Finite Group Databasebeta

← #10690142

Subgroups — #10690142

Order 768.

376 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C3×#3162C211yesmaximal, characteristic
2#83333C211yesmaximal, characteristic
3#71829C211yesmaximal, characteristic
4#71827C212yesmaximal
5#67662C212yesmaximal
6#9006—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C3×#533C2211yescharacteristic
9C3×#501—22no—
10C3×#501C414yes—
11C3×#427—22no—
12C2×C3×#97C2211yescharacteristic
13#5024—24no—
14#5000—22no—
15#4926C2212yes—
16#4925C2212yes—
17#4547—22no—
18#4535—24no—
19#3162D311yescharacteristic
20#2987—31no—
21#1544—31no—
22#1497—32no—
23#1302—32no—
24C22×C3×C8—21no—
25C3×C4⋊Q8C2×C412yes—
26C3×C4⋊Q8D412yes—
27C3×#125—21no—
28C3×#125C2×C412yes—
29C3×C22⋊Q8—22no—
30C3×C22⋊Q8—22no—
31C3×C4×Q8—22no—
32C3×C4×Q8—22no—
33C3×#117—21no—
34C3×#117D412yes—
35C2×C3×C42C2311yescharacteristic
36C3×#107—24no—
37C3×#106—24no—
38C3×#97—21no—
39C3×#97C2×C412yes—
40#830—22no—
41#830—22no—
42C4⋊(C3⋊C8)—24no—
43C4⋊(C3⋊C8)—22no—
44C4⋊(C3⋊C8)D414yes—
45#803—22no—
46#803—22no—
47C4×C3⋊C8—22no—
48C4×C3⋊C8—22no—
49#533C2×D311yescharacteristic
50#533C2×D311yescharacteristic
51#501—22no—
52#501Dic314yes—
53#446—64no—
54#433—62no—
55#432—32no—
56#431—32no—
57#427—22no—
58C2×#97C2×D311yescharacteristic
59#332—62no—
60Q8⋊C8—64no—
61C2×C3×Q8—22no—
62C2×C3×Q8—22no—
63C22×C3×C4C2×D411yescharacteristic
64C22×C3×C4#4112yes—
65C2×C3×C8—24no—
66C2×C3×C8—22no—
67C3×C4⋊C4—24no—
68C3×C4⋊C4—22no—
69C3×C4⋊C4—22no—
70C3×C4⋊C4—22no—
71C3×C4⋊C4—42no—
72C3×C4⋊C4C22⋊C414yes—
73C3×C22⋊C4—22no—
74C3×C22⋊C4—41no—
75C3×C42—22no—
76C3×C42—41no—
77C3×C42C22×C411yescharacteristic
78C3×C42C2×D412yes—
79C3×C42C2×D411yescharacteristic
80C2×C3⋊C8—24no—
81C2×C3⋊C8—42no—
82C22×C8—21no—
83C4⋊Q8C2×Dic312yes—
84C4⋊Q8#6712yes—
85#125—21no—
86#125C2×Dic312yes—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C4×Q8—22no—
90C4×Q8—22no—
91#117—21no—
92#117#6712yes—
93C2×C42C22×D311yescharacteristic
94#107—24no—
95#106—24no—
96C4⋊C8—34no—
97C4⋊C8—64no—
98C4⋊C8—62no—
99C22⋊C8—62no—
100C22⋊C8—62no—
101#97—21no—
102#97—62no—
103#97—62no—
104#97C2×Dic312yes—
105C4×C8—62no—
106C4×C8—62no—
107C23×C3#12311yescharacteristic
108C3×Q8—44no—
109C3×Q8—82no—
110C2×C3×C4—22no—
111C2×C3×C4—24no—
112C2×C3×C4—42no—
113C2×C3×C4—42no—
114C2×C3×C4C2×C22⋊C411yescharacteristic
115C2×C3×C4C4×D412yes—
116C2×C3×C4#12011yescharacteristic
117C2×C3×C4#12112yes—
118C3×C8—24no—
119C3⋊C8—44no—
120C3⋊C8—82no—
121C2×Q8—22no—
122C2×Q8—22no—
123C22×C4#23712yes—
124C22×C4C2×#6711yescharacteristic
125C2×C8—24no—
126C2×C8—22no—
127C2×C8—64no—
128C2×C8—122no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—42no—
134C4⋊C4#21714yes—
135C22⋊C4—22no—
136C22⋊C4—41no—
137C42—22no—
138C42—41no—
139C42D3×D412yes—
140C42C22×Dic311yescharacteristic
141C42C2×#6711yescharacteristic
142C22×C3—22no—
143C22×C3—21no—
144C22×C3#38611yescharacteristic
145C3×C4—22no—
146C3×C4—22no—
147C3×C4—21no—
148C3×C4—44no—
149C3×C4—82no—
150C3×C4#45612yes—
151C23#93511yescharacteristic
152Q8—44no—
153Q8—82no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4D4×Dic312yes—
159C2×C4#93711yescharacteristic
160C2×C4#93912yes—
161C2×C4C2×#21711yescharacteristic
162C8—24no—
163C8—124no—
164C8—242no—
165C2×C3—41no—
166C2×C3#188112yes—
167C2×C3#188611yescharacteristic
168C22—21no—
169C22—22no—
170C22#522011yescharacteristic
171C4—21no—
172C4—22no—
173C4—22no—
174C4—44no—
175C4—82no—
176C4#509512yes—
177C3#900611yescharacteristic, complemented, Sylow
178C2—41no—
179C2#7178411yescharacteristic
180C2#7211612yes—
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3162C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#3162D3
hypocenterC3#9006
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
177. C3#9006semidirect

Maximal subgroups

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

subgroupindex
C3×#31622
#833332
#718292
#718272
#676622
#90063