Finite Group Databasebeta

← #10695877

Subgroups — #10695877

Order 768.

826 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C3×#3195C211yesmaximal, characteristic, complemented
2C2×#5046C212yesmaximal
3#71691C212yesmaximal, complemented
4#69988C211yesmaximal, characteristic
5#67688C211yesmaximal, characteristic, complemented
6#9990—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C2×C3×C22⋊Q8C2212yes—
9C22×#215—22no—
10C3×#483—22no—
11C3×#470—22no—
12C3×#451—22no—
13C3×#407C2211yescharacteristic
14#5207C2211yescharacteristic
15C2×#830C2212yes—
16C2×#823—22no—
17#5046—24no—
18#5046—24no—
19C2×#809—22no—
20#4528—24no—
21#3195D311yescharacteristic, complemented
22C2×#450—32no—
23#1783—31no—
24#1472—32no—
25#1298—31no—
26C24×D3—21no—
27C22×C3×Q8D412yes—
28C23×C3×C4C2311yescharacteristic
29C22×D12—22no—
30C22×D12—21no—
31C22×D12—22no—
32C22×D12—44no—
33C22×D12D412yes—
34C2×C3×Q16—22no—
35C2×C3×M16D412yes—
36C3×C22⋊Q8—22no—
37C3×C22⋊Q8—22no—
38C3×C22⋊Q8—22no—
39C3×C22⋊Q8—22no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4D412yes—
42C2×C3×C22⋊C4—22no—
43C2×#215—28no—
44C2×#215—48no—
45C2×#208—21no—
46C22×C3⋊C8—22no—
47C3×#106—22no—
48C3×#103—24no—
49C3×C22⋊C8—22no—
50#830—22no—
51#830—22no—
52#830D414yes—
53#823—44no—
54#809—28no—
55C23×D4—31no—
56C22×SD16—62no—
57C2×C22⋊Q8C2×D312yes—
58#483—22no—
59#470—22no—
60#451—22no—
61#450—64no—
62#450—64no—
63C2×#102—62no—
64C2×#100—62no—
65#407—31no—
66#407C2×D311yescharacteristic
67C2×C22⋊C8—32no—
68#324—64no—
69C24×C3#4111yescharacteristic
70C23×D3—22no—
71C23×D3—44no—
72C23×D3—42no—
73C23×D3—84no—
74C2×C3×Q8—22no—
75C2×C3×Q8—22no—
76C2×C3×Q8—44no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—42no—
80C22×C3×C4C2×D411yescharacteristic
81C22×C3×C4C2×D412yes—
82C22×C3×C4C2×D412yes—
83C22×C3×C4C2×D411yescharacteristic
84C2×D12—28no—
85C2×D12—24no—
86C2×D12—24no—
87C2×D12—44no—
88C2×D12—48no—
89C2×D12—44no—
90C2×D12—42no—
91C2×D12—41no—
92C2×D12—41no—
93C3×Q16—82no—
94C3×M16—42no—
95C2×C3×C8—22no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—22no—
98C3×C4⋊C4—42no—
99C3×C4⋊C4—42no—
100C3×C22⋊C4—42no—
101C3×C22⋊C4—42no—
102#215—416no—
103#208—42no—
104C2×C3⋊C8—24no—
105C2×C3⋊C8—44no—
106C2×C3⋊C8—41no—
107C25—61no—
108C22×Q8#6712yes—
109C22×D4—32no—
110C22×D4—62no—
111C22×D4—62no—
112C22×D4—61no—
113C22×D4—124no—
114C23×C4C22×D311yescharacteristic
115C2×Q16—22no—
116C2×SD16—68no—
117C2×SD16—128no—
118C2×M16—61no—
119C2×M16#6712yes—
120C22×C8—62no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4#6712yes—
127C2×C22⋊C4—22no—
128#106—22no—
129#103—24no—
130#102—68no—
131#100—124no—
132C22⋊C8—22no—
133C22⋊C8—34no—
134C22⋊C8—62no—
135C22⋊C8—62no—
136C23×C3—24no—
137C23×C3—41no—
138C23×C3#12111yescharacteristic
