Finite Group Databasebeta

← #10697863

Subgroups — #10697863

Order 768.

566 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 113 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#79312C212yesmaximal
2#73308C211yesmaximal, characteristic, complemented
3#70779C212yesmaximal, complemented
4#69395C211yesmaximal, characteristic
5C3×#1276C211yesmaximal, characteristic, complemented
6#10287—31nomaximal, Sylow
7#5477C2211yescharacteristic
8C2×#869C2212yes—
9C3×#431C2212yes—
10C3×C43C2211yescharacteristic
11#4936—22no—
12#4929—24no—
13C2×#805—22no—
14#4924C2211yescharacteristic
15#4841—24no—
16#4738—24no—
17#4693—22no—
18#4681—22no—
19#4481—24no—
20#3206—32no—
21#2862—31no—
22#1962—31no—
23#1548—32no—
24#1276D311yescharacteristic, complemented
25C2×#235—21no—
26C22×Dic6D414yes—
27C2×C3×C42—22no—
28C2×C3×C42D412yes—
29C2×C3×C42C2311yescharacteristic
30#930—22no—
31C2×C4⋊Dic3—22no—
32C2×C4⋊Dic3—22no—
33C2×#208—21no—
34C2×#204—28no—
35#875—21no—
36#874D411yescharacteristic
37C4×D12—22no—
38#869—22no—
39#869—22no—
40#869—22no—
41#869—22no—
42#869D411yescharacteristic
43C3×C4⋊C8D414yes—
44C3×C22⋊C8—22no—
45C3×C4×C8—22no—
46C3⋊#95—28no—
47#816—28no—
48#805—44no—
49C4⋊(C3⋊C8)—22no—
50#532—31no—
51C2×C4⋊Q8—32no—
52#492—62no—
53#487—62no—
54#461—64no—
55#448—64no—
56#431C2×D312yes—
57#423—31no—
58C2×#104—62no—
59#391—64no—
60#382—62no—
61C43C2×D311yescharacteristic
62#333—64no—
63C22×C3×C4—22no—
64C22×C3×C4C2×D411yescharacteristic
65C22×C3×C4C2×D412yes—
66C2×#67—41no—
67C22×Dic3—24no—
68#235—42no—
69#235—42no—
70C2×D12—22no—
71C2×D12—21no—
72C2×C4×D3—41no—
73C2×Dic6—24no—
74C2×Dic6—21no—
75C2×Dic6—24no—
76C2×Dic6—48no—
77C2×C3×C8—24no—
78C3×C42—22no—
79C3×C42—44no—
80C3×C42SD1614yescomplemented
81C3×C42C2×D412yes—
82C3×C42C2×D411yescharacteristic
83C3×C42#4111yescharacteristic
84(C2×D3)⋊C4—42no—
85C4⋊Dic3—22no—
86C4⋊Dic3—42no—
87C4⋊Dic3—42no—
88C4⋊Dic3—42no—
89C4⋊Dic3—42no—
90#208—42no—
91C2×C3⋊C8—41no—
92#204—88no—
93C2×#41—61no—
94C22×Q8—34no—
95C2×SD16—68no—
96C2×M16—61no—
97C4⋊Q8—31no—
98C4⋊Q8—62no—
99C4⋊Q8—62no—
100C4⋊Q8—62no—
101C4⋊Q8—62no—
102#127—31no—
103#124—61no—
104#121—62no—
105C4×D4—62no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4—62no—
108C2×C42—22no—
109C2×C42D1212yes—
110C2×C42C22×D311yescharacteristic
111C4⋊C8—62no—
112C4⋊C8D1214yes—
113#104—124no—
114#103—68no—
115C22⋊C8—22no—
116C4×C8—22no—
117#95—68no—
118C23×C3#12011yescharacteristic
119C22×D3—41no—
120C2×C3×C4—24no—
121C2×C3×C4—24no—
122C2×C3×C4—44no—
123C2×C3×C4#12012yes—
124C2×C3×C4#12112yes—
125C2×C3×C4#12111yescharacteristic
126C2×C3×C4#12711yescharacteristic
127C2×C3×C4C2×SD1612yes—
128C2×C3×C4#13712yes—
129#67—82no—
130C2×Dic3—48no—
131C2×Dic3—41no—
132C2×Dic3—44no—
133D12—44no—
134D12—42no—
135C4×D3—81no—
136C4×D3—81no—
137Dic6—48no—
138Dic6—42no—
139Dic6—84no—
140C3×C8—44no—
141C3⋊C8—81no—
142#41—122no—
143#41—122no—
144C2×Q8—64no—
145C2×Q8—61no—
146C2×Q8—64no—
147C2×Q8—128no—
148C2×D4—62no—
149C2×D4—61no—
150C2×D4—121no—
151C22×C4—22no—
152C22×C4—64no—
153C22×C4—121no—
154C22×C4C2×D1211yescharacteristic
155C22×C4D3×D412yes—
156SD16—248no—
157M16—122no—
158C2×C8—24no—
159C2×C8—121no—
160C4⋊C4—62no—
161C4⋊C4—122no—
162C4⋊C4—122no—
163C4⋊C4—122no—
164C4⋊C4—122no—
165C22⋊C4—122no—
166C42—22no—
167C42—44no—
168C42#20414yescomplemented
169C42C2×D1212yes—
170C42D3×D411yescharacteristic
171C42#23911yescharacteristic
172C22×C3—22no—
173C22×C3#39211yescharacteristic
174C22×C3#44812yes—
175C2×D3—82no—
176C2×D3—81no—
177C3×C4—22no—
178C3×C4—44no—
179C3×C4#45514yes—
180C3×C4#46112yes—
181C3×C4#49211yescharacteristic
182C3×C4#49711yescharacteristic
183Dic3—84no—
184Dic3—81no—
185C23—121no—
186C23#88211yescharacteristic
187Q8—128no—
188Q8—122no—
189Q8—244no—
190D4—124no—
191D4—122no—
192D4—242no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—44no—
196C2×C4—128no—
197C2×C4—121no—
198C2×C4—124no—
199C2×C4—241no—
200C2×C4—241no—
201C2×C4#87411yescharacteristic
202C2×C4#88212yes—
203C2×C4#89512yes—
204C2×C4#89511yescharacteristic
205C2×C4C2×#20412yes—
206C2×C4#90912yes—
207C8—44no—
208C8—241no—
209C2×C3—22no—
210C2×C3#200412yes—
211C2×C3#201311yescharacteristic
212D3—161no—
213C22—22no—
214C22—241no—
215C22—242no—
216C22#467111yescharacteristic
217C22#473812yes—
218C4—22no—
219C4—44no—
220C4—241no—
221C4—244no—
222C4#469311yescharacteristic
223C4#471511yescharacteristic
224C4#482514yes—
225C4#484112yes—
226C3#1028711yescharacteristic, complemented, Sylow
227C2—22no—
228C2—481no—
229C2#6868711yescharacteristic
230C2#6876212yes—
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1276D3
hypocenterC3#10287
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
2. #73308C2semidirect
3. #70779C2semidirect
5. C3×#1276C2semidirect
24. #1276D3semidirect
80. C3×C42SD16semidirect
168. C42#204semidirect
226. C3#10287semidirect

Maximal subgroups

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

subgroupindex
#793122
#733082
#707792
#693952
C3×#12762
#102873