Finite Group Databasebeta

← #10690139

Subgroups — #10690139

Order 768.

376 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#83340C211yesmaximal, characteristic
2#83333C211yesmaximal, characteristic
3C3×#1544C211yesmaximal, characteristic
4#70771C212yesmaximal
5#67742C212yesmaximal
6#9006—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8#5470C2211yescharacteristic
9C3×#431C2212yes—
10#5081—22no—
11#5064C414yes—
12#5024—24no—
13#5000—22no—
14#4998—22no—
15#4926C2212yes—
16C2×#803C2211yescharacteristic
17C3×#332—22no—
18#4456—24no—
19#3162—31no—
20#2987—31no—
21#1544D311yescharacteristic
22#1497—32no—
23#1302—32no—
24C3×C4⋊Q8D412yes—
25C3×#125C2×C412yes—
26C3×C22⋊Q8—22no—
27C3×C4×Q8—22no—
28C3×#117D412yes—
29C2×C3×C42C2311yescharacteristic
30#926—21no—
31#924—22no—
32C22×C3⋊C8—21no—
33#870—21no—
34#869C2×C412yes—
35C4×Dic6—22no—
36C3×C4⋊C8—22no—
37C3×C22⋊C8—22no—
38C3×C4×C8—22no—
39#830—22no—
40#808—24no—
41#807—24no—
42C4⋊(C3⋊C8)—24no—
43C4⋊(C3⋊C8)D414yes—
44#803—22no—
45#803—21no—
46#803—22no—
47#803C2×C412yes—
48C4×C3⋊C8—22no—
49#533—31no—
50#533C2×D311yescharacteristic
51#501—34no—
52#501—62no—
53#446—64no—
54#433—62no—
55#432—32no—
56#431C2×D312yes—
57#427—62no—
58C2×#97—31no—
59#332—22no—
60Q8⋊C8—64no—
61C2×C3×Q8—22no—
62C22×C3×C4C2×D411yescharacteristic
63C22×C3×C4#4112yes—
64C2×Dic6—22no—
65C2×C3×C8—42no—
66C3×C4⋊C4—24no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4C22⋊C414yes—
69C3×C22⋊C4—22no—
70C3×C42—22no—
71C3×C42C22×C411yescharacteristic
72C3×C42C2×D412yes—
73C3×C42C2×D411yescharacteristic
74#217—41no—
75C4⋊Dic3—22no—
76C4⋊Dic3—22no—
77Dic3⋊C4—42no—
78C4×Dic3—41no—
79C2×C3⋊C8—24no—
80C2×C3⋊C8—22no—
81C2×C3⋊C8—24no—
82C22×C8—61no—
83C4⋊Q8—32no—
84C4⋊Q8#6712yes—
85#125—61no—
86#125C4×D312yes—
87C22⋊Q8—22no—
88C22⋊Q8—62no—
89C4×Q8—22no—
90C4×Q8—62no—
91#117—61no—
92#117D1212yes—
93C2×C42C22×D311yescharacteristic
94#107—64no—
95#106—64no—
96C4⋊C8—22no—
97C4⋊C8—34no—
98C4⋊C8—64no—
99C22⋊C8—22no—
100C22⋊C8—62no—
101#97—32no—
102#97—62no—
103#97—62no—
104#97—61no—
105C4×C8—22no—
106C4×C8—62no—
107C23×C3#12311yescharacteristic
108C3×Q8—82no—
109C2×C3×C4—22no—
110C2×C3×C4—24no—
111C2×C3×C4—42no—
112C2×C3×C4C2×C22⋊C411yescharacteristic
113C2×C3×C4C4×D412yes—
114C2×C3×C4#12011yescharacteristic
115C2×C3×C4#12112yes—
116C2×Dic3—42no—
117Dic6—44no—
118C3×C8—82no—
119C3⋊C8—24no—
120C3⋊C8—44no—
121C2×Q8—22no—
122C2×Q8—62no—
123C22×C4C2×D1211yescharacteristic
124C22×C4#23712yes—
125C2×C8—42no—
126C2×C8—64no—
127C2×C8—62no—
128C2×C8—64no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—62no—
132C4⋊C4—62no—
133C4⋊C4—122no—
134C4⋊C4(C2×D3)⋊C414yes—
135C22⋊C4—22no—
136C22⋊C4—121no—
137C42—22no—
138C42—121no—
139C42C2×C4×D311yescharacteristic
140C42D3×D412yes—
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#45612yes—
150Dic3—82no—
151C23#88611yescharacteristic
152Q8—82no—
153Q8—124no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—42no—
157C2×C4—122no—
158C2×C4#67⋊C412yes—
159C2×C4#88211yescharacteristic
160C2×C4C2×(C2×D3)⋊C411yescharacteristic
161C2×C4#93912yes—
162C8—64no—
163C8—82no—
164C8—124no—
165C2×C3—41no—
166C2×C3#188112yes—
167C2×C3#188611yescharacteristic
168C22—22no—
169C22—21no—
170C22#495811yescharacteristic
171C4—22no—
172C4—22no—
173C4—21no—
174C4—44no—
175C4—242no—
176C4#475212yes—
177C3#900611yescharacteristic, complemented, Sylow
178C2—41no—
179C2#7010311yescharacteristic
180C2#7029312yes—
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1544C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#1544D3
hypocenterC3#9006
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1544

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
#833402
#833332
C3×#15442
#707712
#677422
#90063