139C23×C3#12112yes—
140C22×D3—24no—
141C22×D3—48no—
142C22×D3—42no—
143C22×D3—42no—
144C22×D3—84no—
145C22×D3—816no—
146C22×D3—84no—
147C22×D3—84no—
148C3×Q8—44no—
149C3×Q8—44no—
150C2×C3×C4—24no—
151C2×C3×C4—22no—
152C2×C3×C4—41no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—44no—
157C2×C3×C4—82no—
158C2×C3×C4#12012yes—
159C2×C3×C4#12011yescharacteristic
160C2×C3×C4#12711yescharacteristic
161D12—28no—
162D12—44no—
163D12—44no—
164D12—44no—
165C3×C8—42no—
166C3⋊C8—44no—
167C3⋊C8—81no—
168C24—62no—
169C24—124no—
170C24—122no—
171C24—244no—
172C24#23711yescharacteristic
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—44no—
176C2×D4—68no—
177C2×D4—64no—
178C2×D4—64no—
179C2×D4—124no—
180C2×D4—122no—
181C2×D4—124no—
182C2×D4—128no—
183C2×D4—121no—
184C2×D4—121no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4D3×D411yescharacteristic
189C22×C4D3×D412yes—
190C22×C4C2×#6712yes—
191C22×C4C2×#6711yescharacteristic
192Q16—82no—
193SD16—1216no—
194M16—42no—
195M16—122no—
196C2×C8—22no—
197C2×C8—64no—
198C2×C8—124no—
199C2×C8—121no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—21no—
210C22×C3—44no—
211C22×C3#39211yescharacteristic
212C22×C3#47114yes—
213C2×D3—44no—
214C2×D3—816no—
215C2×D3—84no—
216C2×D3—84no—
217C2×D3—84no—
218C3×C4—22no—
219C3×C4—21no—
220C3×C4—82no—
221C3×C4—82no—
222C3×C4B(4,𝔽2)12yes—
223C23—24no—
224C23—41no—
225C23—64no—
226C23—128no—
227C23—122no—
228C23—122no—
229C23—244no—
230C23—244no—
231C23—2416no—
232C23—244no—
233C23#93811yescharacteristic
234C23#93912yes—
235Q8—44no—
236Q8—44no—
237D4—68no—
238D4—124no—
239D4—124no—
240D4—124no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—82no—
249C2×C4#93712yes—
250C2×C4#94011yescharacteristic
251C2×C4#95311yescharacteristic
252C8—42no—
253C8—124no—
254C8—241no—
255C2×C3—24no—
256C2×C3—41no—
257C2×C3#201912yes—
258C2×C3#202111yescharacteristic
259D3—84no—
260C22—22no—
261C22—22no—
262C22—21no—
263C22—22no—
264C22—44no—
265C22—124no—
266C22—2416no—
267C22—244no—
268C22—244no—
269C22—244no—
270C22#520314yes—
271C22#522311yescharacteristic
272C4—22no—
273C4—21no—
274C4—82no—
275C4—82no—
276C4#524412yes—
277C3#999011yescharacteristic, complemented, Sylow
278C2—24no—
279C2—41no—
280C2—244no—
281C2#7185111yescharacteristic
282C2#7210712yes—
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3195C2
socleC22×C3#392
prime radicalC23×D12C22
hypercenter#3195D3
hypocenterC3#9990
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
1. C3×#3195C2semidirect
3. #71691C2semidirect
5. #67688C2semidirect
21. #3195D3semidirect
277. C3#9990semidirect

Maximal subgroups

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

subgroupindex
C3×#31952
C2×#50462
#716912
#699882
#676882
#99903