Finite Group Databasebeta

← #10698212

Subgroups — #10698212

Order 768.

500 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 123 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#74342C211yesmaximal, characteristic, complemented
2C3×#1957C211yesmaximal, characteristic
3#71761C212yesmaximal
4#67655C212yesmaximal, complemented
5#67634C211yesmaximal, characteristic, complemented
6#10351—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8C2×C3×#102—22no—
9C2×C3×C4×C8C2211yescharacteristic
10C3×#384C2211yescharacteristic
11C2×#832—22no—
12C2×#832—22no—
13#4986—22no—
14#4984—22no—
15#4977—22no—
16C4×C4⋊Dic3C2212yes—
17C2×C4⋊(C3⋊C8)C2212yes—
18#2571—31no—
19#1982—32no—
20#1957D311yescharacteristic
21#1296—31no—
22#1279—32no—
23C22×C3×D4—22no—
24C22×C3×D4—21no—
25C22×C3×C8—21no—
26C3×#127—21no—
27C3×#127—22no—
28C3×#127Q812yes—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4D412yes—
31C2×C3×C42C2311yescharacteristic
32C2×C4⋊Dic3—22no—
33C2×C4⋊Dic3D412yes—
34C2×C4⋊Dic3D412yes—
35C2×C4×Dic3—22no—
36C22×C3⋊C8—22no—
37C3×#102—44no—
38C3×C4×C8—22no—
39#832—44no—
40#832—44no—
41C3⋊#95—22no—
42C3⋊#95—24no—
43C3⋊#95—22no—
44C4⋊(C3⋊C8)—22no—
45C4⋊(C3⋊C8)—22no—
46C2×#127C2×D311yescharacteristic
47C2×C4⋊C8—32no—
48C2×#102—22no—
49C2×#102—62no—
50C2×#102—62no—
51C2×C4×C8C2×D311yescharacteristic
52#400—62no—
53#389—62no—
54#384C2×D311yescharacteristic
55#382—62no—
56C4×C4⋊C4—32no—
57C24×C3—41no—
58C2×C3×D4—24no—
59C2×C3×D4—22no—
60C2×C3×D4—42no—
61C2×C3×D4—42no—
62C2×C3×D4—42no—
63C2×C3×D4—41no—
64C2×C3×D4—42no—
65C2×C3×D4—42no—
66C2×C3×D4—42no—
67C2×C3×D4—42no—
68C22×C3×C4—22no—
69C22×C3×C4C2×D412yes—
70C22×C3×C4C2×D411yescharacteristic
71C22×Dic3—22no—
72C22×Dic3—22no—
73C2×C3×C8—22no—
74C2×C3×C8—41no—
75C2×C3×C8—41no—
76C3×C4⋊C4—42no—
77C3×C4⋊C4—42no—
78C3×C4⋊C4—42no—
79C3×C42C2×Q811yescharacteristic
80C3×C42#4111yescharacteristic
81C3×C42#4112yes—
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86C4⋊Dic3—82no—
87C4×Dic3—42no—
88C4×Dic3—42no—
89C2×C3⋊C8—42no—
90C2×C3⋊C8—42no—
91C2×C3⋊C8—42no—
92C22×D4—22no—
93C22×D4—21no—
94C22×C8—21no—
95C22×C8—62no—
96#127—21no—
97#127—22no—
98#127Dic612yes—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—62no—
103C2×C4⋊C4#6712yes—
104C2×C42—62no—
105C2×C42C22×D311yescharacteristic
106C4⋊C8—62no—
107C4⋊C8—62no—
108#102—44no—
109#102—124no—
110#102—124no—
111C4×C8—22no—
112#95—62no—
113#95—64no—
114#95—62no—
115C23×C3—42no—
116C23×C3—82no—
117C23×C3—82no—
118C23×C3—81no—
119C23×C3—81no—
120C23×C3#12011yescharacteristic
121C3×D4—44no—
122C3×D4—44no—
123C3×D4—44no—
124C3×D4—82no—
125C3×D4—82no—
126C3×D4—82no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—44no—
130C2×C3×C4—42no—
131C2×C3×C4C22⋊Q812yes—
132C2×C3×C4C22⋊Q811yescharacteristic
133C2×C3×C4#12412yes—
