Finite Group Databasebeta

← #11723007

Subgroups — #11723007

Order 864.

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

normal subgroups only

This group has 418 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×#6295C211yesmaximal, characteristic
2C2×C3×#627C211yesmaximal, characteristic, complemented
3#88803C212yesmaximal, complemented
4Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C3×#3722C212yesmaximal, complemented
6#63915—31nomaximal
7#63907—32nomaximal
8C22×C3×C3⋊D3C2212yes—
9C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C2×C3×C3⋊Dic3C2211yescharacteristic
11C2×C32×Dic3C2212yes—
12C2×C32×Dic3C2211yescharacteristic
13#6295—22no—
14C3×#627—22no—
15C2×C3×Dic6—32no—
16C2×#627D311yescharacteristic, complemented
17C2×#626—32no—
18C2×#625—31no—
19#3752—32no—
20C4×C3⋊Dic3—31no—
21C3×(C2×D3)⋊C4—32no—
22C3×(C2×D3)⋊C4—32no—
23#3722D312yescomplemented
24(C2×D3)⋊Dic3—32no—
25(C2×D3)⋊Dic3—32no—
26(C2×D3)⋊Dic3—62no—
27Dic32—32no—
28Dic32—61no—
29C22×C33C2311yescharacteristic
30C2×C3×C3⋊D3—22no—
31C2×C3×C3⋊D3—22no—
32C2×C3×C3⋊D3—22no—
33C3⋊(C3⋊Dic3)—21no—
34C3×C3⋊Dic3D412yes—
35C32×Dic3—22no—
36C32×Dic3—21no—
37#947—91no—
38#885—92no—
39C22×C3⋊D3C2×D312yes—
40C22×C3×D3—32no—
41C22×C3×D3—32no—
42C22×C3×D3—62no—
43C2×C32×C4—31no—
44C2×C32×C4—32no—
45C2×C3⋊Dic3—32no—
46C2×C3⋊Dic3—31no—
47C2×C3⋊Dic3—61no—
48C2×C3⋊Dic3—62no—
49C2×C3⋊Dic3—121no—
50C2×C3⋊Dic3C2×D311yescharacteristic
51C2×C3×Dic3—21no—
52C2×C3×Dic3—22no—
53C2×C3×Dic3—32no—
54C2×C3×Dic3—61no—
55C2×C3×Dic3C2×D312yes—
56C2×C3×Dic3C2×D312yes—
57C2×C3×Dic3C2×D312yes—
58C3×Dic6—64no—
59#627—22no—
60#626—64no—
61#625—62no—
62C2×C33C2×D411yescharacteristic
63C2×C33#4111yescharacteristic
64C2×C33#4111yescharacteristic
65C3×C3⋊D3—42no—
66C2×C3×Q8—91no—
67C2×#67—92no—
68C2×D12—91no—
69C2×Dic6—32no—
70C3×C22⋊C4—92no—
71#217—92no—
72(C2×D3)⋊C4—32no—
73(C2×D3)⋊C4—32no—
74(C2×D3)⋊C4—92no—
75(C2×D3)⋊C4—92no—
76(C2×D3)⋊C4—182no—
77C4×Dic3—92no—
78C4×Dic3—91no—
79C4×Dic3—181no—
80C22×C32—21no—
81C22×C32—22no—
82C22×C32—41no—
83C22×C32C22×D312yes—
84C22×C32C22×D311yescharacteristic
85C2×C3⋊D3—22no—
86C2×C3⋊D3—22no—
87C2×C3⋊D3—22no—
88C2×C3×D3—64no—
89C2×C3×D3—62no—
90C2×C3×D3—62no—
91C2×C3×D3—62no—
92C2×C3×D3—62no—
93C2×C3×D3—62no—
94C2×C3×D3—62no—
95C2×C3×D3—122no—
96C2×C3×D3—122no—
97C32×C4—61no—
98C32×C4—62no—
99C3⋊Dic3—61no—
100C3⋊Dic3—62no—
101C3⋊Dic3—62no—
102C3⋊Dic3—122no—
103C3⋊Dic3—122no—
104C3⋊Dic3#6712yes—
105C3×Dic3—22no—
106C3×Dic3—22no—
107C3×Dic3—22no—
108C3×Dic3—22no—
109C3×Dic3—34no—
110C3×Dic3—42no—
111C3×Dic3—62no—
112#124—271noSylow
113C33#12411yescharacteristic, complemented, Sylow
114C23×C3—92no—
115C22×D3—32no—
116C22×D3—32no—
117C22×D3—62no—
118C3×Q8—182no—
119C2×C3×C4—32no—
