Finite Group Databasebeta

← #10697395

Subgroups — #10697395

Order 768.

498 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 81 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#79352C212yesmaximal, complemented
2#79064C211yesmaximal, characteristic
3#70832C212yesmaximal, complemented
4#70515C211yesmaximal, characteristic, complemented
5C3×#1307C211yesmaximal, characteristic, complemented
6#10213—31nomaximal, Sylow
7#5477C2212yes—
8#5470C2211yescharacteristic
9C3×#432C2212yes—
10C3×#423C2211yescharacteristic
11#5135—22no—
12#4932C2211yescharacteristic
13#4849—22no—
14#4848—22no—
15#4843—22no—
16#4842—22no—
17#4733—24no—
18#4715—22no—
19#4714—22no—
20#4555—22no—
21#4470—24no—
22#3253—31no—
23#3216—32no—
24#1995—31no—
25#1505—32no—
26#1307D311yescharacteristic, complemented
27C2×#235—22no—
28C2×C3×M16D412yes—
29C2×C3×C42C2311yescharacteristic
30#930—22no—
31#930—22no—
32#926—21no—
33#924—22no—
34C2×C4×Dic3—21no—
35C2×Dic12—24no—
36C2×D24—22no—
37C2×#204—24no—
38#875—22no—
39#874D412yes—
40C4×D12—22no—
41C4×D12—22no—
42#870—21no—
43#869D412yes—
44#869D412yes—
45C4×Dic6—22no—
46C3×C4⋊C8—22no—
47C3×C4⋊C8—22no—
48C3×C22⋊C8D414yes—
49C3×#97—22no—
50C3⋊#95—22no—
51#821—24no—
52#821—24no—
53#817—22no—
54#816—24no—
55#533—31no—
56#532—32no—
57#497—62no—
58#489—62no—
59#476—62no—
60#469—62no—
61#462—62no—
62#460—62no—
63#459—62no—
64#452—64no—
65#432C2×D312yes—
66#423C2×D311yescharacteristic
67#376—31no—
68#339—62no—
69#327—64no—
70C22×C3×C4C2×D411yescharacteristic
71C22×C3×C4C2×D412yes—
72C2×#67—42no—
73C22×Dic3—41no—
74#235—84no—
75C2×D12—22no—
76C2×D12—22no—
77C2×D12—22no—
78C2×C4×D3—42no—
79C2×Dic6—22no—
80C2×Dic6—22no—
81C3×M16—42no—
82C2×C3×C8—24no—
83C2×C3×C8—22no—
84C3×C42C2×D411yescharacteristic
85C3×C42C2×D412yes—
86C3×C42#4111yescharacteristic
87#217—41no—
88(C2×D3)⋊C4—42no—
89(C2×D3)⋊C4—42no—
90C4⋊Dic3—22no—
91C4⋊Dic3—22no—
92C4⋊Dic3—22no—
93C4⋊Dic3—22no—
94Dic3⋊C4—42no—
95C4×Dic3—22no—
96C4×Dic3—41no—
97C4×Dic3—41no—
98Dic12—84no—
99D24—82no—
100#204—84no—
101C2×#41—62no—
102C2×Q16—64no—
103C2×SD16—64no—
104C2×D8—62no—
105C2×M16D1212yes—
106C4⋊Q8—32no—
107C4⋊Q8—32no—
108#127—32no—
109#125—61no—
110#124—62no—
111C22⋊Q8—62no—
112#121—62no—
113#121—62no—
114C4×Q8—62no—
115C4×D4—62no—
116C4×D4—62no—
117#117—61no—
118C2×C42—61no—
119C2×C42C22×D311yescharacteristic
120#106—62no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123#103—64no—
124#102—64no—
125#102—64no—
126C22⋊C8D1214yes—
127#97—22no—
128#95—62no—
129C23×C3#12711yescharacteristic
130C22×D3—42no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—42no—
134C2×C3×C4#12012yes—
135C2×C3×C4#12011yescharacteristic
136C2×C3×C4#12112yes—
137C2×C3×C4#12111yescharacteristic
138#67—84no—
139C2×Dic3—42no—
140C2×Dic3—42no—
141C2×Dic3—42no—
142C2×Dic3—82no—
143D12—44no—
144D12—42no—
145D12—42no—
146D12—82no—
147C4×D3—84no—
148Dic6—42no—
149Dic6—42no—
150Dic6—44no—
151C3×C8—44no—
152C3×C8—42no—
153#41—244no—
154C2×Q8—62no—
155C2×Q8—62no—
156C2×D4—62no—
157C2×D4—62no—
158C2×D4—62no—
159C2×D4—122no—
160C22×C4—121no—
161C22×C4—122no—
162C22×C4C2×D1211yescharacteristic
163C22×C4C2×D1212yes—
164Q16—244no—
165SD16—244no—
166D8—242no—
167M16—42no—
168C2×C8—24no—
169C2×C8—22no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4—122no—
175C22⋊C4—121no—
176C22⋊C4—122no—
177C22⋊C4—122no—
178C42—62no—
179C42—121no—
180C42—121no—
181C42D3×D411yescharacteristic
182C42D3×D412yes—
183C42#23911yescharacteristic
184C22×C3—21no—
185C22×C3—22no—
186C22×C3#39211yescharacteristic
187C2×D3—82no—
188C2×D3—82no—
189C2×D3—82no—
190C3×C4—21no—
191C3×C4—22no—
192C3×C4—22no—
193C3×C4#45511yescharacteristic
194C3×C4#45611yescharacteristic
195Dic3—82no—
196Dic3—82no—
197Dic3—82no—
198C23—122no—
199C23#87411yescharacteristic
200Q8—122no—
201Q8—122no—
202Q8—124no—
203D4—124no—
204D4—122no—
205D4—122no—
206D4—242no—
207D4—244no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—122no—
212C2×C4—122no—
213C2×C4—122no—
214C2×C4—244no—
215C2×C4—242no—
216C2×C4#88212yes—
217C2×C4#88211yescharacteristic
218C2×C4#89511yescharacteristic
219C2×C4#89512yes—
220C8—44no—
221C8—42no—
222C2×C3—41no—
223C2×C3#200711yescharacteristic
224C2×C3#200812yes—
225D3—162no—
226C22—21no—
227C22—22no—
228C22—242no—
229C22—242no—
230C22—242no—
231C22#467111yescharacteristic
232C4—21no—
233C4—22no—
234C4—22no—
235C4—242no—
236C4—242no—
237C4—242no—
238C4#474611yescharacteristic
239C4#475211yescharacteristic
240C3#1021311yescharacteristic, complemented, Sylow
241C2—41no—
242C2—482no—
243C2#6900911yescharacteristic
244C2#6902812yes—
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1307C2
socleC22×C3#392
prime radical#79064C2
hypercenter#1307D3
hypocenterC3#10213
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
1. #79352C2semidirect
3. #70832C2semidirect
4. #70515C2semidirect
5. C3×#1307C2semidirect
26. #1307D3semidirect
240. C3#10213semidirect

Maximal subgroups

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

subgroupindex
#793522
#790642
#708322
#705152
C3×#13072
#102133