Finite Group Databasebeta

← #10697436

Subgroups — #10697436

Order 768.

542 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 105 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#79305C212yesmaximal
2#78479C211yesmaximal, characteristic, complemented
3#70793C212yesmaximal, complemented
4#69823C211yesmaximal, characteristic
5C3×#1281C211yesmaximal, characteristic, complemented
6#10220—31nomaximal, Sylow
7C2×#874C2211yescharacteristic
8#5470C2212yes—
9C3×#431C2212yes—
10C2×C3×#97C2211yescharacteristic
11C2×#821—22no—
12#5106—22no—
13#4936C2211yescharacteristic
14#4848—24no—
15#4731—24no—
16#4702—22no—
17#4702—22no—
18#4694—22no—
19#4693—22no—
20#4467—24no—
21#3208—32no—
22#3162—31no—
23#1961—31no—
24#1518—32no—
25#1281D311yescharacteristic, complemented
26C22×D12—21no—
27C22×D12—22no—
28C22×C3×C8D412yes—
29C2×C3×C42C2311yescharacteristic
30#926—22no—
31C2×C4⋊Dic3—21no—
32#924—22no—
33#924—22no—
34C2×#204—28no—
35#874—22no—
36#874—21no—
37#874D412yes—
38#870—22no—
39#869D412yes—
40#869D412yes—
41C4×Dic6—22no—
42C4×Dic6—22no—
43C3×C4⋊C8—22no—
44C3×C22⋊C8D414yes—
45C3×#97—22no—
46C3×C4×C8—22no—
47C3⋊#95—22no—
48#821—28no—
49#821—44no—
50#818—24no—
51#817—22no—
52#817—22no—
53#533—32no—
54C2×#127—31no—
55#501—62no—
56#501—62no—
57#492—62no—
58#486—62no—
59#460—64no—
60#450—64no—
61#431C2×D312yes—
62#427—62no—
63C2×#102—62no—
64C2×#97C2×D311yescharacteristic
65#382—31no—
66#328—64no—
67C23×D3—41no—
68C22×C3×C4C2×D412yes—
69C22×C3×C4C2×D411yescharacteristic
70C22×Dic3—41no—
71C2×D12—24no—
72C2×D12—22no—
73C2×D12—42no—
74C2×D12—42no—
75C2×D12—41no—
76C2×D12—42no—
77C2×D12—84no—
78C2×Dic6—22no—
79C2×Dic6—22no—
80C2×C3×C8—22no—
81C2×C3×C8—22no—
82C2×C3×C8—24no—
83C2×C3×C8SD1614yescomplemented
84C3×C42C2×D412yes—
85C3×C42C2×D411yescharacteristic
86C3×C42#4111yescharacteristic
87#217—42no—
88C4⋊Dic3—22no—
89C4⋊Dic3—22no—
90C4⋊Dic3—22no—
91C4⋊Dic3—22no—
92C4⋊Dic3—42no—
93Dic3⋊C4—42no—
94Dic3⋊C4—42no—
95C4×Dic3—42no—
96#204—88no—
97C22×D4—61no—
98C22×D4—62no—
99C2×SD16—68no—
100C22×C8D1212yes—
101C4⋊Q8—32no—
102C4⋊Q8—32no—
103#127—32no—
104#127—62no—
105#127—61no—
106#125—62no—
107C22⋊Q8—62no—
108C22⋊Q8—62no—
109C4×Q8—62no—
110C4×Q8—62no—
111#117—62no—
112C2×C4⋊C4—61no—
113C2×C42C22×D311yescharacteristic
114#107—64no—
115#106—62no—
116#106—62no—
117C4⋊C8—22no—
118#102—68no—
119#102—124no—
120C22⋊C8D1214yes—
121#97—22no—
122C4×C8—22no—
123#95—62no—
124C23×C3#12711yescharacteristic
125C22×D3—42no—
126C22×D3—84no—
127C22×D3—81no—
128C22×D3—81no—
129C2×C3×C4—42no—
130C2×C3×C4#12012yes—
131C2×C3×C4#12011yescharacteristic
132C2×C3×C4#12112yes—
133C2×C3×C4#12111yescharacteristic
134C2×C3×C4C2×SD1612yes—
135C2×C3×C4#13712yes—
136C2×Dic3—42no—
137C2×Dic3—42no—
138C2×Dic3—82no—
139C2×Dic3—81no—
140D12—48no—
141D12—44no—
142D12—84no—
143D12—82no—
144Dic6—44no—
145Dic6—82no—
146C3×C8—24no—
147C3×C8—44no—
148C24—121no—
149C2×Q8—62no—
150C2×Q8—62no—
151C2×D4—64no—
152C2×D4—62no—
153C2×D4—121no—
154C2×D4—122no—
155C2×D4—122no—
156C2×D4—122no—
157C2×D4—244no—
158C22×C4—121no—
159C22×C4C2×D1212yes—
160C22×C4C2×D1211yescharacteristic
161SD16—248no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—24no—
165C2×C8#20414yescomplemented
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—122no—
171C4⋊C4—122no—
172C4⋊C4—122no—
173C22⋊C4—122no—
174C42—122no—
175C42D3×D412yes—
176C42D3×D411yescharacteristic
177C42#23911yescharacteristic
178C22×C3—22no—
179C22×C3#39211yescharacteristic
180C22×C3#49211yescharacteristic
181C22×C3#49711yescharacteristic
182C2×D3—84no—
183C2×D3—82no—
184C2×D3—162no—
185C2×D3—162no—
186C3×C4—22no—
187C3×C4—22no—
188C3×C4#44812yes—
189C3×C4#46112yes—
190Dic3—82no—
191Dic3—82no—
192Dic3—161no—
193C23—122no—
194C23—244no—
195C23—241no—
196C23—241no—
197C23#87411yescharacteristic
198Q8—124no—
199Q8—242no—
200D4—128no—
201D4—124no—
202D4—244no—
203D4—242no—
204C2×C4—42no—
205C2×C4—122no—
206C2×C4—122no—
207C2×C4—242no—
208C2×C4—241no—
209C2×C4#88212yes—
210C2×C4#88211yescharacteristic
211C2×C4#89512yes—
212C2×C4#89511yescharacteristic
213C2×C4C2×#20412yes—
214C2×C4#90912yes—
215C8—24no—
216C8—44no—
217C2×C3—21no—
218C2×C3—21no—
219C2×C3#200812yes—
220C2×C3#201311yescharacteristic
221D3—162no—
222C22—22no—
223C22—242no—
224C22—244no—
225C22—482no—
226C22—482no—
227C22#467111yescharacteristic
228C22#469311yescharacteristic
229C22#471511yescharacteristic
230C4—22no—
231C4—22no—
232C4—242no—
233C4—242no—
234C4—481no—
235C4#473812yes—
236C4#484112yes—
237C3#1022011yescharacteristic, complemented, Sylow
238C2—21no—
239C2—21no—
240C2—482no—
241C2#6868711yescharacteristic
242C2#6902812yes—
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1281C2
socleC22×C3#392
prime radicalC2×#874C22
hypercenter#1281D3
hypocenterC3#10220
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
2. #78479C2semidirect
3. #70793C2semidirect
5. C3×#1281C2semidirect
25. #1281D3semidirect
83. C2×C3×C8SD16semidirect
165. C2×C8#204semidirect
237. C3#10220semidirect

Maximal subgroups

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

subgroupindex
#793052
#784792
#707932
#698232
C3×#12812
#102203