Finite Group Databasebeta

← #10695851

Subgroups — #10695851

Order 768.

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

normal subgroups only

This group has 502 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#78625C212yesmaximal
2#74114C211yesmaximal, characteristic
3#69981C212yesmaximal
4#69534C211yesmaximal, characteristic
5C3×#1306C211yesmaximal, characteristic
6#9985—31nomaximal, Sylow
7C2×#926C2211yescharacteristic
8C2×#924C2212yes—
9C3×#407C2212yes—
10C3×#381C2211yescharacteristic
11#5211C2211yescharacteristic
12#5121—22no—
13#5120—22no—
14#4953—22no—
15#4723—24no—
16#4722—22no—
17#4721—22no—
18#4659—22no—
19#4656—22no—
20#4527—22no—
21#4467—24no—
22#3095—32no—
23#2817—31no—
24#1820—31no—
25#1511—32no—
26#1306D311yescharacteristic
27C23×C3×C4C2311yescharacteristic
28C22×Dic6D412yes—
29C2×C3×M16—22no—
30C2×C3×C22⋊C4D412yes—
31C2×#217—22no—
32C2×#217—21no—
33C2×#217—21no—
34#926—22no—
35#926—22no—
36C2×C4⋊Dic3—22no—
37C2×C4⋊Dic3D412yes—
38C2×C4⋊Dic3D412yes—
39#924—22no—
40#924—22no—
41#924—22no—
42#924—22no—
43C2×Dic3⋊C4—22no—
44C2×C4×Dic3—21no—
45C3×C22⋊C8—22no—
46C3×C22⋊C8D414yes—
47C3×#95—22no—
48C3⋊#95—24no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51#818—24no—
52#817—24no—
53#816—24no—
54#816—24no—
55C2×C22⋊Q8—32no—
56C2×#117—31no—
57#485—64no—
58#484—62no—
59#483—62no—
60#428—62no—
61#419—62no—
62#407C2×D312yes—
63#400—62no—
64#397—62no—
65#395—62no—
66#381—31no—
67#381C2×D311yescharacteristic
68#342—62no—
69#328—64no—
70C24×C3C2×D411yescharacteristic
71C22×C3×C4—22no—
72C22×C3×C4—42no—
73C22×C3×C4C2×D412yes—
74C22×C3×C4C2×D412yes—
75C22×C3×C4C2×D411yescharacteristic
76C22×C3×C4#4111yescharacteristic
77C22×Dic3—22no—
78C22×Dic3—22no—
79C22×Dic3—22no—
80C22×Dic3—41no—
81C2×Dic6—22no—
82C2×Dic6—22no—
83C2×Dic6—44no—
84C3×M16—44no—
85C2×C3×C8—24no—
86C3×C22⋊C4—42no—
87#217—41no—
88#217—42no—
89#217—42no—
90#217—41no—
91#217—42no—
92C4⋊Dic3—22no—
93C4⋊Dic3—22no—
94C4⋊Dic3—22no—
95C4⋊Dic3—22no—
96C4⋊Dic3—82no—
97Dic3⋊C4—42no—
98Dic3⋊C4—42no—
99C4×Dic3—42no—
100C22×Q8—32no—
101C23×C4C22×D311yescharacteristic
102C2×M16—22no—
103C22⋊Q8—62no—
104C22⋊Q8—62no—
105C22⋊Q8—62no—
106C22⋊Q8—62no—
107#117—62no—
108#117—62no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4—62no—
113C2×C22⋊C4—61no—
114C2×C22⋊C4—62no—
115C2×C22⋊C4—61no—
116C2×C22⋊C4D1212yes—
117C2×C42—61no—
118#107—64no—
119#106—64no—
120#103—64no—
121#103—64no—
122C22⋊C8—22no—
123C22⋊C8D1214yes—
124#95—22no—
125#95—64no—
126#95—62no—
127#95—62no—
128C23×C3—22no—
129C23×C3—42no—
130C23×C3#12011yescharacteristic
131C23×C3#12111yescharacteristic
132C23×C3#12711yescharacteristic
133C2×C3×C4—24no—
134C2×C3×C4—42no—
135C2×C3×C4—41no—
136C2×C3×C4—41no—
137C2×C3×C4—44no—
138C2×C3×C4—42no—
139C2×C3×C4#12012yes—
140C2×C3×C4#12112yes—
141C2×Dic3—44no—
142C2×Dic3—42no—
143C2×Dic3—42no—
144C2×Dic3—42no—
145C2×Dic3—44no—
146C2×Dic3—82no—
147C2×Dic3—81no—
148C2×Dic3—82no—
149Dic6—44no—
150Dic6—82no—
151C3×C8—44no—
152C24C2×D1211yescharacteristic
153C2×Q8—62no—
154C2×Q8—62no—
155C2×Q8—124no—
156C22×C4—22no—
157C22×C4—42no—
158C22×C4—62no—
159C22×C4—62no—
160C22×C4—62no—
161C22×C4—121no—
162C22×C4C2×D1212yes—
163C22×C4D3×D412yes—
164C22×C4D3×D411yescharacteristic
165C22×C4#23911yescharacteristic
166M16—44no—
167C2×C8—24no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—122no—
173C4⋊C4—122no—
174C4⋊C4—242no—
175C22⋊C4—42no—
176C22⋊C4—121no—
177C22⋊C4—122no—
178C22⋊C4—121no—
179C22⋊C4—122no—
180C22⋊C4—122no—
181C42—122no—
182C22×C3—22no—
183C22×C3—22no—
184C22×C3—22no—
185C22×C3—21no—
186C22×C3—21no—
187C22×C3—82no—
188C22×C3#39211yescharacteristic
189C22×C3#45812yes—
190C3×C4—22no—
191C3×C4—22no—
192C3×C4—82no—
193Dic3—82no—
194Dic3—82no—
195Dic3—82no—
196Dic3—161no—
197C23—22no—
198C23—42no—
199C23#87411yescharacteristic
200C23#88211yescharacteristic
201C23#89511yescharacteristic
202Q8—124no—
203Q8—242no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—124no—
211C2×C4—122no—
212C2×C4—122no—
213C2×C4—122no—
214C2×C4—124no—
215C2×C4—242no—
216C2×C4—241no—
217C2×C4—242no—
218C2×C4#88212yes—
219C2×C4#89512yes—
220C8—44no—
221C2×C3—22no—
222C2×C3—41no—
223C2×C3—41no—
224C2×C3#200912yes—
225C2×C3#202211yescharacteristic
226C22—22no—
227C22—21no—
228C22—21no—
229C22—22no—
230C22—22no—
231C22—82no—
232C22#467111yescharacteristic
233C22#474212yes—
234C4—22no—
235C4—22no—
236C4—82no—
237C4—242no—
238C4—242no—
239C4—242no—
240C4—481no—
241C3#998511yescharacteristic, complemented, Sylow
242C2—22no—
243C2—41no—
244C2—41no—
245C2#6896611yescharacteristic
246C2#6903112yes—
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1306C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#1306D3
hypocenterC3#9985
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
241. C3#9985semidirect

Maximal subgroups

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

subgroupindex
#786252
#741142
#699812
#695342
C3×#13062
#99853