Finite Group Databasebeta

← #10591344

Subgroups — #10591344

Order 576.

402 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C22×C3×Dic6C211yesmaximal, characteristic
2C2×C3×Dic3⋊C4C211yesmaximal, characteristic
3C2×#3724C212yesmaximal
4#63658C211yesmaximal, characteristic
5#63579C212yesmaximal
6#5230—31nomaximal
7#4979—31nomaximal
8C22×C32×C4C2211yescharacteristic
9C22×C3⋊Dic3C2212yes—
10C22×C3×Dic3C2211yescharacteristic
11C22×C3×Dic3C2212yes—
12C22×C3×Dic3C2211yescharacteristic
13C2×C3×Dic6—22no—
14C2×C3×Dic6—24no—
15C3×Dic3⋊C4—22no—
16#3724—24no—
17C22×C3×Q8—31no—
18C22×Dic6D311yescharacteristic
19C2×C3×C4⋊C4—31no—
20C2×Dic3⋊C4—32no—
21C2×Dic3⋊C4—32no—
22C2×Dic3⋊C4D311yescharacteristic
23C3⋊#95—31no—
24C3⋊#95—31no—
25C3⋊#95—32no—
26C3⋊#95—32no—
27C3⋊#95—61no—
28C23×C32C2311yescharacteristic
29C2×C32×C4—22no—
30C2×C32×C4D412yes—
31C2×C3⋊Dic3—22no—
32C2×C3⋊Dic3—22no—
33C2×C3⋊Dic3Q814yescomplemented
34C2×C3×Dic3—21no—
35C2×C3×Dic3—22no—
36C2×C3×Dic3—22no—
37C2×C3×Dic3—21no—
38C2×C3×Dic3—22no—
39C2×C3×Dic3—21no—
40C2×C3×Dic3D414yes—
41C2×C3×Dic3Q812yescomplemented
42C3×Dic6—28no—
43#395—91noSylow
44C2×C3×Q8—64no—
45C2×C3×Q8—62no—
46C22×C3×C4—21no—
47C22×C3×C4—31no—
48C22×C3×C4—32no—
49C22×C3×C4—31no—
50C22×C3×C4C2×D311yescharacteristic
51C22×C3×C4C2×D311yescharacteristic
52C22×Dic3—32no—
53C22×Dic3—32no—
54C22×Dic3—62no—
55C22×Dic3C2×D311yescharacteristic
56C22×Dic3C2×D311yescharacteristic
57C22×Dic3C2×D312yes—
58C2×Dic6—22no—
59C2×Dic6—24no—
60C3×C4⋊C4—62no—
61Dic3⋊C4—22no—
62Dic3⋊C4—64no—
63Dic3⋊C4—64no—
64C22×C32C2×D412yes—
65C22×C32C2×D411yescharacteristic
66C22×C32C2×Q811yescharacteristic
67C22×C32C2×Q812yes—
68C22×C32#4111yescharacteristic
69C32×C4—22no—
70C3⋊Dic3—24no—
71C3×Dic3—24no—
72C3×Dic3—22no—
73C3×Dic3—41no—
74C22×Q8—31no—
75C2×C4⋊C4—31no—
76C2×C4⋊C4—92no—
77#95—92no—
78#95—91no—
79C23×C3—21no—
80C23×C3C22×D311yescharacteristic
81C23×C3C22×D311yescharacteristic
82C3×Q8—68no—
83C2×C3×C4—22no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—34no—
87C2×C3×C4—32no—
88C2×C3×C4—42no—
89C2×C3×C4—62no—
90C2×C3×C4—61no—
91C2×C3×C4—61no—
92C2×C3×C4—62no—
93C2×C3×C4—62no—
94C2×C3×C4—61no—
95C2×C3×C4#6712yes—
96C2×C3×C4#6712yes—
97C2×Dic3—21no—
98C2×Dic3—21no—
99C2×Dic3—22no—
100C2×Dic3—22no—
101C2×Dic3—22no—
102C2×Dic3—21no—
103C2×Dic3—34no—
104C2×Dic3—34no—
105C2×Dic3—64no—
106C2×Dic3—62no—
107C2×Dic3—62no—
108C2×Dic3—62no—
109C2×Dic3—62no—
110C2×Dic3—122no—
