Finite Group Databasebeta

← #11723035

Subgroups — #11723035

Order 864.

370 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 84 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×C3×#627C211yesmaximal, characteristic
3#88806C212yesmaximal
4Dic3×C3⋊Dic3C211yesmaximal, characteristic
5C3×Dic3⋊Dic3C212yesmaximal
6#63919—31nomaximal
7#63839—32nomaximal
8C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C2×C3×C3⋊Dic3C2212yes—
10C2×C3×C3⋊Dic3C2211yescharacteristic
11C2×C32×Dic3C2212yes—
12C2×C32×Dic3C2211yescharacteristic
13#6296—22no—
14C3×#627—22no—
15C2×C3×Dic6—32no—
16C2×#627—32no—
17C2×#627—31no—
18C2×#627D311yescharacteristic
19#3750—32no—
20C4×C3⋊Dic3—31no—
21C3×C4⋊Dic3—32no—
22C3×Dic3⋊C4—32no—
23#3724—32no—
24#3724—32no—
25#3724—62no—
26Dic3⋊Dic3D312yes—
27Dic32—32no—
28Dic32—61no—
29C22×C33C2311yescharacteristic
30C3⋊(C3⋊Dic3)Q812yescomplemented
31C3×C3⋊Dic3—22no—
32C3×C3⋊Dic3D412yes—
33C32×Dic3—22no—
34C32×Dic3Q812yescomplemented
35#944—91no—
36#888—92no—
37C2×C32×C4—31no—
38C2×C32×C4—32no—
39C2×C3⋊Dic3—32no—
40C2×C3⋊Dic3—31no—
41C2×C3⋊Dic3—61no—
42C2×C3⋊Dic3—62no—
43C2×C3⋊Dic3—121no—
44C2×C3⋊Dic3C2×D312yes—
45C2×C3⋊Dic3C2×D311yescharacteristic
46C2×C3×Dic3—21no—
47C2×C3×Dic3—22no—
48C2×C3×Dic3—32no—
49C2×C3×Dic3—32no—
50C2×C3×Dic3—32no—
51C2×C3×Dic3—61no—
52C2×C3×Dic3—62no—
53C2×C3×Dic3C2×D312yes—
54C2×C3×Dic3C2×D312yes—
55C2×C3×Dic3C2×D312yes—
56C3×Dic6—64no—
57#627—22no—
58#627—64no—
59#627—62no—
60C2×C33C2×D411yescharacteristic
61C2×C33C2×Q811yescharacteristic
62C2×C33C2×Q811yescharacteristic
63C2×C3×Q8—91no—
64C2×Dic6—32no—
65C2×Dic6—92no—
66C2×Dic6—91no—
67C3×C4⋊C4—92no—
68C4⋊Dic3—32no—
69Dic3⋊C4—32no—
70Dic3⋊C4—92no—
71Dic3⋊C4—92no—
72Dic3⋊C4—92no—
73Dic3⋊C4—182no—
74C4×Dic3—92no—
75C4×Dic3—91no—
76C4×Dic3—181no—
77C22×C32—22no—
78C22×C32—21no—
79C22×C32—41no—
80C22×C32C22×D312yes—
81C22×C32C22×D311yescharacteristic
82C32×C4—32no—
83C32×C4—62no—
84C3⋊Dic3—22no—
85C3⋊Dic3—34no—
86C3⋊Dic3—32no—
87C3⋊Dic3—62no—
88C3⋊Dic3—64no—
89C3⋊Dic3—122no—
90C3⋊Dic3#6712yes—
91C3×Dic3—22no—
92C3×Dic3—22no—
93C3×Dic3—22no—
94C3×Dic3—34no—
95C3×Dic3—42no—
96C3×Dic3—62no—
97C3×Dic3—62no—
98C3×Dic3—62no—
99C3×Dic3—122no—
100C3×Dic3Dic614yescomplemented
101C4⋊Q8—271noSylow
102C33C4⋊Q811yescharacteristic, complemented, Sylow
103C3×Q8—182no—
104C2×C3×C4—32no—
105C2×C3×C4—32no—
106C2×C3×C4—32no—
107C2×C3×C4—61no—
