Finite Group Databasebeta

← #10695879

Subgroups — #10695879

Order 768.

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

normal subgroups only

This group has 826 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×#5218C212yesmaximal
2#78646C211yesmaximal, characteristic, complemented
3C3×#1783C211yesmaximal, characteristic
4#69972C212yesmaximal, complemented
5#67688C211yesmaximal, characteristic, complemented
6#9990—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C22×#214—22no—
9C2×#924C2212yes—
10C2×C3×#100—22no—
11C3×#407C2211yescharacteristic
12C3×#407C2211yescharacteristic
13#5218—24no—
14#5218—24no—
15C2×#832—22no—
16C2×#830C2212yes—
17#5137—22no—
18#4739—22no—
19#4721—22no—
20#4469—24no—
21#3195—31no—
22C2×#450—32no—
23#1783D311yescharacteristic
24#1472—32no—
25#1298—31no—
26C25×C3—21no—
27C22×C3×D4—22no—
28C22×C3×D4—21no—
29C22×C3×D4—22no—
30C22×C3×D4—44no—
31C22×C3×D4D412yes—
32C23×C3×C4C2311yescharacteristic
33C22×Dic6D412yes—
34C2×C3×M16—21no—
35C2×C3×M16D412yes—
36C2×#217—22no—
37C2×#214—28no—
38C2×#214—48no—
39C2×C4⋊Dic3D412yes—
40#924—22no—
41#924—22no—
42#924—22no—
43#924—22no—
44C2×Dic3⋊C4—22no—
45C22×C3⋊C8—22no—
46C2×Dic12—22no—
47C3×#100—44no—
48C3×C22⋊C8—22no—
49C3×C22⋊C8—22no—
50#832—28no—
51#830—22no—
52#830D414yes—
53#817—22no—
54#816—24no—
55C23×D4C2×D311yescharacteristic
56C22×SD16—62no—
57C2×C22⋊Q8—32no—
58#483—62no—
59#470—62no—
60#451—62no—
61#450—64no—
62#450—64no—
63C2×#102—62no—
64C2×#100—22no—
65#407C2×D311yescharacteristic
66#407C2×D311yescharacteristic
67C2×C22⋊C8—32no—
68#324—64no—
69C24×C3—22no—
70C24×C3—44no—
71C24×C3—42no—
72C24×C3—84no—
73C24×C3#4111yescharacteristic
74C2×C3×D4—28no—
75C2×C3×D4—24no—
76C2×C3×D4—24no—
77C2×C3×D4—44no—
78C2×C3×D4—42no—
79C2×C3×D4—44no—
80C2×C3×D4—48no—
81C2×C3×D4—41no—
82C2×C3×D4—41no—
83C22×C3×C4—42no—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D412yes—
87C22×C3×C4C2×D411yescharacteristic
88C22×Dic3—22no—
89C22×Dic3—22no—
90C2×Dic6—22no—
91C2×Dic6—22no—
92C2×Dic6—44no—
93C3×M16—42no—
94C3×M16—42no—
95C2×C3×C8—22no—
96C2×C3×C8—41no—
97#217—42no—
98#217—42no—
99#214—416no—
100C4⋊Dic3—22no—
101C4⋊Dic3—22no—
102Dic3⋊C4—42no—
103Dic3⋊C4—42no—
104C2×C3⋊C8—24no—
105C2×C3⋊C8—44no—
106Dic12—82no—
107C25—21no—
108C22×Q8—32no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—21no—
112C22×D4—44no—
113C22×D4D1212yes—
114C23×C4C22×D311yescharacteristic
115C2×Q16—62no—
116C2×SD16—68no—
117C2×SD16—128no—
118C2×M16—21no—
119C2×M16#6712yes—
120C22×C8—62no—
121C22⋊Q8—62no—
122C22⋊Q8—62no—
123C22⋊Q8—62no—
124C22⋊Q8—62no—
125C2×C4⋊C4—32no—
126C2×C4⋊C4—62no—
127C2×C22⋊C4—62no—
128#106—62no—
129#103—64no—
130#102—68no—
131#100—44no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—34no—
135C22⋊C8—62no—
136C23×C3—24no—
137C23×C3—24no—
138C23×C3—41no—
139C23×C3—48no—
140C23×C3—42no—
141C23×C3—42no—
142C23×C3—84no—
143C23×C3—84no—
