Finite Group Databasebeta

← #11723036

Subgroups — #11723036

Order 864.

378 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 90 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×#6296C211yesmaximal, characteristic
2C2×#6293C211yesmaximal, characteristic
3Dic3⋊(C3⋊Dic3)C212yesmaximal
4C3×Dic3⋊Dic3C212yesmaximal
5C3×Dic32C211yesmaximal, characteristic
6#63918—32nomaximal
7#63840—31nomaximal
8C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C2×C3×C3⋊Dic3C2212yes—
10C2×C3×C3⋊Dic3C2211yescharacteristic
11C2×C32×Dic3C2211yescharacteristic
12C2×C32×Dic3C2212yes—
13#6296—22no—
14#6293—22no—
15C2×#634—31no—
16C2×#627—31no—
17C2×#627—32no—
18C2×#627—32no—
19C2×#627—61no—
20C4⋊(C3⋊Dic3)—32no—
21C3×C4⋊Dic3—32no—
22C3×Dic3⋊C4—32no—
23C3×C4×Dic3—32no—
24Dic3⋊Dic3—32no—
25Dic3⋊Dic3—32no—
26Dic3⋊Dic3—62no—
27Dic3⋊Dic3D312yes—
28Dic32D311yescharacteristic
29C22×C33C2311yescharacteristic
30C3⋊(C3⋊Dic3)—21no—
31C3×C3⋊Dic3—22no—
32C3×C3⋊Dic3D412yes—
33C32×Dic3—21no—
34C32×Dic3Q814yescomplemented
35#888—92no—
36#869—91no—
37C2×C32×C4—32no—
38C2×C32×C4—31no—
39C2×C3⋊Dic3—31no—
40C2×C3⋊Dic3—32no—
41C2×C3⋊Dic3—61no—
42C2×C3⋊Dic3—62no—
43C2×C3⋊Dic3—121no—
44C2×C3⋊Dic3C2×D312yes—
45C2×C3⋊Dic3C2×D311yescharacteristic
46#634—62no—
47C2×C3×Dic3—22no—
48C2×C3×Dic3—21no—
49C2×C3×Dic3—32no—
50C2×C3×Dic3—32no—
51C2×C3×Dic3—32no—
52C2×C3×Dic3—61no—
53C2×C3×Dic3—62no—
54C2×C3×Dic3C2×D312yes—
55C2×C3×Dic3C2×D312yes—
56C2×C3×Dic3C2×D312yes—
57#627—64no—
58#627—64no—
59#627—64no—
60#627—62no—
61C2×C33C2×D411yescharacteristic
62C2×C33C2×Q812yes—
63C2×Dic6—91no—
64C2×Dic6—92no—
65C2×Dic6—92no—
66C2×Dic6—91no—
67C2×Dic6—181no—
68C3×C4⋊C4—92no—
69C3×C42—91no—
70C4⋊Dic3—32no—
71C4⋊Dic3—92no—
72C4⋊Dic3—92no—
73C4⋊Dic3—182no—
74Dic3⋊C4—32no—
75Dic3⋊C4—92no—
76C4×Dic3—32no—
77C22×C32—21no—
78C22×C32—22no—
79C22×C32—41no—
80C22×C32C22×D311yescharacteristic
81C22×C32C22×D312yes—
82C32×C4—34no—
83C32×C4—61no—
84C3⋊Dic3—22no—
85C3⋊Dic3—61no—
86C3⋊Dic3—62no—
87C3⋊Dic3—62no—
88C3⋊Dic3—122no—
89C3⋊Dic3—122no—
90C3⋊Dic3D1212yes—
91C3×Dic3—22no—
92C3×Dic3—22no—
93C3×Dic3—24no—
94C3×Dic3—34no—
95C3×Dic3—62no—
96C3×Dic3—62no—
97C3×Dic3—62no—
98C3×Dic3—122no—
99C3×Dic3Dic614yescomplemented
100C3×Dic3Dic614yescomplemented
101C4⋊Q8—271noSylow
102C33C4⋊Q811yescharacteristic, complemented, Sylow
103C2×C3×C4—32no—
104C2×C3×C4—32no—
