Finite Group Databasebeta

← #10692809

Subgroups — #10692809

Order 768.

434 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#83026C211yesmaximal, characteristic
2#83015C211yesmaximal, characteristic
3C3×#1534C211yesmaximal, characteristic
4#70836C212yesmaximal
5#67748C212yesmaximal
6#9440—31nomaximal, Sylow
7C2×C3×#125C2211yescharacteristic
8#5470C2211yescharacteristic
9C3×#432C2212yes—
10#5065—22no—
11#5064C414yescomplemented
12#4980—22no—
13#4980—22no—
14Dic3×C4⋊C4—22no—
15#4964—22no—
16#4932C2212yes—
17C4×#208C2211yescharacteristic
18C3×#330—22no—
19#4467—22no—
20#4467—22no—
21#3166—31no—
22#2371—31no—
23#1534D311yescharacteristic
24#1506—32no—
25#1300—32no—
26C3×#125—22no—
27C3×#125C2×C412yes—
28C3×#125D412yes—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4D412yes—
31C2×C3×C42C2311yescharacteristic
32#926—21no—
33C2×C4⋊Dic3—22no—
34#924—22no—
35C2×C4×Dic3—22no—
36C2×C4×Dic3—22no—
37C2×#208—21no—
38#870—21no—
39#869C2×C412yes—
40C4×Dic6—22no—
41C3×C4⋊C8—22no—
42C3×C22⋊C8—22no—
43C3×#97—22no—
44C3⋊#95—22no—
45C3⋊#95—22no—
46C3⋊#95—22no—
47C3⋊#95D412yes—
48C3⋊#95D412yes—
49#808—24no—
50#807—24no—
51#803C2×C412yes—
52C4×C3⋊C8—21no—
53#533—31no—
54C2×#125C2×D311yescharacteristic
55#501—34no—
56#499—62no—
57#432C2×D312yes—
58#428—62no—
59C4×M16—31no—
60#387—62no—
61#387—62no—
62C4×C4⋊C4—62no—
63#376—32no—
64#330—22no—
65#328—62no—
66#328—62no—
67C22×C3×C4—22no—
68C22×C3×C4C2×D411yescharacteristic
69C22×C3×C4C2×D412yes—
70C22×Dic3—22no—
71C22×Dic3—22no—
72C2×Dic6—22no—
73C2×C3×C8—42no—
74C3×C4⋊C4—24no—
75C3×C4⋊C4—24no—
76C3×C4⋊C4—22no—
77C3×C4⋊C4C22⋊C414yes—
78C3×C42C22×C411yescharacteristic
79C3×C42C2×D411yescharacteristic
80C3×C42#4112yes—
81#217—41no—
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3—44no—
85Dic3⋊C4—42no—
86C4×Dic3—44no—
87C4×Dic3—44no—
88C4×Dic3—41no—
89#208—24no—
90C2×C3⋊C8—22no—
91C2×M16—61no—
92C4⋊Q8—32no—
93#125—22no—
94#125—61no—
95#125C4×D312yes—
96#125#6712yes—
97C22⋊Q8—62no—
98C4×Q8—62no—
99#117—61no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—62no—
102C2×C4⋊C4D1212yes—
103C2×C42—62no—
104C2×C42—62no—
105C2×C42C22×D311yescharacteristic
106#107—64no—
107#106—64no—
108C4⋊C8—22no—
109C22⋊C8—22no—
110#97—22no—
111#97—32no—
112C4×C8—61no—
113#95—32no—
114#95—32no—
115#95—62no—
116#95—62no—
117#95—62no—
118C23×C3#12011yescharacteristic
119C2×C3×C4—22no—
120C2×C3×C4—24no—
121C2×C3×C4—42no—
122C2×C3×C4—42no—
123C2×C3×C4—42no—
124C2×C3×C4C2×C22⋊C411yescharacteristic
125C2×C3×C4C4×D412yes—
126C2×C3×C4#12112yes—
127C2×C3×C4#12311yescharacteristic
128C2×Dic3—24no—
129C2×Dic3—42no—
130C2×Dic3—42no—
131C2×Dic3—42no—
132C2×Dic3—42no—
133C2×Dic3—42no—
134C2×Dic3—42no—
135Dic6—82no—
136C3×C8—82no—
137C3⋊C8—24no—
138C2×Q8—62no—
139C22×C4—22no—
140C22×C4—62no—
141C22×C4—62no—
142C22×C4C2×D1211yescharacteristic
143C22×C4D3×D412yes—
144M16—64no—
145C2×C8—42no—
146C2×C8—62no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—24no—
150C4⋊C4—62no—
151C4⋊C4—62no—
152C4⋊C4—124no—
153C4⋊C4—122no—
154C4⋊C4(C2×D3)⋊C414yes—
155C22⋊C4—121no—
156C42—124no—
157C42—124no—
158C42—121no—
159C42C2×C4×D311yescharacteristic
160C42#23712yes—
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—
170Dic3—44no—
171Dic3—82no—
172Dic3—82no—
173C23#88211yescharacteristic
174Q8—242no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—64no—
181C2×C4—122no—
182C2×C4—122no—
183C2×C4—122no—
184C2×C4—122no—
185C2×C4—122no—
186C2×C4—122no—
187C2×C4#67⋊C412yes—
188C2×C4#88611yescharacteristic
189C2×C4C2×(C2×D3)⋊C411yescharacteristic
190C2×C4#93912yes—
191C8—64no—
192C8—82no—
193C2×C3—22no—
194C2×C3#188611yescharacteristic
195C2×C3#190612yes—
196C22—22no—
197C22#475212yes—
198C22#495811yescharacteristic
199C4—21no—
200C4—21no—
201C4—22no—
202C4—22no—
203C4—44no—
204C4—124no—
205C4—242no—
206C4—242no—
207C3#944011yescharacteristic, complemented, Sylow
208C2—22no—
209C2#7010311yescharacteristic
210C2#7039512yes—
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1534C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1534D3
hypocenterC3#9440
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1534

Complements

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

NH ≅ G/Ntype
11. #5064C4semidirect
207. C3#9440semidirect

Maximal subgroups

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

subgroupindex
#830262
#830152
C3×#15342
#708362
#677482
#94403