111C2×Dic3—122no—
112C2×Dic3Dic612yescomplemented
113C2×Dic3#6714yes—
114Dic6—28no—
115C2×C32#12011yescharacteristic
116C2×C32C22⋊Q812yes—
117C2×C32C22⋊Q811yescharacteristic
118C2×C32C4⋊Q812yes—
119C2×C32C4⋊Q811yescharacteristic
120C2×Q8—64no—
121C2×Q8—62no—
122C22×C4—31no—
123C22×C4—32no—
124C22×C4—31no—
125C22×C4—92no—
126C22×C4D3211yescharacteristic
127C4⋊C4—62no—
128C4⋊C4—184no—
129C22×C3—21no—
130C22×C3—22no—
131C22×C3—21no—
132C22×C3—21no—
133C22×C3—22no—
134C22×C3C2×Dic611yescharacteristic
135C22×C3#23511yescharacteristic
136C22×C3D3×D412yes—
137C22×C3D3×Q811yescharacteristic
138C22×C3D3×Q812yes—
139C22×C3D3×Q812yes—
140C22×C3#23911yescharacteristic
141C22×C3C2×#6711yescharacteristic
142C22×C3C2×#6711yescharacteristic
143C22×C3C2×#6712yes—
144C3×C4—22no—
145C3×C4—22no—
146C3×C4—42no—
147C3×C4—64no—
148C3×C4—62no—
149C3×C4—121no—
150Dic3—22no—
151Dic3—24no—
152Dic3—41no—
153Dic3—64no—
154Dic3—64no—
155Dic3—124no—
156C32#39511yescharacteristic, complemented, Sylow
157C23C2×D3211yescharacteristic
158Q8—68no—
159C2×C4—22no—
160C2×C4—32no—
161C2×C4—34no—
162C2×C4—61no—
163C2×C4—61no—
164C2×C4—62no—
165C2×C4—62no—
166C2×C4—61no—
167C2×C4—62no—
168C2×C4—94no—
169C2×C4—182no—
170C2×C4—182no—
171C2×C4#62512yes—
172C2×C3—22no—
173C2×C3—21no—
174C2×C3—21no—
175C2×C3—22no—
176C2×C3—21no—
177C2×C3#88812yes—
178C2×C3#89612yes—
179C2×C3#92411yescharacteristic
180C2×C3#93711yescharacteristic
181C2×C3#94412yes—
182C2×C3#94411yescharacteristic
183C2×C3#94411yescharacteristic
184C2×C3#94611yescharacteristic
185C2×C3#94612yes—
186C2×C3#95311yescharacteristic
187C22D3×Dic611yescharacteristic
188C22#379712yes—
189C22#379911yescharacteristic
190C22C2×#62511yescharacteristic
191C22D3×#6712yes—
192C4—22no—
193C4—62no—
194C4—64no—
195C4—121no—
196C4—184no—
197C3—21no—
198C3#497911yescharacteristic, complemented
199C3#523011yescharacteristic, complemented
200C2#6383912yes—
201C2#6385611yescharacteristic
202C2#6387311yescharacteristic
203C2#6388812yes—
204C2#6397611yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, prime radical, cosocleC23×C32C23
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#395
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
33. C2×C3⋊Dic3Q8semidirect
41. C2×C3×Dic3Q8semidirect
112. C2×Dic3Dic6semidirect
156. C32#395semidirect
198. C3#4979semidirect
199. C3#5230semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic62
C2×C3×Dic3⋊C42
C2×#37242
#636582
#635792
#52303
#49793