Finite Group Databasebeta

← #11723594

Subgroups — #11723594

Order 864.

592 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 176 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×C32×Dic3C211yesmaximal, characteristic
2#88864C211yesmaximal, characteristic
3C3×#3757C211yesmaximal, characteristic
4Dic3⋊(C3⋊Dic3)C212yesmaximal, complemented
5Dic3×C3⋊Dic3C212yesmaximal, complemented
6#64135—31nomaximal
7#63955—34nomaximal
8C23×C33C2211yescharacteristic
9C2×C3⋊(C3⋊Dic3)C2212yes—
10C2×C3×C3⋊Dic3C2212yes—
11C2×C32×Dic3C414yescomplemented
12C2×C32×Dic3C2211yescharacteristic
13C2×C32×Dic3C2211yescharacteristic
14C22×C32×C4—31no—
15C22×C3×Dic3D314yes—
16#3757—31no—
17#3757—34no—
18#3757—64no—
19#3757D311yescharacteristic
20C4⋊(C3⋊Dic3)—32no—
21C4×C3⋊Dic3—32no—
22C3×#217—34no—
23Dic3⋊Dic3—38no—
24Dic32—38no—
25C22×C33—22no—
26C22×C33C2×C412yes—
27C22×C33C2311yescharacteristic
28C3⋊(C3⋊Dic3)—22no—
29C3×C3⋊Dic3—22no—
30C32×Dic3C2×C412yes—
31C32×Dic3C2×C412yes—
32#926—94no—
33#877—91no—
34C23×C32—24no—
35C23×C32C2×D314yes—
36C23×C32C2×D311yescharacteristic
37C2×C32×C4—34no—
38C2×C32×C4—31no—
39C2×C32×C4—31no—
40C2×C3⋊Dic3—32no—
41C2×C3⋊Dic3—38no—
42C2×C3⋊Dic3—68no—
43C2×C3⋊Dic3C2×D312yes—
44C2×C3×Dic3—38no—
45C2×C3×Dic3Dic3116yescomplemented
46C2×C3×Dic3C2×D314yes—
47C2×C3×Dic3C2×D314yes—
48C2×C33—22no—
49C2×C33C22×C411yescharacteristic
50C2×C33#4112yes—
51C22×C3×C4—34no—
52C22×Dic3C3⋊D311yescharacteristic
53C3×C22⋊C4—91no—
54#217—34no—
55#217—94no—
56#217—91no—
57#217—184no—
58C4⋊Dic3—98no—
59Dic3⋊C4—92no—
60C4×Dic3—98no—
61C4×Dic3—92no—
62C22×C32—22no—
63C22×C32—28no—
64C22×C32—24no—
65C22×C32—28no—
66C22×C32—48no—
67C22×C32C4×D312yes—
68C22×C32C2×Dic318yes—
69C22×C32C22×D314yes—
70C22×C32C22×D311yescharacteristic
71C32×C4—32no—
72C32×C4—32no—
73C3⋊Dic3—22no—
74C3⋊Dic3—62no—
75C3⋊Dic3—68no—
76C3⋊Dic3—128no—
77C3×Dic3—68no—
78C3×Dic3C2×Dic318yes—
79C3×Dic3C2×Dic318yes—
80#117—271noSylow
81C33#11711yescharacteristic, complemented, Sylow
82C23×C3—24no—
83C23×C3D3214yes—
84C23×C3C2×C3⋊D311yescharacteristic
85C2×C3×C4—316no—
86C2×C3×C4—34no—
87C2×C3×C4—34no—
88C2×C3×C4—92no—
89C2×Dic3—38no—
90C2×Dic3—98no—
91C2×Dic3—92no—
92C2×Dic3—188no—
93C2×Dic3C3⋊Dic314yescomplemented
94C2×Dic3C2×C3⋊D311yescharacteristic
95C2×Dic3C2×C3⋊D311yescharacteristic
96C2×C32—22no—
97C2×C32—24no—
98C2×C32—28no—
99C2×C32—28no—
100C2×C32—48no—
101C2×C32C2×C4×D311yescharacteristic
102C2×C32#23518yes—
103C2×C32#23712yes—
104C2×C32C22×Dic314yes—
105C22×C4—31no—
106C4⋊C4—272no—
107C22⋊C4—91no—
108C22⋊C4—271no—
109C42—272no—
110C22×C3—28no—
111C22×C3—22no—
112C22×C3—28no—
113C22×C3—24no—
114C22×C3—48no—
115C22×C3D3×Dic318yes—
116C22×C3C2×C3⋊Dic312yes—
117C22×C3C2×D3214yes—
118C22×C3C22×C3⋊D311yescharacteristic
119C3×C4—38no—
120C3×C4—38no—
121C3×C4—182no—
122Dic3—68no—
123Dic3—188no—
124Dic3—182no—
125Dic3—368no—
126Dic3C2×C3⋊Dic312yes—
127Dic3C2×C3⋊Dic312yes—
128C32—24no—
129C32#87711yescharacteristic, complemented
130C32#92614yescomplemented
131C23D3×C3⋊D311yescharacteristic
132C2×C4—34no—
133C2×C4—31no—
134C2×C4—31no—
135C2×C4—92no—
136C2×C4—272no—
137C2×C3—22no—
138C2×C3—28no—
139C2×C3—24no—
140C2×C3—28no—
141C2×C3—48no—
142C2×C3C2×D3×Dic314yes—
143C2×C3#380418yes—
144C2×C3#382812yes—
145C2×C3C22×C3⋊Dic311yescharacteristic
146C22—22no—
147C22D3×C3⋊Dic312yes—
148C22C2×D3×C3⋊D311yescharacteristic
149C4—32no—
150C4—32no—
151C4—182no—
152C4—542no—
153C3—24no—
154C3#6395514yescomplemented
155C3#6413511yescharacteristic, complemented
156C2—22no—
157C2C2×D3×C3⋊Dic311yescharacteristic
158C2#8903112yes—
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutatorC2×C33C22×C4
Fitting, prime radicalC23×C33C22
socle, cosocleC22×C33C23
hypercenterC23D3×C3⋊D3
hypocenterC33#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. Dic3⋊(C3⋊Dic3)C2semidirect
5. Dic3×C3⋊Dic3C2semidirect
11. C2×C32×Dic3C4semidirect
45. C2×C3×Dic3Dic3semidirect
81. C33#117semidirect
93. C2×Dic3C3⋊Dic3semidirect
129. C32#877semidirect
130. C32#926semidirect
154. C3#63955semidirect
155. C3#64135semidirect

Maximal subgroups

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

subgroupindex
C22×C32×Dic32
#888642
C3×#37572
Dic3⋊(C3⋊Dic3)2
Dic3×C3⋊Dic32
#641353
#639553