Finite Group Databasebeta

← #10692811

Subgroups — #10692811

Order 768.

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

normal subgroups only

This group has 434 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×#2371C211yesmaximal, characteristic
2#83015C211yesmaximal, characteristic
3#71736C211yesmaximal, characteristic
4#69433C212yesmaximal
5#67615C212yesmaximal
6#9440—31nomaximal, Sylow
7C2×C3×#125C2211yescharacteristic
8#5470C2211yescharacteristic
9C3×#387—22no—
10C3×#387—22no—
11C3×C4×C4⋊C4—22no—
12C3×#376C2212yes—
13#5065—22no—
14#5064C414yescomplemented
15#4964—22no—
16#4926C2212yes—
17C4×#208C2211yescharacteristic
18#4543—22no—
19#4452—22no—
20#4452—22no—
21#3166—31no—
22#2371D311yescharacteristic
23#1534—31no—
24#1506—32no—
25#1300—32no—
26C3×#125—22no—
27C3×#125C2×C412yes—
28C3×#125D412yes—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4D412yes—
32C2×C3×C42—22no—
33C2×C3×C42—22no—
34C2×C3×C42C2311yescharacteristic
35#926—21no—
36#924—22no—
37C2×#208—21no—
38#870—21no—
39#869C2×C412yes—
40C4×Dic6—22no—
41C3×#95—22no—
42C3×#95—22no—
43C3×#95—22no—
44C3×#95D412yes—
45C3×#95D412yes—
46#830—22no—
47#808—24no—
48#807—24no—
49C4⋊(C3⋊C8)—22no—
50#803—22no—
51#803C2×C412yes—
52C4×C3⋊C8—21no—
53#533—31no—
54C2×#125C2×D311yescharacteristic
55#501—34no—
56#499—62no—
57#432—32no—
58#428—62no—
59C4×M16—31no—
60#387—22no—
61#387—22no—
62C4×C4⋊C4—22no—
63#376C2×D312yes—
64#330—62no—
65#328—62no—
66#328—62no—
67C22×C3×C4—22no—
68C22×C3×C4—22no—
69C22×C3×C4—22no—
70C22×C3×C4C2×D411yescharacteristic
71C22×C3×C4C2×D412yes—
72C2×Dic6—22no—
73C3×C4⋊C4—24no—
74C3×C4⋊C4—24no—
75C3×C4⋊C4—22no—
76C3×C4⋊C4—44no—
77C3×C4⋊C4C22⋊C414yes—
78C3×C42—44no—
79C3×C42—44no—
80C3×C42C22×C411yescharacteristic
81C3×C42C2×D411yescharacteristic
82C3×C42#4112yes—
83#217—41no—
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86Dic3⋊C4—42no—
87C4×Dic3—41no—
88#208—24no—
89C2×C3⋊C8—22no—
90C2×C3⋊C8—42no—
91C2×M16—61no—
92C4⋊Q8—32no—
93#125—22no—
94#125—61no—
95#125C4×D312yes—
96#125D1212yes—
97C22⋊Q8—62no—
98C4×Q8—62no—
99#117—61no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4#6712yes—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42C22×D311yescharacteristic
106#107—64no—
107#106—64no—
108C4⋊C8—62no—
109C22⋊C8—62no—
110#97—32no—
111#97—62no—
112C4×C8—61no—
113#95—22no—
114#95—22no—
115#95—22no—
116#95#6712yes—
117#95#6712yes—
118C23×C3#12011yescharacteristic
119C2×C3×C4—22no—
120C2×C3×C4—24no—
121C2×C3×C4—24no—
122C2×C3×C4—42no—
123C2×C3×C4—42no—
124C2×C3×C4—42no—
125C2×C3×C4—42no—
126C2×C3×C4—42no—
127C2×C3×C4—42no—
128C2×C3×C4—42no—
129C2×C3×C4—42no—
130C2×C3×C4C2×C22⋊C411yescharacteristic
131C2×C3×C4C4×D412yes—
132C2×C3×C4#12112yes—
133C2×C3×C4#12311yescharacteristic
134C2×Dic3—42no—
135Dic6—82no—
136C3⋊C8—24no—
137C3⋊C8—82no—
138C2×Q8—62no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4C2×#6711yescharacteristic
143C22×C4C2×#6712yes—
144M16—64no—
145C2×C8—62no—
146C2×C8—122no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—24no—
150C4⋊C4—44no—
151C4⋊C4—62no—
152C4⋊C4—62no—
153C4⋊C4—122no—
154C4⋊C4(C2×D3)⋊C414yes—
155C22⋊C4—121no—
156C42—44no—
157C42—44no—
158C42—121no—
159C42C2×C4×D311yescharacteristic
160C42C2×D1211yescharacteristic
161C42#23512yes—
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—44no—
171C3×C4—82no—
172Dic3—82no—
173C23#95311yescharacteristic
174Q8—242no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—122no—
187C2×C4C2×(C2×D3)⋊C411yescharacteristic
188C2×C4C4×#6712yes—
189C2×C4#92911yescharacteristic
190C2×C4#93012yes—
191C8—64no—
192C8—242no—
193C2×C3—22no—
194C2×C3#188611yescharacteristic
195C2×C3#190612yes—
196C22—22no—
197C22#521311yescharacteristic
198C22#524612yes—
199C4—22no—
200C4—22no—
201C4—21no—
202C4—21no—
203C4—44no—
204C4—44no—
205C4—82no—
206C4—242no—
207C3#944011yescharacteristic, complemented, Sylow
208C2—22no—
209C2#7164512yes—
210C2#7166011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2371C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#2371D3
hypocenterC3#9440
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2371

Complements

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

NH ≅ G/Ntype
14. #5064C4semidirect
207. C3#9440semidirect

Maximal subgroups

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

subgroupindex
C3×#23712
#830152
#717362
#694332
#676152
#94403