105C2×C3×C4—32no—
106C2×C3×C4—62no—
107C2×C3×C4—61no—
108C2×C3×C4—91no—
109C2×C3×C4—92no—
110C2×Dic3—32no—
111C2×Dic3—32no—
112C2×Dic3—32no—
113C2×Dic3—61no—
114C2×Dic3—62no—
115C2×Dic3—92no—
116C2×Dic3—91no—
117C2×Dic3—181no—
118C2×Dic3—182no—
119C2×Dic3—361no—
120C2×Dic3D3211yescharacteristic
121C2×Dic3D3212yes—
122Dic6—182no—
123Dic6—184no—
124Dic6—184no—
125Dic6—182no—
126Dic6—184no—
127C2×C32—22no—
128C2×C32—22no—
129C2×C32—22no—
130C2×C32—22no—
131C2×C32—21no—
132C2×C32—41no—
133C2×C32—42no—
134C2×C32C2×Dic612yes—
135C2×C32C2×Dic612yes—
136C2×C32C2×D1211yescharacteristic
137C2×C32D3×D412yes—
138C2×C32D3×Q812yes—
139C2×Q8—271no—
140C2×Q8—271no—
141C4⋊C4—92no—
142C4⋊C4—272no—
143C42—91no—
144C22×C3—21no—
145C22×C3—22no—
146C22×C3—41no—
147C22×C3C2×D3211yescharacteristic
148C22×C3C2×D3212yes—
149C3×C4—34no—
150C3×C4—34no—
151C3×C4—64no—
152C3×C4—62no—
153C3×C4—62no—
154C3×C4—92no—
155C3×C4—182no—
156Dic3—21no—
157Dic3—34no—
158Dic3—62no—
159Dic3—62no—
160Dic3—62no—
161Dic3—122no—
162Dic3—182no—
163Dic3—182no—
164Dic3—181no—
165Dic3—362no—
166Dic3—362no—
167Dic3#62714yescomplemented
168C32—21no—
169C32—22no—
170C32—41no—
171C32#86911yescharacteristic, complemented
172C32#88812yescomplemented
173Q8—542no—
174Q8—542no—
175C2×C4—32no—
176C2×C4—31no—
177C2×C4—91no—
178C2×C4—92no—
179C2×C4—271no—
180C2×C3—21no—
181C2×C3—22no—
182C2×C3—22no—
183C2×C3—22no—
184C2×C3—22no—
185C2×C3—41no—
186C2×C3—42no—
187C2×C3D3×Dic612yes—
188C2×C3D3×Dic612yes—
189C2×C3D3×D1212yes—
190C2×C3C2×#62712yes—
191C2×C3#381111yescharacteristic
192C22D3311yescharacteristic
193C4—34no—
194C4—61no—
195C4—92no—
196C4—182no—
197C4—541no—
198C3—21no—
199C3—22no—
200C3—41no—
201C3#6384011yescharacteristic, complemented
202C3#6391812yescomplemented
203C2#8895511yescharacteristic
204C2D3×#62712yes—
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, prime radical, cosocleC22×C33C23
hypocenterC33C4⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
34. C32×Dic3Q8semidirect
99. C3×Dic3Dic6semidirect
100. C3×Dic3Dic6semidirect
102. C33C4⋊Q8semidirect
167. Dic3#627semidirect
171. C32#869semidirect
172. C32#888semidirect
201. C3#63840semidirect
202. C3#63918semidirect

Maximal subgroups

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

subgroupindex
C2×#62962
C2×#62932
Dic3⋊(C3⋊Dic3)2
C3×Dic3⋊Dic32
C3×Dic322
#639183
#638403