134C2×C3×C4#12411yescharacteristic
135C2×C3×C4#13512yes—
136C2×C3×C4#13512yes—
137C2×C3×C4AGL(1,ℤ/8)12yes—
138C2×C3×C4AGL(1,ℤ/8)12yes—
139C2×Dic3—42no—
140C2×Dic3—42no—
141C2×Dic3—42no—
142C2×Dic3—42no—
143C2×Dic3—42no—
144C2×Dic3—42no—
145C3×C8—42no—
146C3⋊C8—82no—
147C24—41no—
148C2×D4—24no—
149C2×D4—22no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—41no—
158C22×C4—22no—
159C22×C4—62no—
160C22×C4—62no—
161C22×C4D3×D412yes—
162C22×C4C2×#6711yescharacteristic
163C2×C8—22no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—122no—
167C2×C8—122no—
168C2×C8—122no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4—62no—
175C4⋊C4—62no—
176C4⋊C4—242no—
177C42—122no—
178C42—122no—
179C42C2×Dic611yescharacteristic
180C42#23511yescharacteristic
181C42#23712yes—
182C22×C3—84no—
183C22×C3—82no—
184C22×C3—161no—
185C22×C3—161no—
186C22×C3—162no—
187C22×C3#39311yescharacteristic
188C22×C3#44912yes—
189C22×C3#44912yes—
190C22×C3#45311yescharacteristic
191C22×C3#45411yescharacteristic
192C3×C4—21no—
193C3×C4—21no—
194C3×C4—82no—
195C3×C4#47812yes—
196C3×C4#47812yes—
197C3×C4#48812yes—
198C3×C4#49012yes—
199Dic3—82no—
200Dic3—82no—
201C23—42no—
202C23—82no—
203C23—82no—
204C23—81no—
205C23—81no—
206C23#93711yescharacteristic
207D4—44no—
208D4—44no—
209D4—44no—
210D4—82no—
211D4—82no—
212D4—82no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—122no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4—122no—
221C2×C4—122no—
222C2×C4—122no—
223C2×C4#87812yes—
224C2×C4#88512yes—
225C2×C4#91112yes—
226C2×C4#91212yes—
227C2×C4#91412yes—
228C2×C4#91612yes—
229C2×C4#92411yescharacteristic
230C2×C4#93611yescharacteristic
231C8—42no—
232C8—242no—
233C2×C3—162no—
234C2×C3#162011yescharacteristic
235C2×C3#162411yescharacteristic
236C2×C3#162812yes—
237C2×C3#202512yes—
238C2×C3#202711yescharacteristic
239C22—82no—
240C22—84no—
241C22—162no—
242C22—161no—
243C22—161no—
244C22#494711yescharacteristic
245C22#506211yescharacteristic
246C22#507711yescharacteristic
247C22#515812yes—
248C22#517212yes—
249C4—21no—
250C4—21no—
251C4—82no—
252C4—242no—
253C4—242no—
254C4#476312yes—
255C4#476412yes—
256C4#476612yes—
257C4#476812yes—
258C3#1035111yescharacteristic, complemented, Sylow
259C2—162no—
260C2#6964311yescharacteristic
261C2#7016412yes—
262C2#7119712yes—
263C2#7126511yescharacteristic
264C2#7128411yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1957C2
socleC23×C3#120
prime radicalC2×C3×#127C22
hypercenter#1957D3
hypocenterC3#10351
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
1. #74342C2semidirect
4. #67655C2semidirect
5. #67634C2semidirect
258. C3#10351semidirect

Maximal subgroups

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

subgroupindex
#743422
C3×#19572
#717612
#676552
#676342
#103513