Finite Group Databasebeta

← #10692724

Subgroups — #10692724

Order 768.

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

normal subgroups only

This group has 534 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×#2380C211yesmaximal, characteristic
2#83159C211yesmaximal, characteristic
3#71760C211yesmaximal, characteristic
4#69566C212yesmaximal, complemented
5#67633C212yesmaximal, complemented
6#9425—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8#5470C2211yescharacteristic
9C3×#386—22no—
10C3×#386—24no—
11C3×#383C2212yes—
12#5215—22no—
13#5071—22no—
14#5070C414yescomplemented
15#4926C2212yes—
16C4×#208C2211yescharacteristic
17#4544—22no—
18#4452—24no—
19#3138—31no—
20#2380D311yescharacteristic
21#1555—31no—
22#1515—32no—
23#1288—32no—
24C22×C3×D4—22no—
25C22×C3×D4D412yes—
26C23×C3×C4—22no—
27C3×#127—22no—
28C3×#127C2×C412yes—
29C3×#127D412yes—
30C2×C3×C22⋊C4—24no—
31C2×C3×C22⋊C4—22no—
32C2×C3×C42C2311yescharacteristic
33#926—21no—
34#924—22no—
35C2×#208—21no—
36#870—21no—
37#869C2×C412yescomplemented
38C4×Dic6—22no—
39C3×#95—22no—
40C3×#95—22no—
41C3×#95D414yes—
42#832—28no—
43#830—22no—
44C4⋊(C3⋊C8)—22no—
45#803—21no—
46#803—22no—
47C4×C3⋊C8C2×C412yescomplemented
48#533—31no—
49C2×#127C2×D311yescharacteristic
50#490—62no—
51#486—34no—
52#432—32no—
53#418—62no—
54C4×M16—31no—
55#386—22no—
56#386—24no—
57#383C2×D312yes—
58#331—62no—
59#328—64no—
60C24×C3—22no—
61C2×C3×D4—24no—
62C2×C3×D4—24no—
63C2×C3×D4—22no—
64C2×C3×D4—42no—
65C2×C3×D4—42no—
66C2×C3×D4—82no—
67C2×C3×D4—82no—
68C2×C3×D4C22⋊C414yes—
69C22×C3×C4—24no—
70C22×C3×C4—84no—
71C22×C3×C4—82no—
72C22×C3×C4C2×D411yescharacteristic
73C22×C3×C4C2×D412yes—
74C2×Dic6—22no—
75C3×C22⋊C4—84no—
76C3×C22⋊C4—82no—
77C3×C42C22×C411yescharacteristic
78C3×C42C2×D411yescharacteristic
79C3×C42#4112yes—
80#217—41no—
81C4⋊Dic3—22no—
82C4⋊Dic3—22no—
83Dic3⋊C4—42no—
84C4×Dic3—41no—
85#208—42no—
86C2×C3⋊C8—22no—
87C2×C3⋊C8—42no—
88C22×D4—22no—
89C22×D4#6712yes—
90C23×C4—22no—
91C2×M16—61no—
92C4⋊Q8—32no—
93#127—22no—
94#127C4×D312yes—
95#127D1212yes—
96#125—61no—
97C22⋊Q8—62no—
98C4×Q8—62no—
99#117—61no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—22no—
102C2×C42C22×D311yescharacteristic
103C4⋊C8—62no—
104#102—68no—
105C22⋊C8—62no—
106#97—61no—
107#97—62no—
108C4×C8—32no—
109#95—22no—
110#95—22no—
111#95#6714yes—
112C23×C3—24no—
113C23×C3—44no—
114C23×C3—44no—
115C23×C3—42no—
116C23×C3—42no—
117C23×C3#12011yescharacteristic
118C3×D4—44no—
119C3×D4—82no—
120C3×D4—84no—
121C3×D4—84no—
122C2×C3×C4—22no—
123C2×C3×C4—42no—
124C2×C3×C4—48no—
125C2×C3×C4—44no—
126C2×C3×C4—88no—
127C2×C3×C4C2×C22⋊C411yescharacteristic
128C2×C3×C4C4×D412yes—
129C2×C3×C4#12112yes—
130C2×C3×C4#12311yescharacteristic
131C2×Dic3—42no—
132Dic6—82no—
133C3⋊C8—42no—
134C3⋊C8—82no—
135C24—22no—
136C2×Q8—62no—
137C2×D4—24no—
138C2×D4—22no—
139C2×D4—24no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—82no—
143C2×D4—82no—
144C2×D4(C2×D3)⋊C414yes—
145C22×C4—24no—
146C22×C4—84no—
147C22×C4—82no—
148C22×C4C2×#6711yescharacteristic
149C22×C4C2×#6712yes—
150M16—122no—
151C2×C8—62no—
152C2×C8—122no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4—122no—
156C22⋊C4—84no—
157C22⋊C4—82no—
158C22⋊C4—121no—
159C42—121no—
160C42C2×C4×D311yescharacteristic
161C42C2×D1211yescharacteristic
162C42#23512yes—
163C22×C3—22no—
164C22×C3—44no—
165C22×C3—48no—
166C22×C3—82no—
167C22×C3—82no—
168C22×C3—84no—
169C22×C3#38611yescharacteristic
170C22×C3#45512yes—
171C3×C4—21no—
172C3×C4—21no—
173C3×C4—22no—
174C3×C4—22no—
175C3×C4—84no—
176Dic3—82no—
177C23—24no—
178C23—44no—
179C23—44no—
180C23—42no—
181C23—42no—
182C23#95311yescharacteristic
183Q8—242no—
184D4—44no—
185D4—82no—
186D4—84no—
187D4—84no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—48no—
191C2×C4—44no—
192C2×C4—88no—
193C2×C4—122no—
194C2×C4C2×(C2×D3)⋊C411yescharacteristic
195C2×C4C4×#6712yes—
196C2×C4#92911yescharacteristic
197C2×C4#93012yes—
198C8—122no—
199C8—242no—
200C2×C3—22no—
201C2×C3—84no—
202C2×C3#188511yescharacteristic
203C2×C3#190412yes—
204C22—22no—
205C22—44no—
206C22—48no—
207C22—82no—
208C22—82no—
209C22—84no—
210C22#521311yescharacteristic
211C22#524212yes—
212C4—21no—
213C4—21no—
214C4—22no—
215C4—22no—
216C4—84no—
217C4—242no—
218C3#942511yescharacteristic, complemented, Sylow
219C2—22no—
220C2—84no—
221C2#7164911yescharacteristic
222C2#7165112yes—
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2380C2
socleC22×C3#386
prime radicalC2×C3×#127C22
cosocleC2×C3×C42C23
hypercenter#2380D3
hypocenterC3#9425
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2380

Complements

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

NH ≅ G/Ntype
4. #69566C2semidirect
5. #67633C2semidirect
14. #5070C4semidirect
37. #869C2×C4semidirect
47. C4×C3⋊C8C2×C4semidirect
218. C3#9425semidirect

Maximal subgroups

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

subgroupindex
C3×#23802
#831592
#717602
#695662
#676332
#94253