Finite Group Databasebeta

← #11723018

Subgroups — #11723018

Order 864.

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

normal subgroups only

This group has 322 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
1#88815C211yesmaximal, characteristic
2Dic3⋊(C3⋊Dic3)C212yesmaximal
3Dic3×C3⋊Dic3C211yesmaximal, characteristic
4C3×#3724C212yesmaximal
5C3×#3724C211yesmaximal, characteristic
6#63870—31nomaximal
7#63845—32nomaximal
8C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C2×C3×C3⋊Dic3C2212yes—
10C2×C3×C3⋊Dic3C2211yescharacteristic
11C2×C32×Dic3C2211yescharacteristic
12C2×C32×Dic3C2212yes—
13C4⋊(C3⋊Dic3)—32no—
14C4×C3⋊Dic3—31no—
15C3×Dic3⋊C4—32no—
16C3×Dic3⋊C4—32no—
17C3×Dic3⋊C4—32no—
18#3724—31no—
19#3724D312yes—
20#3724D311yescharacteristic
21Dic3⋊Dic3—32no—
22Dic3⋊Dic3—32no—
23Dic3⋊Dic3—32no—
24Dic3⋊Dic3—62no—
25Dic32—32no—
26Dic32—61no—
27C22×C33C2311yescharacteristic
28C3⋊(C3⋊Dic3)—21no—
29C3×C3⋊Dic3—22no—
30C3×C3⋊Dic3Q812yes—
31C32×Dic3—21no—
32C32×Dic3—22no—
33#890—91no—
34#889—92no—
35C2×C32×C4—31no—
36C2×C32×C4—32no—
37C2×C3⋊Dic3—32no—
38C2×C3⋊Dic3—31no—
39C2×C3⋊Dic3—61no—
40C2×C3⋊Dic3—62no—
41C2×C3⋊Dic3—121no—
42C2×C3⋊Dic3C2×D312yes—
43C2×C3⋊Dic3C2×D311yescharacteristic
44C2×C3×Dic3—21no—
45C2×C3×Dic3—22no—
46C2×C3×Dic3—32no—
47C2×C3×Dic3—32no—
48C2×C3×Dic3—32no—
49C2×C3×Dic3—61no—
50C2×C3×Dic3—62no—
51C2×C3×Dic3C2×D312yes—
52C2×C3×Dic3C2×D312yes—
53C2×C3×Dic3C2×D312yes—
54C2×C33C2×Q811yescharacteristic
55C2×C33#4111yescharacteristic
56C2×C33#4111yescharacteristic
57C3×C4⋊C4—92no—
58C3×C4⋊C4—91no—
59C4⋊Dic3—91no—
60C4⋊Dic3—92no—
61C4⋊Dic3—92no—
62C4⋊Dic3—182no—
63Dic3⋊C4—32no—
64Dic3⋊C4—32no—
65Dic3⋊C4—32no—
66Dic3⋊C4—92no—
67Dic3⋊C4—92no—
68C4×Dic3—92no—
69C4×Dic3—91no—
70C4×Dic3—181no—
71C22×C32—22no—
72C22×C32—21no—
73C22×C32—41no—
74C22×C32C22×D311yescharacteristic
75C22×C32C22×D312yes—
76C32×C4—62no—
77C32×C4—61no—
78C3⋊Dic3—22no—
79C3⋊Dic3—61no—
80C3⋊Dic3—62no—
81C3⋊Dic3—62no—
82C3⋊Dic3—122no—
83C3⋊Dic3—122no—
84C3⋊Dic3Dic612yes—
85C3×Dic3—22no—
86C3×Dic3—24no—
87C3×Dic3—22no—
88C3×Dic3—22no—
89C3×Dic3—22no—
90C3×Dic3—34no—
91C3×Dic3—62no—
92C3×Dic3—62no—
93C3×Dic3—62no—
94C3×Dic3—122no—
95#125—271noSylow
96C33#12511yescharacteristic, complemented, Sylow
97C2×C3×C4—32no—
98C2×C3×C4—32no—
99C2×C3×C4—32no—
100C2×C3×C4—61no—
101C2×C3×C4—62no—
102C2×C3×C4—91no—
103C2×C3×C4—92no—
104C2×Dic3—32no—
105C2×Dic3—32no—
106C2×Dic3—32no—
107C2×Dic3—61no—
108C2×Dic3—62no—
109C2×Dic3—92no—
110C2×Dic3—91no—
111C2×Dic3—181no—
112C2×Dic3—182no—
113C2×Dic3—361no—
114C2×Dic3D3211yescharacteristic
115C2×Dic3D3212yes—
116C2×C32—22no—
117C2×C32—21no—
118C2×C32—21no—
119C2×C32—22no—
120C2×C32—22no—
121C2×C32—21no—
122C2×C32—41no—
123C2×C32—41no—
124C2×C32—41no—
125C2×C32C2×Dic611yescharacteristic
126C2×C32#23512yes—
127C2×C32#23712yes—
128C2×C32#23711yescharacteristic
129C2×C32D3×Q812yes—
130C2×C32#23911yescharacteristic
131C4⋊C4—92no—
132C4⋊C4—91no—
133C4⋊C4—272no—
134C4⋊C4—271no—
135C42—271no—
136C22×C3—21no—
137C22×C3—22no—
138C22×C3—41no—
139C22×C3C2×D3212yes—
140C22×C3C2×D3211yescharacteristic
141C3×C4—62no—
142C3×C4—64no—
143C3×C4—62no—
144C3×C4—62no—
145C3×C4—62no—
146C3×C4—92no—
147C3×C4—182no—
148Dic3—22no—
149Dic3—21no—
150Dic3—34no—
151Dic3—62no—
152Dic3—62no—
153Dic3—62no—
154Dic3—122no—
155Dic3—182no—
156Dic3—181no—
157Dic3—182no—
158Dic3—362no—
159Dic3—362no—
160C32—22no—
161C32—21no—
162C32—41no—
163C32#88912yescomplemented
164C32#89011yescharacteristic, complemented
165C2×C4—31no—
166C2×C4—32no—
167C2×C4—91no—
168C2×C4—92no—
169C2×C4—271no—
170C2×C3—21no—
171C2×C3—22no—
172C2×C3—22no—
173C2×C3—21no—
174C2×C3—22no—
175C2×C3—21no—
176C2×C3—41no—
177C2×C3—41no—
178C2×C3—41no—
179C2×C3D3×Dic612yes—
180C2×C3#379711yescharacteristic
181C2×C3#379811yescharacteristic
182C2×C3#379912yes—
183C2×C3#380512yes—
184C2×C3#380511yescharacteristic
185C22D3311yescharacteristic
186C4—61no—
187C4—62no—
188C4—92no—
189C4—182no—
190C4—541no—
191C3—22no—
192C3—21no—
193C3—41no—
194C3#6384512yescomplemented
195C3#6387011yescharacteristic, complemented
196C2#8895711yescharacteristic
197C2#8896311yescharacteristic
198C2#8896511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, prime radical, cosocleC22×C33C23
hypocenterC33#125
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
96. C33#125semidirect
163. C32#889semidirect
164. C32#890semidirect
194. C3#63845semidirect
195. C3#63870semidirect

Maximal subgroups

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

subgroupindex
#888152
Dic3⋊(C3⋊Dic3)2
Dic3×C3⋊Dic32
C3×#37242
#638703
#638453