Finite Group Databasebeta

← #11722993

Subgroups — #11722993

Order 864.

466 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 74 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×#6292C211yesmaximal, characteristic
2C2×C3×#624C211yesmaximal, characteristic, complemented
3#88814C211yesmaximal, characteristic, complemented
4#88804C212yesmaximal, complemented
5C3×#3724C212yesmaximal, complemented
6#63897—31nomaximal
7#63850—32nomaximal
8C22×C3⋊(C3⋊D3)C2211yescharacteristic
9C22×C3×C3⋊D3C2211yescharacteristic
10C2×C3×C3⋊Dic3C2212yes—
11C2×C32×Dic3C2212yes—
12C2×C32×Dic3C2211yescharacteristic
13#6292—22no—
14C3×#624—22no—
15C2×C4⋊(C3⋊D3)—31no—
16C2×C3×C4×D3—32no—
17C2×#626—32no—
18C2×#626—61no—
19C2×#624D311yescharacteristic, complemented
20#3752—32no—
21C3×Dic3⋊C4—32no—
22C3×Dic3⋊C4—32no—
23#3724D312yescomplemented
24#3722—31no—
25#3722—32no—
26#3722—32no—
27#3722—62no—
28(C2×D3)⋊Dic3—32no—
29C22×C33C2311yescharacteristic
30C2×C3⋊(C3⋊D3)—21no—
31C2×C3⋊(C3⋊D3)—22no—
32C2×C3×C3⋊D3—22no—
33C2×C3×C3⋊D3D412yes—
34C3×C3⋊Dic3—22no—
35C32×Dic3—21no—
36C32×Dic3—22no—
37#929—91no—
38#894—92no—
39C22×C3⋊D3—31no—
40C22×C3⋊D3—32no—
41C22×C3⋊D3—62no—
42C22×C3⋊D3—61no—
43C22×C3⋊D3—121no—
44C22×C3⋊D3C2×D311yescharacteristic
45C22×C3×D3—32no—
46C22×C3×D3—61no—
47C2×C32×C4—31no—
48C2×C32×C4—32no—
49C2×C3⋊Dic3C2×D312yes—
50C4⋊(C3⋊D3)—62no—
51C2×C3×Dic3—21no—
52C2×C3×Dic3—22no—
53C2×C3×Dic3—32no—
54C2×C3×Dic3—32no—
55C2×C3×Dic3—62no—
56C2×C3×Dic3C2×D312yes—
57C2×C3×Dic3C2×D312yes—
58C2×C3×Dic3C2×D312yes—
59C3×C4×D3—64no—
60#626—64no—
61#626—64no—
62#624—22no—
63C2×C33C2×D411yescharacteristic
64C2×C33#4112yes—
65C3⋊(C3⋊D3)—41no—
66C3×C3⋊D3—22no—
67C22×C3×C4—91no—
68C2×#67—91no—
69C2×D12—92no—
70C2×D12—181no—
71C2×C4×D3—32no—
72C3×C4⋊C4—92no—
73#217—91no—
74(C2×D3)⋊C4—92no—
75(C2×D3)⋊C4—92no—
76(C2×D3)⋊C4—92no—
77(C2×D3)⋊C4—92no—
78(C2×D3)⋊C4—182no—
79Dic3⋊C4—32no—
80Dic3⋊C4—32no—
81C22×C32—22no—
82C22×C32—21no—
83C22×C32—41no—
84C22×C32C22×D311yescharacteristic
85C22×C32C22×D312yes—
86C2×C3⋊D3—22no—
87C2×C3⋊D3—62no—
88C2×C3⋊D3—61no—
89C2×C3⋊D3—62no—
90C2×C3⋊D3—64no—
91C2×C3⋊D3—62no—
92C2×C3⋊D3—62no—
93C2×C3⋊D3—62no—
94C2×C3⋊D3—122no—
95C2×C3⋊D3—122no—
96C2×C3⋊D3—122no—
97C2×C3⋊D3—122no—
98C2×C3⋊D3—124no—
99C2×C3⋊D3#6712yes—
100C2×C3×D3—34no—
101C2×C3×D3—62no—
102C2×C3×D3—62no—
103C2×C3×D3—62no—
104C2×C3×D3—122no—
105C32×C4—61no—
106C32×C4—62no—
107C3⋊Dic3—22no—
108C3×Dic3—22no—
109C3×Dic3—22no—
110C3×Dic3—22no—
111C3×Dic3—22no—
112C3×Dic3—42no—
113C3×Dic3—62no—
114C3×Dic3—62no—
115C3×Dic3—122no—
116#123—271noSylow
117C33#12311yescharacteristic, complemented, Sylow
118C23×C3—91no—
119C22×D3—32no—
120C22×D3—61no—
121C22×D3—92no—
122C22×D3—91no—
123C22×D3—182no—
