Finite Group Databasebeta

← #10692812

Subgroups — #10692812

Order 768.

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

normal subgroups only

This group has 434 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×#3166C211yesmaximal, characteristic
2#83026C211yesmaximal, characteristic
3#71836C212yesmaximal
4#71736C211yesmaximal, characteristic
5#67665C212yesmaximal
6#9440—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×C3×#125C2211yescharacteristic
9C3×#501C414yescomplemented
10C3×#499—22no—
11C3×#428—22no—
12C3×C4×M16C2211yescharacteristic
13#4980—22no—
14#4980—22no—
15Dic3×C4⋊C4—22no—
16#4932C2212yes—
17#4926C2212yes—
18#4543—22no—
19#4538—22no—
20#4538—22no—
21#3166D311yescharacteristic
22#2371—31no—
23#1534—31no—
24#1506—32no—
25#1300—32no—
26C2×C3×M16—21no—
27C3×C4⋊Q8C2×C412yes—
28C3×#125—21no—
29C3×#125—22no—
30C3×#125C2×C412yes—
31C3×#125D412yes—
32C3×C22⋊Q8—22no—
33C3×C4×Q8—22no—
34C3×#117—21no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4D412yes—
37C2×C3×C42C2311yescharacteristic
38C2×C4⋊Dic3—22no—
39C2×C4×Dic3—22no—
40C2×C4×Dic3—22no—
41C3×#107—24no—
42C3×#106—24no—
43C3×#97C2×C412yes—
44C3×C4×C8—21no—
45C3⋊#95—22no—
46C3⋊#95—22no—
47C3⋊#95—22no—
48C3⋊#95D412yes—
49C3⋊#95D412yes—
50#830—22no—
51C4⋊(C3⋊C8)—22no—
52#803—22no—
53#533C2×D311yescharacteristic
54C2×#125C2×D311yescharacteristic
55#501Dic314yescomplemented
56#499—22no—
57#432—32no—
58#428—22no—
59C4×M16C2×D311yescharacteristic
60#387—62no—
61#387—62no—
62C4×C4⋊C4—62no—
63#376—32no—
64#330—62no—
65#328—62no—
66#328—62no—
67C2×C3×Q8—22no—
68C22×C3×C4—22no—
69C22×C3×C4C2×D412yes—
70C22×C3×C4C2×D411yescharacteristic
71C22×Dic3—22no—
72C22×Dic3—22no—
73C3×M16—24no—
74C2×C3×C8—22no—
75C3×C4⋊C4—24no—
76C3×C4⋊C4—24no—
77C3×C4⋊C4—22no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4—22no—
80C3×C4⋊C4—42no—
81C3×C4⋊C4C22⋊C414yes—
82C3×C22⋊C4—41no—
83C3×C42—41no—
84C3×C42C22×C411yescharacteristic
85C3×C42C2×D411yescharacteristic
86C3×C42#4112yes—
87C4⋊Dic3—44no—
88C4×Dic3—44no—
89C4×Dic3—44no—
90C2×C3⋊C8—42no—
91C2×M16—21no—
92C4⋊Q8C2×Dic312yes—
93#125—22no—
94#125—21no—
95#125C2×Dic312yes—
96#125#6712yes—
97C22⋊Q8—22no—
98C4×Q8—22no—
99#117—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—62no—
102C2×C4⋊C4#6712yes—
103C2×C42—62no—
104C2×C42—62no—
105C2×C42C22×D311yescharacteristic
106#107—24no—
107#106—24no—
108C4⋊C8—62no—
109C22⋊C8—62no—
110#97—62no—
111#97C2×Dic312yes—
112C4×C8—21no—
113#95—32no—
114#95—32no—
115#95—62no—
116#95—62no—
117#95—62no—
118C23×C3#12011yescharacteristic
119C3×Q8—82no—
120C2×C3×C4—22no—
121C2×C3×C4—24no—
122C2×C3×C4—42no—
123C2×C3×C4—42no—
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—24no—
131C2×Dic3—42no—
132C2×Dic3—42no—
133C2×Dic3—42no—
134C2×Dic3—42no—
135C2×Dic3—42no—
136C3×C8—24no—
137C3⋊C8—82no—
138C2×Q8—22no—
139C22×C4—22no—
140C22×C4—62no—
141C22×C4—62no—
142C22×C4D3×D412yes—
143C22×C4C2×#6711yescharacteristic
144M16—24no—
145C2×C8—22no—
146C2×C8—122no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—24no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—42no—
153C4⋊C4—124no—
154C4⋊C4#21714yes—
155C22⋊C4—41no—
156C42—41no—
157C42—124no—
158C42—124no—
159C42#23712yes—
160C42C22×Dic311yescharacteristic
161C42C2×#6711yescharacteristic
162C22×C3—22no—
163C22×C3#38611yescharacteristic
164C22×C3#45612yes—
165C3×C4—21no—
166C3×C4—21no—
167C3×C4—22no—
168C3×C4—22no—
169C3×C4—44no—
170C3×C4—82no—
171Dic3—44no—
172Dic3—82no—
173C23#93711yescharacteristic
174Q8—82no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—64no—
182C2×C4—122no—
183C2×C4—122no—
184C2×C4—122no—
185C2×C4—122no—
186C2×C4—122no—
187C2×C4D4×Dic312yes—
188C2×C4#93511yescharacteristic
189C2×C4#93912yes—
190C2×C4C2×#21711yescharacteristic
191C8—24no—
192C8—242no—
193C2×C3—22no—
194C2×C3#188611yescharacteristic
195C2×C3#190612yes—
196C22—22no—
197C22#509512yes—
198C22#522011yescharacteristic
199C4—21no—
200C4—21no—
201C4—22no—
202C4—22no—
203C4—44no—
204C4—82no—
205C4—124no—
206C4—242no—
207C3#944011yescharacteristic, complemented, Sylow
208C2—22no—
209C2#7178411yescharacteristic
210C2#7211412yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3166C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#3166D3
hypocenterC3#9440
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3166

Complements

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

NH ≅ G/Ntype
9. C3×#501C4semidirect
55. #501Dic3semidirect
207. C3#9440semidirect

Maximal subgroups

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

subgroupindex
C3×#31662
#830262
#718362
#717362
#676652
#94403