Finite Group Databasebeta

← #11723010

Subgroups — #11723010

Order 864.

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

normal subgroups only

This group has 426 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×#6293C211yesmaximal, characteristic, complemented
3#88803C212yesmaximal, complemented
4C3×#3722C212yesmaximal, complemented
5C3×Dic32C211yesmaximal, characteristic, complemented
6#63913—31nomaximal
7#63852—32nomaximal
8C22×C3×C3⋊D3C2212yes—
9C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C2×C3×C3⋊Dic3C2211yescharacteristic
11C2×C32×Dic3C2212yes—
12C2×C32×Dic3C2211yescharacteristic
13#6295—22no—
14#6293—22no—
15C2×#634—31no—
16C2×#627—32no—
17C2×#627—61no—
18C2×#626—32no—
19C2×#625—31no—
20#3752—32no—
21C3×(C2×D3)⋊C4—32no—
22C3×(C2×D3)⋊C4—32no—
23C3×C4×Dic3—32no—
24#3722D312yescomplemented
25(C2×D3)⋊Dic3—32no—
26(C2×D3)⋊Dic3—32no—
27(C2×D3)⋊Dic3—62no—
28Dic32D311yescharacteristic, complemented
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—21no—
36C32×Dic3—22no—
37#885—92no—
38#875—91no—
39C22×C3⋊D3C2×D312yes—
40C22×C3×D3—32no—
41C22×C3×D3—32no—
42C22×C3×D3—62no—
43C2×C32×C4—32no—
44C2×C32×C4—31no—
45C2×C3⋊Dic3—31no—
46C2×C3⋊Dic3—32no—
47C2×C3⋊Dic3—61no—
48C2×C3⋊Dic3—62no—
49C2×C3⋊Dic3—121no—
50C2×C3⋊Dic3C2×D311yescharacteristic
51#634—62no—
52C2×C3×Dic3—21no—
53C2×C3×Dic3—22no—
54C2×C3×Dic3—32no—
55C2×C3×Dic3—61no—
56C2×C3×Dic3C2×D312yes—
57C2×C3×Dic3C2×D312yes—
58C2×C3×Dic3C2×D312yes—
59#627—64no—
60#627—64no—
61#626—64no—
62#625—62no—
63C2×C33C2×D411yescharacteristic
64C2×C33#4112yes—
65C3×C3⋊D3—42no—
66C2×#67—92no—
67C2×D12—91no—
68C2×Dic6—92no—
69C2×Dic6—91no—
70C2×Dic6—181no—
71C3×C22⋊C4—92no—
72C3×C42—91no—
73#217—92no—
74(C2×D3)⋊C4—32no—
75(C2×D3)⋊C4—32no—
76(C2×D3)⋊C4—92no—
77(C2×D3)⋊C4—92no—
78(C2×D3)⋊C4—182no—
79C4×Dic3—32no—
80C22×C32—21no—
81C22×C32—22no—
82C22×C32—41no—
83C22×C32C22×D311yescharacteristic
84C22×C32C22×D312yes—
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⋊Dic3D1212yes—
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—
118C2×C3×C4—32no—
119C2×C3×C4—32no—
120C2×C3×C4—32no—
121C2×C3×C4—61no—
122C2×C3×C4—62no—
123C2×C3×C4—91no—
124#67—184no—
125C2×Dic3—32no—
126C2×Dic3—61no—
127C2×Dic3—91no—
128C2×Dic3—92no—
129C2×Dic3—181no—
130C2×Dic3—182no—
131C2×Dic3—361no—
132C2×Dic3D3212yes—
133C2×Dic3D3211yescharacteristic
134D12—182no—
135Dic6—182no—
136Dic6—184no—
137Dic6—184no—
138C2×C32—22no—
139C2×C32—22no—
140C2×C32—22no—
141C2×C32—22no—
142C2×C32—21no—
143C2×C32—41no—
144C2×C32—42no—
145C2×C32C2×D1211yescharacteristic
146C2×C32#23512yes—
147C2×C32#23512yes—
148C2×C32D3×D412yes—
149C2×C32#23712yes—
150C3⋊D3—42no—
151C3×D3—124no—
152C3×D3—122no—
153C3×D3—122no—
154C2×Q8—271no—
155C2×D4—271no—
156C22⋊C4—92no—
157C22⋊C4—272no—
158C42—91no—
159C22×C3—21no—
160C22×C3—22no—
161C22×C3—41no—
162C22×C3—182no—
163C22×C3—182no—
164C22×C3—182no—
165C22×C3C2×D3211yescharacteristic
166C22×C3C2×D3212yes—
167C2×D3—64no—
168C2×D3—62no—
169C2×D3—62no—
170C2×D3—62no—
171C2×D3—62no—
172C2×D3—62no—
173C2×D3—62no—
174C2×D3—122no—
175C2×D3—122no—
176C3×C4—62no—
177C3×C4—62no—
178C3×C4—62no—
179C3×C4—62no—
180C3×C4—92no—
181C3×C4—122no—
182Dic3—22no—
183Dic3—21no—
184Dic3—34no—
185Dic3—62no—
186Dic3—182no—
187Dic3—182no—
188Dic3—181no—
189Dic3—362no—
190Dic3—362no—
191C32—22no—
192C32—21no—
193C32—41no—
194C32#87511yescharacteristic, complemented
195C32#88512yescomplemented
196C23—92no—
197Q8—542no—
198D4—542no—
199C2×C4—32no—
200C2×C4—31no—
201C2×C4—91no—
202C2×C4—271no—
203C2×C3—21no—
204C2×C3—22no—
205C2×C3—22no—
206C2×C3—22no—
207C2×C3—22no—
208C2×C3—41no—
209C2×C3—42no—
210C2×C3—362no—
211C2×C3#379512yes—
212C2×C3#379812yes—
213C2×C3D3×D1212yes—
214C2×C3#380412yes—
215C2×C3#381111yescharacteristic
216D3—124no—
217D3—122no—
218D3—122no—
219C22—182no—
220C22—182no—
221C22—182no—
222C22D3311yescharacteristic
223C4—61no—
224C4—62no—
225C4—92no—
226C4—541no—
227C3—22no—
228C3—21no—
229C3—41no—
230C3#6385212yescomplemented
231C3#6391311yescharacteristic, complemented
232C2—362no—
233C2#8895511yescharacteristic
234C2#8896012yes—
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×#6293C2semidirect
3. #88803C2semidirect
4. C3×#3722C2semidirect
5. C3×Dic32C2semidirect
24. #3722D3semidirect
28. Dic32D3semidirect
113. C33#124semidirect
194. C32#875semidirect
195. C32#885semidirect
230. C3#63852semidirect
231. C3#63913semidirect

Maximal subgroups

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

subgroupindex
C2×#62952
C2×#62932
#888032
C3×#37222
C3×Dic322
#639133
#638523