Finite Group Databasebeta

← #10695848

Subgroups — #10695848

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
1C3×#2817C211yesmaximal, characteristic
2#82727C212yesmaximal
3#71689C212yesmaximal
4#69534C211yesmaximal, characteristic
5#67629C211yesmaximal, characteristic
6#9985—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2212yes—
8C2×#926C2211yescharacteristic
9C3×#400—22no—
10C3×#397—22no—
11C3×#395—22no—
12C3×#381C2211yescharacteristic
13#5225—22no—
14#5211C2211yescharacteristic
15#5207C2212yes—
16#5045—24no—
17#5044—22no—
18#5043—22no—
19#4964—22no—
20#4538—24no—
21#4527—22no—
22#3095—32no—
23#2817D311yescharacteristic
24#1820—31no—
25#1511—32no—
26#1306—31no—
27C22×C3×Q8D412yes—
28C23×C3×C4C2311yescharacteristic
29C3×C22⋊Q8—22no—
30C3×C22⋊Q8—22no—
31C3×C22⋊Q8—22no—
32C3×C22⋊Q8—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4D412yes—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4D412yes—
38C2×#217—21no—
39C2×#217—21no—
40#926—22no—
41#926—22no—
42C2×C4⋊Dic3D412yes—
43C2×C4×Dic3—21no—
44C2×#208—22no—
45C3×#95—24no—
46C3×#95—22no—
47C3×#95—22no—
48#835—24no—
49#835—24no—
50C3⋊#95—22no—
51#830—22no—
52#830D414yes—
53#808—24no—
54#807—24no—
55C2×C22⋊Q8C2×D312yes—
56C2×#117—31no—
57#485—64no—
58#484—62no—
59#483—62no—
60#428—62no—
61#419—62no—
62#407—32no—
63#400—22no—
64#397—22no—
65#395—22no—
66#381—31no—
67#381C2×D311yescharacteristic
68#342—62no—
69#328—64no—
70C24×C3C2×D411yescharacteristic
71C2×C3×Q8—22no—
72C2×C3×Q8—22no—
73C2×C3×Q8—44no—
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—42no—
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4C2×D412yes—
80C22×C3×C4C2×D412yes—
81C22×C3×C4#4111yescharacteristic
82C22×Dic3—22no—
83C22×Dic3—41no—
84C3×C4⋊C4—22no—
85C3×C4⋊C4—22no—
86C3×C4⋊C4—42no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4—82no—
89C3×C22⋊C4—42no—
90C3×C22⋊C4—42no—
91C3×C22⋊C4—42no—
92#217—41no—
93#217—41no—
94#217—42no—
95C4⋊Dic3—22no—
96C4⋊Dic3—22no—
97C4×Dic3—42no—
98#208—44no—
99C2×C3⋊C8—24no—
100C22×Q8#6712yes—
101C23×C4C22×D311yescharacteristic
102C2×M16—62no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C22⋊Q8—22no—
107#117—62no—
108#117—62no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—32no—
112C2×C4⋊C4#6712yes—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—61no—
115C2×C22⋊C4—61no—
116C2×C22⋊C4#6712yes—
117C2×C42—61no—
118#107—64no—
119#106—64no—
120#103—64no—
121#103—64no—
122C22⋊C8—34no—
123C22⋊C8—62no—
124#95—24no—
125#95—22no—
126#95—22no—
127#95—62no—
128C23×C3—22no—
129C23×C3—42no—
130C23×C3#12011yescharacteristic
131C23×C3#12111yescharacteristic
132C23×C3#12711yescharacteristic
133C3×Q8—44no—
134C3×Q8—82no—
135C2×C3×C4—24no—
136C2×C3×C4—41no—
137C2×C3×C4—42no—
138C2×C3×C4—41no—
139C2×C3×C4—44no—
140C2×C3×C4—44no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4—42no—
144C2×C3×C4—44no—
145C2×C3×C4#12012yes—
146C2×C3×C4#12112yes—
147C2×Dic3—42no—
148C2×Dic3—82no—
149C2×Dic3—81no—
150C2×Dic3—82no—
151C3⋊C8—44no—
152C24C2×#6711yescharacteristic
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—44no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—42no—
160C22×C4—62no—
161C22×C4—121no—
162C22×C4D3×D411yescharacteristic
163C22×C4D3×D412yes—
164C22×C4#23711yescharacteristic
165C22×C4C2×#6712yes—
166M16—124no—
167C2×C8—64no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4—82no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—121no—
179C22⋊C4—121no—
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—
193C3×C4—82no—
194C3×C4—82no—
195Dic3—82no—
196Dic3—161no—
197C23—22no—
198C23—42no—
199C23#93811yescharacteristic
200C23#94011yescharacteristic
201C23#95311yescharacteristic
202Q8—44no—
203Q8—82no—
204C2×C4—24no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—122no—
215C2×C4—242no—
216C2×C4—241no—
217C2×C4—242no—
218C2×C4#93712yes—
219C2×C4#93912yes—
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#522311yescharacteristic
233C22#524312yes—
234C4—22no—
235C4—22no—
236C4—82no—
237C4—82no—
238C4—82no—
239C4—242no—
240C4—481no—
241C3#998511yescharacteristic, complemented, Sylow
242C2—22no—
243C2—41no—
244C2—41no—
245C2#7184911yescharacteristic
246C2#7186912yes—
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2817C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#2817D3
hypocenterC3#9985
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2817

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
C3×#28172
#827272
#716892
#695342
#676292
#99853