteristic
262C4—92no—
263C4—182no—
264C4—541no—
265C3—21no—
266C3—22no—
267C3—41no—
268C3#6397012yescomplemented
269C3#6398011yescharacteristic, complemented
270C2—62no—
271C2—122no—
272C2D3×#62511yescharacteristic
273C2#8895211yescharacteristic
274C2#8896311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
hypocenterC33#121
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×D3×C3⋊Dic3C2semidirect
2. C2×C3×#625C2semidirect
3. C2×C3×#625C2semidirect
4. #88815C2semidirect
5. (C2×D3)⋊(C3⋊Dic3)C2semidirect
20. C2×#625D3semidirect
21. C2×#625D3semidirect
36. C2×C32×D3D4semidirect
39. C3×C3⋊Dic3D4semidirect
113. C2×C3×D3#67semidirect
120. C3⋊Dic3#67semidirect
127. C33#121semidirect
207. C2×D3#625semidirect
223. C32#938semidirect
224. C32#939semidirect
268. C3#63970semidirect
269. C3#63980semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C3⋊Dic32
C2×C3×#6252
#888152
(C2×D3)⋊(C3⋊Dic3)2
#639803
#639703