Finite Group Databasebeta

← #10695850

Subgroups — #10695850

Order 768.

502 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 79 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#82220C212yesmaximal
2#74114C211yesmaximal, characteristic
3C3×#1820C211yesmaximal, characteristic
4#69533C212yesmaximal
5#67629C211yesmaximal, characteristic
6#9985—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#924C2212yes—
9C3×#381C2211yescharacteristic
10C3×#381C2211yescharacteristic
11#5207C2212yes—
12#5004—22no—
13#4996—24no—
14#4995—22no—
15#4971—22no—
16#4970—22no—
17#4953—22no—
18#4659—22no—
19#4656—22no—
20C3×#342—22no—
21#4452—24no—
22#3095—32no—
23#2817—31no—
24#1820D311yescharacteristic
25#1511—32no—
26#1306—31no—
27C23×C3×C4C2311yescharacteristic
28C22×Dic6D412yes—
29C3×#117—22no—
30C3×#117—22no—
31C2×C3×C4⋊C4D412yes—
32C2×C3×C22⋊C4—21no—
33C2×C3×C22⋊C4—21no—
34C2×C3×C22⋊C4D412yes—
35C2×C3×C42—21no—
36C2×#217—22no—
37C2×C4⋊Dic3—22no—
38C2×C4⋊Dic3D412yes—
39#924—22no—
40#924—22no—
41#924—22no—
42#924—22no—
43C2×Dic3⋊C4—22no—
44C2×#208—22no—
45C3×#95—22no—
46C3×#95—22no—
47C3⋊#95—24no—
48C3⋊#95—22no—
49#830—22no—
50#830D414yes—
51#810—24no—
52#810—24no—
53#808—24no—
54#807—24no—
55C2×C22⋊Q8—32no—
56C2×#117C2×D311yescharacteristic
57#485—64no—
58#484—62no—
59#483—62no—
60#428—62no—
61#419—62no—
62#407—32no—
63#400—62no—
64#397—62no—
65#395—62no—
66#381C2×D311yescharacteristic
67#381C2×D311yescharacteristic
68#342—22no—
69#328—64no—
70C24×C3C2×D411yescharacteristic
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4—42no—
74C22×C3×C4—41no—
75C22×C3×C4C2×D411yescharacteristic
76C22×C3×C4C2×D412yes—
77C22×C3×C4C2×D412yes—
78C22×C3×C4#4111yescharacteristic
79C22×Dic3—22no—
80C22×Dic3—22no—
81C2×Dic6—22no—
82C2×Dic6—22no—
83C2×Dic6—44no—
84C3×C4⋊C4—22no—
85C3×C4⋊C4—22no—
86C3×C22⋊C4—41no—
87C3×C22⋊C4—41no—
88C3×C22⋊C4—42no—
89C3×C22⋊C4—42no—
90C3×C42—42no—
91#217—42no—
92#217—42no—
93C4⋊Dic3—22no—
94C4⋊Dic3—22no—
95C4⋊Dic3—82no—
96Dic3⋊C4—42no—
97Dic3⋊C4—42no—
98#208—44no—
99C2×C3⋊C8—24no—
100C22×Q8—32no—
101C23×C4C22×D311yescharacteristic
102C2×M16—62no—
103C22⋊Q8—62no—
104C22⋊Q8—62no—
105C22⋊Q8—62no—
106C22⋊Q8—62no—
107#117—22no—
108#117—22no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4D1212yes—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—62no—
116C2×C22⋊C4#6712yes—
117C2×C42—21no—
118#107—64no—
119#106—64no—
120#103—64no—
121#103—64no—
122C22⋊C8—34no—
123C22⋊C8—62no—
124#95—22no—
125#95—22no—
126#95—64no—
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—41no—
135C2×C3×C4—42no—
136C2×C3×C4—41no—
137C2×C3×C4—44no—
138C2×C3×C4—42no—
139C2×C3×C4—42no—
140C2×C3×C4—82no—
141C2×C3×C4—81no—
142C2×C3×C4—82no—
143C2×C3×C4#12012yes—
144C2×C3×C4#12112yes—
145C2×Dic3—44no—
146C2×Dic3—42no—
147C2×Dic3—42no—
148C2×Dic3—44no—
149Dic6—44no—
150Dic6—82no—
151C3⋊C8—44no—
152C24C2×#6711yescharacteristic
153C2×Q8—62no—
154C2×Q8—62no—
155C2×Q8—124no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—62no—
161C22×C4—62no—
162C22×C4C2×D1211yescharacteristic
163C22×C4#23511yescharacteristic
164C22×C4D3×D412yes—
165C22×C4D3×D412yes—
166M16—124no—
167C2×C8—64no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—122no—
173C4⋊C4—122no—
174C4⋊C4—242no—
175C22⋊C4—41no—
176C22⋊C4—41no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—122no—
180C22⋊C4—122no—
181C42—42no—
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—
193C3×C4—82no—
194C3×C4—161no—
195Dic3—82no—
196Dic3—82no—
197C23—22no—
198C23—42no—
199C23#93011yescharacteristic
200C23#93711yescharacteristic
201C23#94011yescharacteristic
202Q8—124no—
203Q8—242no—
204C2×C4—24no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4—81no—
213C2×C4—82no—
214C2×C4—124no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—124no—
218C2×C4#88212yes—
219C2×C4#88412yes—
220C8—124no—
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#496011yescharacteristic
233C22#505312yes—
234C4—22no—
235C4—22no—
236C4—82no—
237C4—82no—
238C4—161no—
239C4—242no—
240C4—242no—
241C3#998511yescharacteristic, complemented, Sylow
242C2—22no—
243C2—41no—
244C2—41no—
245C2#6953911yescharacteristic
246C2#7005312yes—
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1820C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#1820D3
hypocenterC3#9985
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

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
#822202
#741142
C3×#18202
#695332
#676292
#99853