Finite Group Databasebeta

← #11723032

Subgroups — #11723032

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×C3×#627C212yesmaximal
2#88815C211yesmaximal, characteristic
3#88806C212yesmaximal
4Dic3×C3⋊Dic3C211yesmaximal, characteristic
5C3×#3724C211yesmaximal, characteristic
6#63873—31nomaximal
7#63839—32nomaximal
8C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C2×C3×C3⋊Dic3C2212yes—
10C2×C3×C3⋊Dic3C2211yescharacteristic
11C2×C32×Dic3C2212yes—
12C2×C32×Dic3C2211yescharacteristic
13C3×#627—24no—
14C2×C3×Dic6—32no—
15C2×C3×Dic6—32no—
16C2×#627D312yes—
17#3750—32no—
18C4×C3⋊Dic3—31no—
19C3×Dic3⋊C4—32no—
20#3724—31no—
21#3724—32no—
22#3724—32no—
23#3724—62no—
24#3724D311yescharacteristic
25Dic3⋊Dic3—32no—
26Dic32—32no—
27Dic32—61no—
28C22×C33C2311yescharacteristic
29C3⋊(C3⋊Dic3)Q812yescomplemented
30C3×C3⋊Dic3—22no—
31C3×C3⋊Dic3Q812yescomplemented
32C32×Dic3—22no—
33C32×Dic3D412yes—
34#944—92no—
35#888—91no—
36C2×C32×C4—31no—
37C2×C32×C4—32no—
38C2×C3⋊Dic3—32no—
39C2×C3⋊Dic3—31no—
40C2×C3⋊Dic3—61no—
41C2×C3⋊Dic3—62no—
42C2×C3⋊Dic3—121no—
43C2×C3⋊Dic3C2×D312yes—
44C2×C3⋊Dic3C2×D311yescharacteristic
45C2×C3×Dic3—21no—
46C2×C3×Dic3—22no—
47C2×C3×Dic3—32no—
48C2×C3×Dic3—32no—
49C2×C3×Dic3—32no—
50C2×C3×Dic3—61no—
51C2×C3×Dic3—62no—
52C2×C3×Dic3C2×D312yes—
53C2×C3×Dic3C2×D312yes—
54C2×C3×Dic3C2×D312yes—
55C3×Dic6—64no—
56C3×Dic6—64no—
57#627—24no—
58C2×C33C2×D411yescharacteristic
59C2×C33C2×Q811yescharacteristic
60C2×C33C2×Q811yescharacteristic
61C2×C3×Q8—92no—
62C2×Dic6—32no—
63C2×Dic6—32no—
64C3×C4⋊C4—91no—
65C4⋊Dic3—91no—
66Dic3⋊C4—32no—
67Dic3⋊C4—92no—
68Dic3⋊C4—92no—
69Dic3⋊C4—92no—
70Dic3⋊C4—92no—
71Dic3⋊C4—182no—
72C4×Dic3—92no—
73C4×Dic3—91no—
74C4×Dic3—181no—
75C22×C32—22no—
76C22×C32—21no—
77C22×C32—41no—
78C22×C32C22×D311yescharacteristic
79C22×C32C22×D312yes—
80C32×C4—32no—
81C32×C4—62no—
82C3⋊Dic3—22no—
83C3⋊Dic3—32no—
84C3⋊Dic3—34no—
85C3⋊Dic3—62no—
86C3⋊Dic3—64no—
87C3⋊Dic3—122no—
88C3⋊Dic3Dic612yescomplemented
89C3×Dic3—22no—
90C3×Dic3—22no—
91C3×Dic3—22no—
92C3×Dic3—34no—
93C3×Dic3—42no—
94C3×Dic3—62no—
95C3×Dic3—62no—
96C3×Dic3—62no—
97C3×Dic3—122no—
98C3×Dic3#6714yes—
99C4⋊Q8—271noSylow
100C33C4⋊Q811yescharacteristic, complemented, Sylow
101C3×Q8—184no—
102C2×C3×C4—32no—
103C2×C3×C4—32no—
104C2×C3×C4—32no—
105C2×C3×C4—61no—
106C2×C3×C4—62no—
107C2×C3×C4—91no—
108C2×C3×C4—92no—
109C2×Dic3—32no—
110C2×Dic3—32no—
111C2×Dic3—32no—
112C2×Dic3—61no—
113C2×Dic3—62no—
114C2×Dic3—92no—
115C2×Dic3—91no—
116C2×Dic3—181no—
117C2×Dic3—182no—
118C2×Dic3—361no—
119C2×Dic3D3212yes—
120C2×Dic3D3211yescharacteristic
121Dic6—64no—
122Dic6—64no—
123C2×C32—22no—
124C2×C32—21no—
125C2×C32—22no—
126C2×C32—21no—
127C2×C32—21no—
128C2×C32—22no—
129C2×C32—41no—
130C2×C32—41no—
131C2×C32—41no—
132C2×C32C2×Dic611yescharacteristic
133C2×C32D3×D411yescharacteristic
134C2×C32D3×Q812yes—
135C2×C32D3×Q811yescharacteristic
136C2×C32D3×Q812yes—
137C2×C32C2×#6712yes—
138C2×Q8—92no—
139C4⋊C4—91no—
140C4⋊C4—272no—
141C4⋊C4—271no—
142C42—271no—
143C22×C3—22no—
144C22×C3—21no—
145C22×C3—41no—
146C22×C3C2×D3212yes—
147C22×C3C2×D3211yescharacteristic
148C3×C4—34no—
149C3×C4—62no—
150C3×C4—62no—
151C3×C4—62no—
152C3×C4—92no—
153C3×C4—122no—
154C3×C4—182no—
155Dic3—22no—
156Dic3—34no—
157Dic3—62no—
158Dic3—62no—
159Dic3—62no—
160Dic3—94no—
161Dic3—92no—
162Dic3—122no—
163Dic3—182no—
164Dic3—184no—
165Dic3—362no—
166Dic3#62512yes—
167C32—22no—
168C32—21no—
169C32—41no—
170C32#88811yescharacteristic, complemented
171C32#94412yescomplemented
172Q8—184no—
173C2×C4—31no—
174C2×C4—32no—
175C2×C4—91no—
176C2×C4—92no—
177C2×C4—271no—
178C2×C3—22no—
179C2×C3—21no—
180C2×C3—21no—
181C2×C3—22no—
182C2×C3—21no—
183C2×C3—22no—
184C2×C3—41no—
185C2×C3—41no—
186C2×C3—41no—
187C2×C3D3×Dic612yes—
188C2×C3#379711yescharacteristic
189C2×C3#379712yes—
190C2×C3#379711yescharacteristic
191C2×C3C2×#62511yescharacteristic
192C2×C3D3×#6712yes—
193C22D3311yescharacteristic
194C4—32no—
195C4—62no—
196C4—92no—
197C4—182no—
198C4—272no—
199C3—21no—
200C3—22no—
201C3—41no—
202C3#6383912yescomplemented
203C3#6387311yescharacteristic, complemented
204C2D3×#62511yescharacteristic
205C2#8895711yescharacteristic
206C2#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
29. C3⋊(C3⋊Dic3)Q8semidirect
31. C3×C3⋊Dic3Q8semidirect
88. C3⋊Dic3Dic6semidirect
100. C33C4⋊Q8semidirect
170. C32#888semidirect
171. C32#944semidirect
202. C3#63839semidirect
203. C3#63873semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#6272
#888152
#888062
Dic3×C3⋊Dic32
C3×#37242
#638733
#638393