Finite Group Databasebeta

← #10698325

Subgroups — #10698325

Order 768.

402 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 91 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#73398C211yesmaximal, characteristic, complemented
2C3×#1963C211yesmaximal, characteristic
3#71811C212yesmaximal
4#67715C212yesmaximal, complemented
5#67614C211yesmaximal, characteristic, complemented
6#10372—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8C3×#423C2211yescharacteristic
9C2×C3×#104—22no—
10C3×#376C2211yescharacteristic
11C4×C4⋊Dic3C2212yes—
12C4×Dic3⋊C4—22no—
13#4925C2212yes—
14#4660—22no—
15#4657—22no—
16#4537—22no—
17#4537—22no—
18#2402—31no—
19#1963D311yescharacteristic
20#1502—32no—
21#1303—32no—
22#1274—31no—
23C2×C3×#41—21no—
24C2×C3×M16—21no—
25C3×C4⋊Q8Q811yescharacteristic
26C3×#127Q811yescharacteristic
27C3×#124—21no—
28C3×#121—22no—
29C3×C4×D4—22no—
30C2×C3×C42D412yes—
31C2×C3×C42C2311yescharacteristic
32C2×C4⋊Dic3—22no—
33C2×Dic3⋊C4—22no—
34C2×Dic3⋊C4—22no—
35C2×C4×Dic3—22no—
36C3×C4⋊C8—22no—
37C3×#104—44no—
38C3×#95—22no—
39C3⋊#95—22no—
40C3⋊#95—22no—
41C3⋊#95D412yes—
42C3⋊#95D412yes—
43#830—22no—
44C4⋊(C3⋊C8)—22no—
45C4×C3⋊C8—22no—
46#532C2×D311yescharacteristic
47#431—32no—
48#423C2×D311yescharacteristic
49C2×#104—22no—
50#400—62no—
51#398—62no—
52C4×C4⋊C4—32no—
53C4×C4⋊C4—62no—
54#376C2×D311yescharacteristic
55#327—62no—
56#327—62no—
57C3×#41—42no—
58C3×#41—42no—
59C2×C3×Q8—21no—
60C2×C3×D4—21no—
61C2×C3×D4—22no—
62C2×C3×D4—41no—
63C22×C3×C4—22no—
64C22×C3×C4—41no—
65C22×C3×C4C2×D412yes—
66C22×C3×C4C2×D411yescharacteristic
67C22×Dic3—22no—
68C22×Dic3—22no—
69C3×M16—42no—
70C2×C3×C8—41no—
71C3×C4⋊C4—22no—
72C3×C22⋊C4—42no—
73C3×C42—22no—
74C3×C42—22no—
75C3×C42C2×Q811yescharacteristic
76C3×C42#4111yescharacteristic
77C3×C42#4112yes—
78C4⋊Dic3—44no—
79Dic3⋊C4—44no—
80Dic3⋊C4—44no—
81C4×Dic3—42no—
82C4×Dic3—42no—
83C2×C3⋊C8—42no—
84C2×#41—21no—
85C2×M16—21no—
86C4⋊Q8Dic611yescharacteristic
87#127Dic611yescharacteristic
88#124—21no—
89#121—22no—
90C4×D4—22no—
91C2×C4⋊C4—62no—
92C2×C4⋊C4—62no—
93C2×C4⋊C4—62no—
94C2×C42—62no—
95C2×C42#6712yes—
96C2×C42C22×D311yescharacteristic
97C4⋊C8—22no—
98C4⋊C8—62no—
99#104—44no—
100C22⋊C8—62no—
101C4×C8—62no—
102#95—22no—
103#95—32no—
104#95—32no—
105#95—62no—
106#95—62no—
107C23×C3—41no—
108C23×C3#12011yescharacteristic
109C3×Q8—42no—
110C3×D4—42no—
111C3×D4—82no—
112C3×D4—82no—
113C2×C3×C4—42no—
114C2×C3×C4—42no—
115C2×C3×C4—42no—
116C2×C3×C4—41no—
117C2×C3×C4—42no—
118C2×C3×C4—81no—
119C2×C3×C4—81no—
120C2×C3×C4C22⋊Q812yes—
121C2×C3×C4C22⋊Q811yescharacteristic
122C2×C3×C4#12412yes—
123C2×C3×C4#12411yescharacteristic
124C2×C3×C4#13512yes—
125C2×C3×C4AGL(1,ℤ/8)12yes—
126C2×Dic3—24no—
127C2×Dic3—42no—
128C2×Dic3—42no—
129C2×Dic3—42no—
130C2×Dic3—42no—
131C2×Dic3—42no—
132C3×C8—81no—
133C3⋊C8—82no—
134#41—42no—
135#41—42no—
136C2×Q8—21no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—41no—
140C22×C4—22no—
141C22×C4—41no—
142C22×C4—62no—
143C22×C4—62no—
144C22×C4D3×D412yes—
145C22×C4C2×#6711yescharacteristic
146M16—42no—
147C2×C8—41no—
148C2×C8—122no—
149C4⋊C4—22no—
150C4⋊C4—124no—
151C4⋊C4—124no—
152C4⋊C4—124no—
153C22⋊C4—42no—
154C42—22no—
155C42—22no—
156C42—122no—
157C42—122no—
158C42C2×Dic611yescharacteristic
159C42#23511yescharacteristic
160C42#23712yes—
161C22×C3—22no—
162C22×C3—81no—
163C22×C3—82no—
164C22×C3#39311yescharacteristic
165C22×C3#44912yes—
166C3×C4—22no—
167C3×C4—22no—
168C3×C4—82no—
169C3×C4—81no—
170C3×C4#47812yes—
171C3×C4#48811yescharacteristic
172C3×C4#49011yescharacteristic
173Dic3—44no—
174Dic3—82no—
175C23—41no—
176C23#93711yescharacteristic
177Q8—42no—
178D4—42no—
179D4—82no—
180D4—82no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—64no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—122no—
190C2×C4—122no—
191C2×C4—122no—
192C2×C4—122no—
193C2×C4—122no—
194C2×C4#87812yes—
195C2×C4#88512yes—
196C2×C4#92411yescharacteristic
197C2×C4#93212yes—
198C2×C4#93611yescharacteristic
199C2×C4#95012yes—
200C8—81no—
201C8—242no—
202C2×C3—22no—
203C2×C3—161no—
204C2×C3#202511yescharacteristic
205C2×C3#203112yes—
206C22—22no—
207C22—81no—
208C22—82no—
209C22#494711yescharacteristic
210C22#503712yes—
211C4—22no—
212C4—22no—
213C4—82no—
214C4—81no—
215C4—124no—
216C4—242no—
217C4#500912yes—
218C4#505611yescharacteristic
219C4#507111yescharacteristic
220C3#1037211yescharacteristic, complemented, Sylow
221C2—22no—
222C2—161no—
223C2#6963911yescharacteristic
224C2#7039412yes—
subgroupG/H
centerC2×C4#936
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1963C2
socleC22×C3#393
prime radicalC3×#532C22
hypercenter#1963D3
hypocenterC3#10372
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
1. #73398C2semidirect
4. #67715C2semidirect
5. #67614C2semidirect
220. C3#10372semidirect

Maximal subgroups

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

subgroupindex
#733982
C3×#19632
#718112
#677152
#676142
#103723