108C2×C3×C4—62no—
109C2×C3×C4—91no—
110C2×C3×C4—92no—
111C2×Dic3—32no—
112C2×Dic3—32no—
113C2×Dic3—32no—
114C2×Dic3—61no—
115C2×Dic3—62no—
116C2×Dic3—92no—
117C2×Dic3—91no—
118C2×Dic3—181no—
119C2×Dic3—182no—
120C2×Dic3—361no—
121C2×Dic3D3212yes—
122C2×Dic3D3211yescharacteristic
123Dic6—64no—
124Dic6—184no—
125Dic6—182no—
126C2×C32—22no—
127C2×C32—22no—
128C2×C32—21no—
129C2×C32—21no—
130C2×C32—21no—
131C2×C32—22no—
132C2×C32—41no—
133C2×C32—41no—
134C2×C32—41no—
135C2×C32C2×Dic612yes—
136C2×C32D3×D412yes—
137C2×C32D3×Q811yescharacteristic
138C2×C32D3×Q812yes—
139C2×C32D3×Q811yescharacteristic
140C2×C32C2×#6711yescharacteristic
141C2×Q8—91no—
142C2×Q8—271no—
143C4⋊C4—92no—
144C4⋊C4—272no—
145C42—271no—
146C22×C3—21no—
147C22×C3—22no—
148C22×C3—41no—
149C22×C3C2×D3212yes—
150C22×C3C2×D3211yescharacteristic
151C3×C4—34no—
152C3×C4—62no—
153C3×C4—62no—
154C3×C4—62no—
155C3×C4—92no—
156C3×C4—122no—
157C3×C4—182no—
158Dic3—22no—
159Dic3—34no—
160Dic3—62no—
161Dic3—62no—
162Dic3—62no—
163Dic3—94no—
164Dic3—92no—
165Dic3—122no—
166Dic3—182no—
167Dic3—184no—
168Dic3—362no—
169Dic3#62712yescomplemented
170C32—22no—
171C32—21no—
172C32—41no—
173C32#88812yescomplemented
174C32#94411yescharacteristic, complemented
175Q8—182no—
176Q8—542no—
177C2×C4—31no—
178C2×C4—32no—
179C2×C4—91no—
180C2×C4—92no—
181C2×C4—271no—
182C2×C3—21no—
183C2×C3—22no—
184C2×C3—21no—
185C2×C3—22no—
186C2×C3—22no—
187C2×C3—21no—
188C2×C3—41no—
189C2×C3—41no—
190C2×C3—41no—
191C2×C3D3×Dic612yes—
192C2×C3#379712yes—
193C2×C3#379711yescharacteristic
194C2×C3#380211yescharacteristic
195C2×C3C2×#62711yescharacteristic
196C2×C3D3×#6712yes—
197C22D3311yescharacteristic
198C4—32no—
199C4—62no—
200C4—92no—
201C4—182no—
202C4—272no—
203C3—22no—
204C3—21no—
205C3—41no—
206C3#6383912yescomplemented
207C3#6391911yescharacteristic, complemented
208C2#8895311yescharacteristic
209C2D3×#62711yescharacteristic
210C2#8895811yescharacteristic
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
30. C3⋊(C3⋊Dic3)Q8semidirect
34. C32×Dic3Q8semidirect
100. C3×Dic3Dic6semidirect
102. C33C4⋊Q8semidirect
169. Dic3#627semidirect
173. C32#888semidirect
174. C32#944semidirect
206. C3#63839semidirect
207. C3#63919semidirect

Maximal subgroups

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

subgroupindex
C2×#62962
C2×C3×#6272
#888062
Dic3×C3⋊Dic32
C3×Dic3⋊Dic32
#639193
#638393