Finite Group Databasebeta

← #10695867

Subgroups — #10695867

Order 768.

750 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#5227C212yesmaximal, complemented
2#78641C211yesmaximal, characteristic
3C3×#1784C211yesmaximal, characteristic, complemented
4#69970C212yesmaximal, complemented
5#67688C211yesmaximal, characteristic, complemented
6#9988—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C22×#215—22no—
9C2×#930C2212yes—
10C2×C3×#101—22no—
11C3×#407C2211yescharacteristic
12C3×#407C2211yescharacteristic
13#5227—24no—
14#5227—24no—
15C2×#835—22no—
16C2×#830C2212yes—
17#5135—22no—
18#4735—22no—
19#4732—22no—
20#4468—24no—
21#3192—31no—
22C2×#448—32no—
23#1784D311yescharacteristic, complemented
24#1471—32no—
25#1298—31no—
26C22×C3×Q8—21no—
27C22×C3×Q8—22no—
28C22×C3×Q8—22no—
29C22×C3×Q8—44no—
30C22×C3×Q8D412yes—
31C23×C3×C4—21no—
32C23×C3×C4C2311yescharacteristic
33C22×#67—22no—
34C22×D12D412yes—
35C2×C3×M16—21no—
36C2×C3×M16D412yes—
37C2×#215—28no—
38C2×#215—48no—
39#930—22no—
40#930—22no—
41#930—22no—
42#930—22no—
43C2×(C2×D3)⋊C4—22no—
44C2×C4⋊Dic3D412yes—
45C22×C3⋊C8—22no—
46C2×D24—22no—
47C3×#101—44no—
48C3×C22⋊C8—22no—
49C3×C22⋊C8—22no—
50#835—28no—
51#830—22no—
52#830D414yes—
53#821—24no—
54#817—22no—
55C23×Q8C2×D311yescharacteristic
56C22×SD16—62no—
57C2×#121—32no—
58#481—62no—
59#469—62no—
60#448—64no—
61#448—64no—
62#447—62no—
63C2×#103—62no—
64C2×#101—22no—
65#407C2×D311yescharacteristic
66#407C2×D311yescharacteristic
67C2×C22⋊C8—32no—
68#323—64no—
69C24×C3#4111yescharacteristic
70C23×D3—22no—
71C2×C3×Q8—24no—
72C2×C3×Q8—28no—
73C2×C3×Q8—24no—
74C2×C3×Q8—44no—
75C2×C3×Q8—44no—
76C2×C3×Q8—42no—
77C2×C3×Q8—48no—
78C2×C3×Q8—41no—
79C2×C3×Q8—41no—
80C22×C3×C4—22no—
81C22×C3×C4—42no—
82C22×C3×C4—44no—
83C22×C3×C4—42no—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D412yes—
87C22×C3×C4C2×D411yescharacteristic
88C2×#67—42no—
89C2×#67—42no—
90C2×#67—84no—
91C22×Dic3—22no—
92C2×D12—22no—
93C2×D12—22no—
94C2×D12—44no—
95C3×M16—42no—
96C3×M16—42no—
97C2×C3×C8—22no—
98C2×C3×C8—41no—
99#215—416no—
100(C2×D3)⋊C4—42no—
101(C2×D3)⋊C4—42no—
102C4⋊Dic3—22no—
103C4⋊Dic3—22no—
104C2×C3⋊C8—24no—
105C2×C3⋊C8—44no—
106D24—82no—
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—44no—
111C22×Q8D1212yes—
112C22×D4—32no—
113C22×D4—62no—
114C23×C4—21no—
115C23×C4C22×D311yescharacteristic
116C2×SD16—68no—
117C2×SD16—128no—
118C2×D8—62no—
119C2×M16—21no—
120C2×M16#6712yes—
121C22×C8—62no—
122#121—62no—
123#121—62no—
124#121—62no—
125#121—62no—
126C2×C4⋊C4—32no—
127C2×C22⋊C4—62no—
128#106—62no—
129#103—68no—
130#102—64no—
131#101—44no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—34no—
135C22⋊C8—62no—
136C23×C3—24no—
137C23×C3—41no—
138C23×C3#12112yes—
139C23×C3#12111yescharacteristic
140C22×D3—42no—
141C22×D3—44no—
