Finite Group Databasebeta

← #10692725

Subgroups — #10692725

Order 768.

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

normal subgroups only

This group has 534 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×#3138C211yesmaximal, characteristic
2#83174C211yesmaximal, characteristic
3#71837C212yesmaximal, complemented
4#71760C211yesmaximal, characteristic
5#67666C212yesmaximal, complemented
6#9425—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×C3×#127C2211yescharacteristic
9C3×#490—22no—
10C3×#486C414yescomplemented
11C3×#418—22no—
12C3×C4×M16C2211yescharacteristic
13#5220—22no—
14#5220—24no—
15#4937C2212yes—
16#4926C2212yes—
17#4544—22no—
18#4538—24no—
19#3138D311yescharacteristic
20#2380—31no—
21#1555—31no—
22#1515—32no—
23#1288—32no—
24C22×C3×D4—22no—
25C22×C3×D4D412yes—
26C23×Dic3—22no—
27C2×C3×M16—21no—
28C3×C4⋊Q8C2×C412yescomplemented
29C3×#127—22no—
30C3×#127C2×C412yes—
31C3×#127D412yes—
32C3×#125—21no—
33C3×C22⋊Q8—22no—
34C3×C4×Q8—22no—
35C3×#117—21no—
36C2×C3×C42C2311yescharacteristic
37C2×#217—24no—
38C2×#217—22no—
39C3×#102—28no—
40C3×#97—21no—
41C3×C4×C8C2×C412yescomplemented
42C3⋊#95—22no—
43C3⋊#95—22no—
44C3⋊#95D414yes—
45#830—22no—
46C4⋊(C3⋊C8)—22no—
47#803—22no—
48#533C2×D311yescharacteristic
49C2×#127C2×D311yescharacteristic
50#490—22no—
51#486Dic314yescomplemented
52#432—32no—
53#418—22no—
54C4×M16C2×D311yescharacteristic
55#386—64no—
56#386—62no—
57#383—32no—
58#331—62no—
59#328—64no—
60C24×C3—22no—
61C2×C3×Q8—22no—
62C2×C3×D4—24no—
63C2×C3×D4—24no—
64C2×C3×D4—22no—
65C2×C3×D4—42no—
66C2×C3×D4—42no—
67C2×C3×D4—82no—
68C2×C3×D4—82no—
69C2×C3×D4C22⋊C414yes—
70C22×C3×C4C2×D412yes—
71C22×C3×C4C2×D411yescharacteristic
72C22×Dic3—24no—
73C22×Dic3—84no—
74C22×Dic3—82no—
75C3×M16—42no—
76C2×C3×C8—22no—
77C3×C4⋊C4—22no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4—42no—
80C3×C22⋊C4—41no—
81C3×C42—41no—
82C3×C42C22×C411yescharacteristic
83C3×C42C2×D411yescharacteristic
84C3×C42#4112yes—
85#217—84no—
86#217—82no—
87C2×C3⋊C8—42no—
88C22×D4—22no—
89C22×D4#6712yes—
90C23×C4—62no—
91C2×M16—21no—
92C4⋊Q8C2×Dic312yescomplemented
93#127—22no—
94#127C2×Dic312yes—
95#127#6712yes—
96#125—21no—
97C22⋊Q8—22no—
98C4×Q8—22no—
99#117—21no—
100C2×C22⋊C4—64no—
101C2×C22⋊C4—62no—
102C2×C42C22×D311yescharacteristic
103C4⋊C8—62no—
104#102—28no—
105C22⋊C8—62no—
106#97—21no—
107#97—62no—
108C4×C8C2×Dic312yescomplemented
109#95—34no—
110#95—62no—
111#95—62no—
112C23×C3—24no—
113C23×C3—44no—
114C23×C3—44no—
115C23×C3—42no—
116C23×C3—42no—
117C23×C3#12011yescharacteristic
118C3×Q8—82no—
119C3×D4—44no—
120C3×D4—82no—
121C3×D4—84no—
122C3×D4—84no—
123C2×C3×C4—22no—
124C2×C3×C4—42no—
125C2×C3×C4—42no—
126C2×C3×C4C2×C22⋊C411yescharacteristic
127C2×C3×C4C4×D412yes—
128C2×C3×C4#12112yes—
129C2×C3×C4#12311yescharacteristic
130C2×Dic3—48no—
131C2×Dic3—44no—
132C2×Dic3—88no—
133C3×C8—42no—
134C3⋊C8—82no—
135C24—22no—
136C2×Q8—22no—
137C2×D4—24no—
138C2×D4—22no—
139C2×D4—24no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—82no—
143C2×D4—82no—
144C2×D4#21714yes—
145C22×C4—64no—
146C22×C4—244no—
147C22×C4—242no—
148C22×C4D3×D412yes—
149C22×C4C2×#6711yescharacteristic
150M16—42no—
151C2×C8—22no—
152C2×C8—122no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C22⋊C4—41no—
157C22⋊C4—244no—
158C22⋊C4—242no—
159C42—41no—
160C42#23712yes—
161C42C22×Dic311yescharacteristic
162C42C2×#6711yescharacteristic
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—82no—
176Dic3—84no—
177C23—24no—
178C23—44no—
179C23—44no—
180C23—42no—
181C23—42no—
182C23#93711yescharacteristic
183Q8—82no—
184D4—44no—
185D4—82no—
186D4—84no—
187D4—84no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—128no—
192C2×C4—124no—
193C2×C4—248no—
194C2×C4D4×Dic312yes—
195C2×C4#93511yescharacteristic
196C2×C4#93912yes—
197C2×C4C2×#21711yescharacteristic
198C8—42no—
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#506112yes—
211C22#522011yescharacteristic
212C4—21no—
213C4—21no—
214C4—22no—
215C4—22no—
216C4—82no—
217C4—244no—
218C3#942511yescharacteristic, complemented, Sylow
219C2—22no—
220C2—84no—
221C2#7172311yescharacteristic
222C2#7208912yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3138C2
socleC22×C3#386
prime radicalC2×C3×#127C22
cosocleC2×C3×C42C23
hypercenter#3138D3
hypocenterC3#9425
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
3. #71837C2semidirect
5. #67666C2semidirect
10. C3×#486C4semidirect
28. C3×C4⋊Q8C2×C4semidirect
41. C3×C4×C8C2×C4semidirect
51. #486Dic3semidirect
92. C4⋊Q8C2×Dic3semidirect
108. C4×C8C2×Dic3semidirect
218. C3#9425semidirect

Maximal subgroups

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

subgroupindex
C3×#31382
#831742
#718372
#717602
#676662
#94253