Finite Group Databasebeta

← #10692710

Subgroups — #10692710

Order 768.

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

normal subgroups only

This group has 384 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×#3139C211yesmaximal, characteristic
2#83333C211yesmaximal, characteristic
3#71823C211yesmaximal, characteristic
4#71776C212yesmaximal
5#67670C212yesmaximal
6#9422—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×C3×C4⋊Q8C2211yescharacteristic
9C3×#491—22no—
10C3×#487C414yes—
11C3×#419—22no—
12C3×C4×M16C2211yescharacteristic
13#5024—24no—
14#5000—22no—
15#4926C2212yes—
16#4926C2212yes—
17#4547—24no—
18#4543—22no—
19#3139D311yescharacteristic
20#2987—31no—
21#1556—32no—
22#1537—31no—
23#1291—32no—
24C22×C3×Q8—21no—
25C2×C3×M16—21no—
26C3×C4⋊Q8—22no—
27C3×C4⋊Q8—21no—
28C3×C4⋊Q8C2×C412yes—
29C3×C4⋊Q8C2×C412yes—
30C3×#125D412yes—
31C3×C22⋊Q8—22no—
32C3×C4×Q8—22no—
33C3×#117D412yes—
34C2×C3×C4⋊C4—22no—
35C2×C3×C42C2311yescharacteristic
36C3×#103—28no—
37C3×#97—21no—
38C3×C4×C8C2×C412yes—
39#830—22no—
40#830—22no—
41C4⋊(C3⋊C8)—24no—
42C4⋊(C3⋊C8)—22no—
43C4⋊(C3⋊C8)D414yes—
44#803—22no—
45#803—22no—
46#803—22no—
47C4×C3⋊C8—22no—
48#533C2×D311yescharacteristic
49C2×C4⋊Q8C2×D311yescharacteristic
50#491—22no—
51#487Dic314yes—
52#446—64no—
53#433—62no—
54#432—32no—
55#432—32no—
56#419—22no—
57C4×M16C2×D311yescharacteristic
58#332—64no—
59#330—62no—
60C2×C3×Q8—22no—
61C2×C3×Q8—22no—
62C2×C3×Q8—44no—
63C2×C3×Q8—41no—
64C22×C3×C4—41no—
65C22×C3×C4C2×D411yescharacteristic
66C22×C3×C4#4112yes—
67C3×M16—42no—
68C2×C3×C8—22no—
69C3×C4⋊C4—22no—
70C3×C4⋊C4—24no—
71C3×C4⋊C4—24no—
72C3×C4⋊C4—42no—
73C3×C4⋊C4C22⋊C414yes—
74C3×C22⋊C4—22no—
75C3×C42—22no—
76C3×C42C22×C411yescharacteristic
77C3×C42C2×D412yes—
78C3×C42C2×D411yescharacteristic
79C2×C3⋊C8—24no—
80C2×C3⋊C8—42no—
81C22×Q8—21no—
82C2×M16—21no—
83C4⋊Q8—22no—
84C4⋊Q8—21no—
85C4⋊Q8C2×Dic312yes—
86C4⋊Q8C2×Dic312yes—
87#125#6712yes—
88C22⋊Q8—22no—
89C4×Q8—22no—
90#117#6712yes—
91C2×C4⋊C4—22no—
92C2×C42C22×D311yescharacteristic
93C4⋊C8—34no—
94C4⋊C8—64no—
95C4⋊C8—62no—
96#103—28no—
97C22⋊C8—62no—
98C22⋊C8—62no—
99#97—21no—
100#97—62no—
101#97—62no—
102#97—62no—
103C4×C8—62no—
104C4×C8C2×Dic312yes—
105C23×C3#12311yescharacteristic
106C3×Q8—44no—
107C3×Q8—82no—
108C3×Q8—82no—
109C2×C3×C4—22no—
110C2×C3×C4—24no—
111C2×C3×C4—42no—
112C2×C3×C4—42no—
113C2×C3×C4—82no—
114C2×C3×C4C2×C22⋊C411yescharacteristic
115C2×C3×C4C4×D412yes—
116C2×C3×C4#12011yescharacteristic
117C2×C3×C4#12112yes—
118C3×C8—42no—
119C3⋊C8—44no—
120C3⋊C8—82no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—44no—
124C2×Q8—41no—
125C22×C4—41no—
126C22×C4#23712yes—
127C22×C4C2×#6711yescharacteristic
128M16—42no—
129C2×C8—22no—
130C2×C8—64no—
131C2×C8—122no—
132C4⋊C4—22no—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—42no—
136C4⋊C4#21714yes—
137C22⋊C4—22no—
138C42—22no—
139C42D3×D412yes—
140C42C22×Dic311yescharacteristic
141C42C2×#6711yescharacteristic
142C22×C3—21no—
143C22×C3—22no—
144C22×C3#38611yescharacteristic
145C3×C4—21no—
146C3×C4—22no—
147C3×C4—22no—
148C3×C4—44no—
149C3×C4—82no—
150C3×C4#45412yes—
151C23#93511yescharacteristic
152Q8—44no—
153Q8—82no—
154Q8—82no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—82no—
160C2×C4D4×Dic312yes—
161C2×C4#93711yescharacteristic
162C2×C4#93912yes—
163C2×C4C2×#21711yescharacteristic
164C8—42no—
165C8—124no—
166C8—242no—
167C2×C3—41no—
168C2×C3#188811yescharacteristic
169C2×C3#190212yes—
170C22—21no—
171C22—22no—
172C22#522011yescharacteristic
173C4—21no—
174C4—22no—
175C4—22no—
176C4—44no—
177C4—82no—
178C4#506212yes—
179C3#942211yescharacteristic, complemented, Sylow
180C2—41no—
181C2#7172411yescharacteristic
182C2#7210212yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3139C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#3139D3
hypocenterC3#9422
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
179. C3#9422semidirect

Maximal subgroups

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

subgroupindex
C3×#31392
#833332
#718232
#717762
#676702
#94223