142C22×D3—82no—
143C22×D3—82no—
144C3×Q8—28no—
145C3×Q8—44no—
146C3×Q8—44no—
147C3×Q8—44no—
148C2×C3×C4—24no—
149C2×C3×C4—22no—
150C2×C3×C4—24no—
151C2×C3×C4—41no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—48no—
155C2×C3×C4—42no—
156C2×C3×C4#12012yes—
157C2×C3×C4#12011yescharacteristic
158C2×C3×C4#12711yescharacteristic
159#67—84no—
160#67—84no—
161C2×Dic3—42no—
162C2×Dic3—82no—
163D12—44no—
164D12—44no—
165C3×C8—42no—
166C3×C8—81no—
167C3⋊C8—44no—
168C24—62no—
169C24#23511yescharacteristic
170C2×Q8—24no—
171C2×Q8—28no—
172C2×Q8—24no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—42no—
176C2×Q8—48no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×D4—62no—
180C2×D4—62no—
181C2×D4—124no—
182C2×D4—122no—
183C2×D4—122no—
184C2×D4—244no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—62no—
190C22×C4C2×D1211yescharacteristic
191C22×C4D3×D412yes—
192C22×C4D3×D412yes—
193C22×C4C2×#6711yescharacteristic
194SD16—1216no—
195D8—242no—
196M16—42no—
197M16—42no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—64no—
201C2×C8—124no—
202C4⋊C4—62no—
203C4⋊C4—62no—
204C22⋊C4—122no—
205C22⋊C4—122no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—21no—
210C22×C3—44no—
211C22×C3#39211yescharacteristic
212C22×C3#47114yes—
213C2×D3—82no—
214C2×D3—82no—
215C2×D3—82no—
216C2×D3—164no—
217C3×C4—22no—
218C3×C4—21no—
219C3×C4—44no—
220C3×C4B(4,𝔽2)12yes—
221Dic3—82no—
222C23—24no—
223C23—41no—
224C23—122no—
225C23—124no—
226C23—242no—
227C23—242no—
228C23#88412yes—
229C23#93011yescharacteristic
230Q8—28no—
231Q8—44no—
232Q8—44no—
233Q8—44no—
234D4—124no—
235D4—124no—
236D4—244no—
237D4—244no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—48no—
245C2×C4—42no—
246C2×C4—122no—
247C2×C4—242no—
248C2×C4#88212yes—
249C2×C4#93711yescharacteristic
250C2×C4#94011yescharacteristic
251C8—42no—
252C8—81no—
253C8—124no—
254C2×C3—24no—
255C2×C3—41no—
256C2×C3#201912yes—
257C2×C3#202111yescharacteristic
258D3—162no—
259C22—22no—
260C22—22no—
261C22—21no—
262C22—22no—
263C22—44no—
264C22—242no—
265C22—242no—
266C22—242no—
267C22—484no—
268C22#475414yes—
269C22#496011yescharacteristic
270C4—21no—
271C4—22no—
272C4—44no—
273C4—242no—
274C4#503212yes—
275C3#998811yescharacteristic, complemented, Sylow
276C2—24no—
277C2—41no—
278C2—482no—
279C2#6950011yescharacteristic
280C2#7011712yes—
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1784C2
socleC22×C3#392
prime radical#78641C2
hypercenter#1784D3
hypocenterC3#9988
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
1. C2×#5227C2semidirect
3. C3×#1784C2semidirect
4. #69970C2semidirect
5. #67688C2semidirect
23. #1784D3semidirect
275. C3#9988semidirect

Maximal subgroups

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

subgroupindex
C2×#52272
#786412
C3×#17842
#699702
#676882
#99883