Finite Group Databasebeta

← #10698213

Subgroups — #10698213

Order 768.

500 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 121 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
2#70457C212yesmaximal
3#69580C211yesmaximal, characteristic
4C3×#1296C211yesmaximal, characteristic, complemented
5#67704C212yesmaximal, complemented
6#10351—31nomaximal, Sylow
7C2×#874C2211yescharacteristic
8C2×C3×C4⋊C8C2212yes—
9C3×#384C2211yescharacteristic
10C2×#821—22no—
11C2×#821—22no—
12#4986—22no—
13#4984—22no—
14#4977—22no—
15C2×#809—22no—
16C4×C4⋊Dic3C2212yes—
17C2×C4×C3⋊C8C2211yescharacteristic
18#2571—31no—
19#1982—32no—
20#1957—31no—
21#1296D311yescharacteristic, complemented
22#1279—32no—
23C22×D12—21no—
24C22×D12—22no—
25C22×C3×C8—22no—
26C2×C3×C4⋊C4—22no—
27C2×C3×C4⋊C4D412yes—
28C2×C3×C42C2311yescharacteristic
29C2×C4⋊Dic3—22no—
30C2×C4⋊Dic3D412yes—
31C2×C4⋊Dic3D412yes—
32C2×C4×Dic3—22no—
33C22×C3⋊C8—21no—
34#874—22no—
35#874—21no—
36#874Q812yes—
37C3×C4⋊C8—22no—
38C3×C4⋊C8—22no—
39C3⋊#95—22no—
40C3⋊#95—24no—
41C3⋊#95—22no—
42#821—44no—
43#821—44no—
44#809—44no—
45C4×C3⋊C8—22no—
46C2×#127—31no—
47C2×C4⋊C8C2×D312yes—
48C2×#102—62no—
49C2×#102—62no—
50C2×#102—62no—
51C2×C4×C8—31no—
52#400—62no—
53#389—62no—
54#384C2×D311yescharacteristic
55#382—62no—
56C4×C4⋊C4—32no—
57C23×D3—41no—
58C22×C3×C4—22no—
59C22×C3×C4C2×D411yescharacteristic
60C22×C3×C4C2×D412yes—
61C22×Dic3—22no—
62C22×Dic3—22no—
63C2×D12—24no—
64C2×D12—22no—
65C2×D12—42no—
66C2×D12—42no—
67C2×D12—42no—
68C2×D12—42no—
69C2×D12—42no—
70C2×D12—42no—
71C2×D12—41no—
72C2×D12—42no—
73C2×C3×C8—42no—
74C2×C3×C8—42no—
75C2×C3×C8—42no—
76C3×C4⋊C4—42no—
77C3×C4⋊C4—42no—
78C3×C4⋊C4—42no—
79C3×C42C2×Q811yescharacteristic
80C3×C42#4112yes—
81C3×C42#4111yescharacteristic
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86C4⋊Dic3—82no—
87C4×Dic3—42no—
88C4×Dic3—42no—
89C2×C3⋊C8—22no—
90C2×C3⋊C8—41no—
91C2×C3⋊C8—41no—
92C22×D4—61no—
93C22×D4—62no—
94C22×C8—22no—
95C22×C8—61no—
96#127—32no—
97#127—62no—
98#127—61no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—62no—
103C2×C4⋊C4D1212yes—
104C2×C42—62no—
105C2×C42C22×D311yescharacteristic
106C4⋊C8—22no—
107C4⋊C8—22no—
108#102—124no—
109#102—124no—
110#102—124no—
111C4×C8—62no—
112#95—62no—
113#95—64no—
114#95—62no—
115C23×C3#12011yescharacteristic
116C22×D3—42no—
117C22×D3—82no—
118C22×D3—82no—
119C22×D3—81no—
120C22×D3—81no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—44no—
124C2×C3×C4—42no—
125C2×C3×C4C22⋊Q812yes—
126C2×C3×C4C22⋊Q811yescharacteristic
127C2×C3×C4#12411yescharacteristic
128C2×C3×C4#12412yes—
129C2×C3×C4#13512yes—
130C2×C3×C4#13512yes—
131C2×C3×C4AGL(1,ℤ/8)12yes—
132C2×C3×C4AGL(1,ℤ/8)12yes—
133C2×Dic3—42no—
134C2×Dic3—42no—
135C2×Dic3—42no—
136C2×Dic3—42no—
137C2×Dic3—42no—
138C2×Dic3—42no—
139D12—44no—
140D12—44no—
141D12—44no—
142D12—82no—
143D12—82no—
144D12—82no—
145C3×C8—82no—
146C3⋊C8—42no—
147C24—121no—
148C2×D4—64no—
149C2×D4—62no—
150C2×D4—122no—
151C2×D4—122no—
152C2×D4—121no—
153C2×D4—122no—
154C2×D4—122no—
155C2×D4—122no—
156C2×D4—122no—
157C2×D4—122no—
158C22×C4—22no—
159C22×C4—62no—
160C22×C4—62no—
161C22×C4C2×D1211yescharacteristic
162C22×C4D3×D412yes—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—62no—
167C2×C8—121no—
168C2×C8—121no—
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—
179C42#23512yes—
180C42#23711yescharacteristic
181C42D3×Q811yescharacteristic
182C22×C3#39311yescharacteristic
183C22×C3#44912yes—
184C22×C3#44912yes—
185C22×C3#45311yescharacteristic
186C22×C3#45411yescharacteristic
187C2×D3—84no—
188C2×D3—82no—
189C2×D3—161no—
190C2×D3—161no—
191C2×D3—162no—
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—122no—
202C23—241no—
203C23—241no—
204C23—242no—
205C23—242no—
206C23#88211yescharacteristic
207D4—124no—
208D4—124no—
209D4—124no—
210D4—242no—
211D4—242no—
212D4—242no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—122no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4—122no—
221C2×C4—122no—
222C2×C4—122no—
223C2×C4#87511yescharacteristic
224C2×C4#88512yes—
225C2×C4#89612yes—
226C2×C4#89711yescharacteristic
227C2×C4#91112yes—
228C2×C4#91412yes—
229C2×C4#91612yes—
230C2×C4#91912yes—
231C8—82no—
232C8—122no—
233C2×C3#162011yescharacteristic
234C2×C3#162411yescharacteristic
235C2×C3#162812yes—
236C2×C3#202512yes—
237C2×C3#202711yescharacteristic
238D3—162no—
239C22—244no—
240C22—242no—
241C22—481no—
242C22—482no—
243C22—481no—
244C22#467311yescharacteristic
245C22#474711yescharacteristic
246C22#475311yescharacteristic
247C22#478112yes—
248C22#481012yes—
249C4—21no—
250C4—21no—
251C4—82no—
252C4—242no—
253C4—242no—
254C4#476412yes—
255C4#480012yes—
256C4#487112yes—
257C4#489112yes—
258C3#1035111yescharacteristic, complemented, Sylow
259C2—482no—
260C2#6840211yescharacteristic
261C2#6843312yes—
262C2#6849111yescharacteristic
263C2#6889912yes—
264C2#6899511yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socleC23×C3#120
prime radicalC2×#874C22
hypercenter#1296D3
hypocenterC3#10351
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
1. #74342C2semidirect
4. C3×#1296C2semidirect
5. #67704C2semidirect
21. #1296D3semidirect
258. C3#10351semidirect

Maximal subgroups

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

subgroupindex
#743422
#704572
#695802
C3×#12962
#677042
#103513