144C23×C3—816no—
145C23×C3—84no—
146C23×C3#12112yes—
147C23×C3#12111yescharacteristic
148C3×D4—28no—
149C3×D4—44no—
150C3×D4—44no—
151C3×D4—44no—
152C2×C3×C4—24no—
153C2×C3×C4—22no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4#12012yes—
157C2×C3×C4#12011yescharacteristic
158C2×C3×C4#12711yescharacteristic
159C2×Dic3—42no—
160C2×Dic3—42no—
161C2×Dic3—44no—
162C2×Dic3—82no—
163Dic6—44no—
164Dic6—44no—
165C3×C8—42no—
166C3×C8—81no—
167C3⋊C8—44no—
168C24—22no—
169C24—44no—
170C24—42no—
171C24—84no—
172C24#23511yescharacteristic
173C2×Q8—62no—
174C2×Q8—62no—
175C2×Q8—124no—
176C2×D4—28no—
177C2×D4—24no—
178C2×D4—24no—
179C2×D4—44no—
180C2×D4—42no—
181C2×D4—44no—
182C2×D4—48no—
183C2×D4—41no—
184C2×D4—41no—
185C22×C4—42no—
186C22×C4—62no—
187C22×C4—62no—
188C22×C4C2×D1211yescharacteristic
189C22×C4D3×D412yes—
190C22×C4D3×D412yes—
191C22×C4C2×#6711yescharacteristic
192Q16—242no—
193SD16—1216no—
194M16—42no—
195M16—42no—
196C2×C8—22no—
197C2×C8—41no—
198C2×C8—64no—
199C2×C8—124no—
200C4⋊C4—62no—
201C4⋊C4—62no—
202C4⋊C4—122no—
203C4⋊C4—122no—
204C22⋊C4—122no—
205C22⋊C4—122no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—21no—
210C22×C3—44no—
211C22×C3—44no—
212C22×C3—816no—
213C22×C3—84no—
214C22×C3—84no—
215C22×C3—84no—
216C22×C3#39211yescharacteristic
217C22×C3#47114yes—
218C3×C4—22no—
219C3×C4—21no—
220C3×C4B(4,𝔽2)12yes—
221Dic3—82no—
222Dic3—82no—
223C23—24no—
224C23—24no—
225C23—41no—
226C23—48no—
227C23—42no—
228C23—42no—
229C23—84no—
230C23—84no—
231C23—816no—
232C23—84no—
233C23#88412yes—
234C23#93011yescharacteristic
235Q8—124no—
236Q8—124no—
237D4—28no—
238D4—44no—
239D4—44no—
240D4—44no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—122no—
246C2×C4—122no—
247C2×C4—124no—
248C2×C4—242no—
249C2×C4#88212yes—
250C2×C4#93711yescharacteristic
251C2×C4#94011yescharacteristic
252C8—42no—
253C8—81no—
254C8—124no—
255C2×C3—24no—
256C2×C3—41no—
257C2×C3—84no—
258C2×C3#201912yes—
259C2×C3#202111yescharacteristic
260C22—22no—
261C22—22no—
262C22—21no—
263C22—22no—
264C22—44no—
265C22—44no—
266C22—816no—
267C22—84no—
268C22—84no—
269C22—84no—
270C22#474814yes—
271C22#496011yescharacteristic
272C4—21no—
273C4—22no—
274C4—242no—
275C4—242no—
276C4#503212yes—
277C3#999011yescharacteristic, complemented, Sylow
278C2—24no—
279C2—41no—
280C2—84no—
281C2#6950011yescharacteristic
282C2#7009112yes—
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1783C2
socleC22×C3#392
prime radicalC23×C3×D4C22
hypercenter#1783D3
hypocenterC3#9990
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1783

Complements

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

NH ≅ G/Ntype
2. #78646C2semidirect
4. #69972C2semidirect
5. #67688C2semidirect
277. C3#9990semidirect

Maximal subgroups

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

subgroupindex
C2×#52182
#786462
C3×#17832
#699722
#676882
#99903