124C22×D3—181no—
125C22×D3—361no—
126C2×C3×C4—32no—
127C2×C3×C4—32no—
128C2×C3×C4—32no—
129C2×C3×C4—61no—
130C2×C3×C4—62no—
131C2×C3×C4—92no—
132C2×C3×C4—182no—
133#67—182no—
134C2×Dic3—32no—
135C2×Dic3—32no—
136C2×Dic3—62no—
137C2×Dic3D3211yescharacteristic
138C2×Dic3D3212yes—
139D12—184no—
140D12—184no—
141C4×D3—64no—
142C2×C32—22no—
143C2×C32—22no—
144C2×C32—22no—
145C2×C32—22no—
146C2×C32—21no—
147C2×C32—41no—
148C2×C32—42no—
149C2×C32#23512yes—
150C2×C32#23512yes—
151C2×C32D3×D412yes—
152C2×C32#23912yes—
153C2×C32C2×#6711yescharacteristic
154C3⋊D3—22no—
155C3⋊D3—124no—
156C3⋊D3—122no—
157C3⋊D3—124no—
158C3⋊D3—122no—
159C3⋊D3—121no—
160C3×D3—64no—
161C3×D3—122no—
162C2×D4—271no—
163C22×C4—91no—
164C4⋊C4—92no—
165C22⋊C4—272no—
166C22⋊C4—271no—
167C22×C3—22no—
168C22×C3—21no—
169C22×C3—41no—
170C22×C3—92no—
171C22×C3—182no—
172C22×C3C2×D3212yes—
173C22×C3C2×D3211yescharacteristic
174C2×D3—34no—
175C2×D3—62no—
176C2×D3—62no—
177C2×D3—62no—
178C2×D3—122no—
179C2×D3—182no—
180C2×D3—184no—
181C2×D3—182no—
182C2×D3—182no—
183C2×D3—182no—
184C2×D3—182no—
185C2×D3—181no—
186C2×D3—362no—
187C2×D3—362no—
188C2×D3—362no—
189C2×D3—362no—
190C2×D3—364no—
191C3×C4—62no—
192C3×C4—62no—
193C3×C4—62no—
194C3×C4—62no—
195C3×C4—122no—
196C3×C4—182no—
197Dic3—22no—
198Dic3—21no—
199Dic3—62no—
200Dic3—62no—
201Dic3—122no—
202C32—22no—
203C32—21no—
204C32—41no—
205C32#89412yescomplemented
206C32#92911yescharacteristic, complemented
207C23—91no—
208C23—271no—
209D4—542no—
210C2×C4—32no—
211C2×C4—31no—
212C2×C4—92no—
213C2×C4—182no—
214C2×C3—21no—
215C2×C3—22no—
216C2×C3—22no—
217C2×C3—22no—
218C2×C3—22no—
219C2×C3—41no—
220C2×C3—42no—
221C2×C3—182no—
222C2×C3#379812yes—
223C2×C3#379912yes—
224C2×C3#379912yes—
225C2×C3D3×#6712yes—
226C2×C3#381111yescharacteristic
227D3—64no—
228D3—122no—
229D3—362no—
230D3—364no—
231D3—364no—
232D3—362no—
233D3—361no—
234C22—92no—
235C22—182no—
236C22—541no—
237C22—542no—
238C22D3311yescharacteristic
239C4—62no—
240C4—61no—
241C4—182no—
242C3—21no—
243C3—22no—
244C3—41no—
245C3#6385012yescomplemented
246C3#6389711yescharacteristic, complemented
247C2—182no—
248C2—1081no—
249C2#8895211yescharacteristic
250C2#8896512yes—
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC2×#6292C2
hypocenterC33#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3×#624C2semidirect
3. #88814C2semidirect
4. #88804C2semidirect
5. C3×#3724C2semidirect
19. C2×#624D3semidirect
23. #3724D3semidirect
117. C33#123semidirect
205. C32#894semidirect
206. C32#929semidirect
245. C3#63850semidirect
246. C3#63897semidirect

Maximal subgroups

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

subgroupindex
C2×#62922
C2×C3×#6242
#888142
#888042
C3×#37242
#638973
#638503