120C2×C3×C4—32no—
121C2×C3×C4—32no—
122C2×C3×C4—61no—
123C2×C3×C4—62no—
124C2×C3×C4—91no—
125#67—184no—
126C2×Dic3—32no—
127C2×Dic3—61no—
128C2×Dic3—91no—
129C2×Dic3—92no—
130C2×Dic3—181no—
131C2×Dic3—182no—
132C2×Dic3—361no—
133C2×Dic3D3211yescharacteristic
134C2×Dic3D3212yes—
135D12—182no—
136Dic6—64no—
137C2×C32—22no—
138C2×C32—21no—
139C2×C32—22no—
140C2×C32—21no—
141C2×C32—21no—
142C2×C32—22no—
143C2×C32—41no—
144C2×C32—41no—
145C2×C32—41no—
146C2×C32#23512yes—
147C2×C32D3×D412yes—
148C2×C32#23712yes—
149C2×C32#23911yescharacteristic
150C2×C32#23911yescharacteristic
151C2×C32C2×#6711yescharacteristic
152C3⋊D3—42no—
153C3×D3—124no—
154C3×D3—122no—
155C3×D3—122no—
156C2×Q8—91no—
157C2×D4—271no—
158C22⋊C4—92no—
159C22⋊C4—272no—
160C42—271no—
161C22×C3—21no—
162C22×C3—22no—
163C22×C3—41no—
164C22×C3—182no—
165C22×C3—182no—
166C22×C3—182no—
167C22×C3C2×D3212yes—
168C22×C3C2×D3211yescharacteristic
169C2×D3—64no—
170C2×D3—62no—
171C2×D3—62no—
172C2×D3—62no—
173C2×D3—62no—
174C2×D3—62no—
175C2×D3—62no—
176C2×D3—122no—
177C2×D3—122no—
178C3×C4—62no—
179C3×C4—62no—
180C3×C4—62no—
181C3×C4—62no—
182C3×C4—92no—
183C3×C4—122no—
184Dic3—21no—
185Dic3—22no—
186Dic3—34no—
187Dic3—62no—
188Dic3—182no—
189Dic3—181no—
190Dic3—182no—
191Dic3—362no—
192Dic3—362no—
193C32—22no—
194C32—21no—
195C32—41no—
196C32#88512yescomplemented
197C32#94711yescharacteristic, complemented
198C23—92no—
199Q8—182no—
200D4—542no—
201C2×C4—31no—
202C2×C4—32no—
203C2×C4—91no—
204C2×C4—271no—
205C2×C3—21no—
206C2×C3—22no—
207C2×C3—21no—
208C2×C3—22no—
209C2×C3—21no—
210C2×C3—22no—
211C2×C3—41no—
212C2×C3—41no—
213C2×C3—41no—
214C2×C3—362no—
215C2×C3#379612yes—
216C2×C3#379811yescharacteristic
217C2×C3#379912yes—
218C2×C3#380211yescharacteristic
219C2×C3#380511yescharacteristic
220C2×C3D3×#6712yes—
221D3—124no—
222D3—122no—
223D3—122no—
224C22—182no—
225C22—182no—
226C22—182no—
227C22D3311yescharacteristic
228C4—62no—
229C4—61no—
230C4—92no—
231C4—541no—
232C3—22no—
233C3—21no—
234C3—41no—
235C3#6390712yescomplemented
236C3#6391511yescharacteristic, complemented
237C2—362no—
238C2#8895311yescharacteristic
239C2#8896411yescharacteristic
240C2#8896511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC2×#6295C2
hypocenterC33#124
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3×#627C2semidirect
3. #88803C2semidirect
4. Dic3×C3⋊Dic3C2semidirect
5. C3×#3722C2semidirect
16. C2×#627D3semidirect
23. #3722D3semidirect
113. C33#124semidirect
196. C32#885semidirect
197. C32#947semidirect
235. C3#63907semidirect
236. C3#63915semidirect

Maximal subgroups

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

subgroupindex
C2×#62952
C2×C3×#6272
#888032
Dic3×C3⋊Dic32
C3×#37222
#639153
#639073