Finite Group Databasebeta

← #10692730

Subgroups — #10692730

Order 768.

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

normal subgroups only

This group has 422 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#83231C211yesmaximal, characteristic
3#71837C212yesmaximal
4#71777C211yesmaximal, characteristic
5#67666C212yesmaximal
6#9426—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×C3×C4⋊Q8C2211yescharacteristic
9C3×#491—22no—
10C3×#487C414yescomplemented
11C3×#419—22no—
12C3×C4×M16C2211yescharacteristic
13#5229—22no—
14#4980—24no—
15#4937C2212yes—
16#4926C2212yes—
17#4546—22no—
18#4538—24no—
19#3139D311yescharacteristic
20#2381—31no—
21#1558—31no—
22#1515—32no—
23#1288—32no—
24C22×C3×Q8D412yes—
25C2×C3×M16—21no—
26C3×C4⋊Q8—22no—
27C3×C4⋊Q8C2×C412yes—
28C3×C4⋊Q8C2×C412yes—
29C3×C4⋊Q8D412yes—
30C3×#125—21no—
31C3×C22⋊Q8—22no—
32C3×C4×Q8—22no—
33C3×#117—21no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C42C2311yescharacteristic
36C2×C4⋊Dic3—22no—
37C2×C4×Dic3—22no—
38C3×#103—28no—
39C3×#97—21no—
40C3×C4×C8C2×C412yes—
41C3⋊#95—24no—
42C3⋊#95—22no—
43C3⋊#95—22no—
44C3⋊#95D414yes—
45#830—22no—
46C4⋊(C3⋊C8)—22no—
47#803—22no—
48#533C2×D311yescharacteristic
49C2×C4⋊Q8C2×D311yescharacteristic
50#491—22no—
51#487Dic314yescomplemented
52#432—32no—
53#419—22no—
54C4×M16C2×D311yescharacteristic
55#391—62no—
56#387—64no—
57#383—32no—
58#333—62no—
59#328—64no—
60C2×C3×Q8—22no—
61C2×C3×Q8—22no—
62C2×C3×Q8—42no—
63C2×C3×Q8—42no—
64C2×C3×Q8C22⋊C414yes—
65C22×C3×C4—22no—
66C22×C3×C4C2×D412yes—
67C22×C3×C4C2×D411yescharacteristic
68C22×Dic3—24no—
69C3×M16—42no—
70C2×C3×C8—22no—
71C3×C4⋊C4—24no—
72C3×C4⋊C4—24no—
73C3×C4⋊C4—22no—
74C3×C4⋊C4—22no—
75C3×C4⋊C4—42no—
76C3×C22⋊C4—41no—
77C3×C42—41no—
78C3×C42C22×C411yescharacteristic
79C3×C42C2×D411yescharacteristic
80C3×C42#4112yes—
81C4⋊Dic3—82no—
82C4×Dic3—82no—
83C2×C3⋊C8—42no—
84C22×Q8#6712yes—
85C2×M16—21no—
86C4⋊Q8—22no—
87C4⋊Q8C2×Dic312yes—
88C4⋊Q8C2×Dic312yes—
89C4⋊Q8#6712yes—
90#125—21no—
91C22⋊Q8—22no—
92C4×Q8—22no—
93#117—21no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—62no—
96C2×C42—62no—
97C2×C42C22×D311yescharacteristic
98C4⋊C8—62no—
99#103—28no—
100C22⋊C8—62no—
101#97—21no—
102#97—62no—
103C4×C8C2×Dic312yes—
104#95—34no—
105#95—64no—
106#95—62no—
107#95—62no—
108C23×C3#12011yescharacteristic
109C3×Q8—44no—
110C3×Q8—82no—
111C3×Q8—82no—
112C2×C3×C4—22no—
113C2×C3×C4—24no—
114C2×C3×C4—42no—
115C2×C3×C4—42no—
116C2×C3×C4—42no—
117C2×C3×C4—42no—
118C2×C3×C4C2×C22⋊C411yescharacteristic
119C2×C3×C4C4×D412yes—
120C2×C3×C4#12112yes—
121C2×C3×C4#12311yescharacteristic
122C2×Dic3—48no—
123C2×Dic3—44no—
124C3×C8—42no—
125C3⋊C8—82no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8#21714yes—
131C22×C4—22no—
132C22×C4—64no—
133C22×C4D3×D412yes—
134C22×C4C2×#6711yescharacteristic
135M16—42no—
136C2×C8—22no—
137C2×C8—122no—
138C4⋊C4—24no—
139C4⋊C4—24no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—42no—
143C4⋊C4—242no—
144C22⋊C4—41no—
145C42—41no—
146C42—242no—
147C42#23712yes—
148C42C22×Dic311yescharacteristic
149C42C2×#6711yescharacteristic
150C22×C3—22no—
151C22×C3#38611yescharacteristic
152C22×C3#45512yes—
153C3×C4—21no—
154C3×C4—21no—
155C3×C4—22no—
156C3×C4—22no—
157C3×C4—44no—
158C3×C4—82no—
159Dic3—84no—
160C23#93711yescharacteristic
161Q8—44no—
162Q8—82no—
163Q8—82no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—128no—
171C2×C4—124no—
172C2×C4D4×Dic312yes—
173C2×C4#93511yescharacteristic
174C2×C4#93912yes—
175C2×C4C2×#21711yescharacteristic
176C8—42no—
177C8—242no—
178C2×C3—22no—
179C2×C3#188511yescharacteristic
180C2×C3#190512yes—
181C22—22no—
182C22#506112yes—
183C22#522011yescharacteristic
184C4—21no—
185C4—21no—
186C4—22no—
187C4—22no—
188C4—44no—
189C4—82no—
190C4—244no—
191C3#942611yescharacteristic, complemented, Sylow
192C2—22no—
193C2#7172311yescharacteristic
194C2#7210112yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3139C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#3139D3
hypocenterC3#9426
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
10. C3×#487C4semidirect
51. #487Dic3semidirect
191. C3#9426semidirect

Maximal subgroups

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

subgroupindex
C3×#31392
#832312
#718372
#717772
#676662
#94263