Finite Group Databasebeta

← #10692729

Subgroups — #10692729

Order 768.

422 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2381C211yesmaximal, characteristic
2#83198C211yesmaximal, characteristic
3#71777C211yesmaximal, characteristic
4#69566C212yesmaximal
5#67633C212yesmaximal
6#9426—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8#5470C2211yescharacteristic
9C3×#391—22no—
10C3×#387—24no—
11C3×#383C2212yes—
12#5225—22no—
13#5082—22no—
14#5079C414yescomplemented
15#4926C2212yes—
16C4×#208C2211yescharacteristic
17#4546—22no—
18#4452—24no—
19#3139—31no—
20#2381D311yescharacteristic
21#1558—31no—
22#1515—32no—
23#1288—32no—
24C22×C3×Q8D412yes—
25C3×C4⋊Q8—22no—
26C3×C4⋊Q8C2×C412yes—
27C3×C4⋊Q8D412yes—
28C2×C3×C4⋊C4—22no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C42—22no—
31C2×C3×C42C2311yescharacteristic
32#926—21no—
33#924—22no—
34C2×#208—21no—
35#870—21no—
36#869C2×C412yes—
37C4×Dic6—22no—
38C3×#95—24no—
39C3×#95—22no—
40C3×#95—22no—
41C3×#95D414yes—
42#835—28no—
43#830—22no—
44C4⋊(C3⋊C8)—22no—
45#803—21no—
46#803—22no—
47C4×C3⋊C8C2×C412yes—
48#533—31no—
49C2×C4⋊Q8C2×D311yescharacteristic
50#491—62no—
51#487—34no—
52#432—32no—
53#419—62no—
54C4×M16—31no—
55#391—22no—
56#387—24no—
57#383C2×D312yes—
58#333—62no—
59#328—64no—
60C2×C3×Q8—22no—
61C2×C3×Q8—42no—
62C2×C3×Q8—42no—
63C2×C3×Q8C22⋊C414yes—
64C22×C3×C4—22no—
65C22×C3×C4—24no—
66C22×C3×C4C2×D411yescharacteristic
67C22×C3×C4C2×D412yes—
68C2×Dic6—22no—
69C3×C4⋊C4—24no—
70C3×C4⋊C4—24no—
71C3×C4⋊C4—82no—
72C3×C42—82no—
73C3×C42C22×C411yescharacteristic
74C3×C42C2×D411yescharacteristic
75C3×C42#4112yes—
76#217—41no—
77C4⋊Dic3—22no—
78C4⋊Dic3—22no—
79Dic3⋊C4—42no—
80C4×Dic3—41no—
81#208—42no—
82C2×C3⋊C8—22no—
83C2×C3⋊C8—42no—
84C22×Q8#6712yes—
85C2×M16—61no—
86C4⋊Q8—22no—
87C4⋊Q8—32no—
88C4⋊Q8C4×D312yes—
89C4⋊Q8D1212yes—
90#125—61no—
91C22⋊Q8—62no—
92C4×Q8—62no—
93#117—61no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C42—22no—
97C2×C42C22×D311yescharacteristic
98C4⋊C8—62no—
99#103—68no—
100C22⋊C8—62no—
101#97—61no—
102#97—62no—
103C4×C8—32no—
104#95—24no—
105#95—22no—
106#95—22no—
107#95#6714yes—
108C23×C3#12011yescharacteristic
109C3×Q8—44no—
110C3×Q8—82no—
111C2×C3×C4—22no—
112C2×C3×C4—24no—
113C2×C3×C4—42no—
114C2×C3×C4—42no—
115C2×C3×C4—42no—
116C2×C3×C4—48no—
117C2×C3×C4—44no—
118C2×C3×C4C2×C22⋊C411yescharacteristic
119C2×C3×C4C4×D412yes—
120C2×C3×C4#12112yes—
121C2×C3×C4#12311yescharacteristic
122C2×Dic3—42no—
123Dic6—82no—
124C3⋊C8—42no—
125C3⋊C8—82no—
126C2×Q8—22no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—62no—
130C2×Q8(C2×D3)⋊C414yes—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4C2×#6711yescharacteristic
134C22×C4C2×#6712yes—
135M16—122no—
136C2×C8—62no—
137C2×C8—122no—
138C4⋊C4—24no—
139C4⋊C4—24no—
140C4⋊C4—62no—
141C4⋊C4—62no—
142C4⋊C4—82no—
143C4⋊C4—122no—
144C22⋊C4—121no—
145C42—82no—
146C42—121no—
147C42C2×C4×D311yescharacteristic
148C42C2×D1211yescharacteristic
149C42#23512yes—
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—84no—
159Dic3—82no—
160C23#95311yescharacteristic
161Q8—44no—
162Q8—82no—
163Q8—242no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—48no—
170C2×C4—44no—
171C2×C4—122no—
172C2×C4C2×(C2×D3)⋊C411yescharacteristic
173C2×C4C4×#6712yes—
174C2×C4#92911yescharacteristic
175C2×C4#93012yes—
176C8—122no—
177C8—242no—
178C2×C3—22no—
179C2×C3#188511yescharacteristic
180C2×C3#190512yes—
181C22—22no—
182C22#521311yescharacteristic
183C22#524212yes—
184C4—21no—
185C4—21no—
186C4—22no—
187C4—22no—
188C4—44no—
189C4—84no—
190C4—242no—
191C3#942611yescharacteristic, complemented, Sylow
192C2—22no—
193C2#7164911yescharacteristic
194C2#7166212yes—
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2381C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#2381D3
hypocenterC3#9426
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2381

Complements

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

NH ≅ G/Ntype
14. #5079C4semidirect
191. C3#9426semidirect

Maximal subgroups

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

subgroupindex
C3×#23812
#831982
#717772
#695662
#676332
#94263