Finite Group Databasebeta

← #10943841

Subgroups — #10943841

Order 768.

3798 conjugacy classes of subgroups; only the 323 normal subgroups are shown.

normal subgroups only

This group has 3798 conjugacy classes of subgroups — too many to draw. Only its 323 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
1C24×Dic6C211yesmaximal, characteristic
2C3×#2848C211yesmaximal, characteristic
3C2×#5050C216yesmaximal, complemented
4#72126C211yesmaximal, characteristic
5#69525C216yesmaximal, complemented
6#32882—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C24×Dic3—21no—
9C23×Dic6—23no—
10C23×Dic6—224no—
11C23×Dic6C2216yes—
12C2×C3×C22⋊Q8C2213yes—
13C22×#216C2214yescomplemented
14C3×#385—23no—
15C3×#380C2213yes—
16C3×#379—21no—
17C2×#830C2216yes—
18#5050—224no—
19C2×#810C22112yescomplemented
20C24×Q8—31no—
21C2×#451—36no—
22#2848D311yescharacteristic
23#1787—36no—
24#1779—31no—
25C25×C3D411yescharacteristic
26C22×C3×Q8C2311yescharacteristic
27C23×C3×C4—212no—
28C23×C3×C4D413yes—
29C23×C3×C4C2313yes—
30C23×Dic3—224no—
31C23×Dic3—26no—
32C22×Dic6—224no—
33C22×Dic6—248no—
34C22×Dic6—2128no—
35C22×Dic6—212no—
36C22×Dic6—26no—
37C22×Dic6—248no—
38C22×Dic6D4124yes—
39C22×Dic6C2314yes—
40C3×C22⋊Q8—212no—
41C2×C3×C4⋊C4—23no—
42C2×C3×C4⋊C4C2313yes—
43C2×C3×C22⋊C4—23no—
44C2×C3×C22⋊C4—23no—
45C2×#216—224no—
46C22×C3⋊C8C2314yes—
47C3×#95—23no—
48#830—212no—
49#810—224no—
50C23×Q8—36no—
51C23×Q8—624no—
52C23×Q8—63no—
53C24×C4—61no—
54C24×C4C2×D311yescharacteristic
55C22×Q16—34no—
56C2×C22⋊Q8C2×D313yes—
57#451—624no—
58C2×#103—312no—
59C2×C22⋊C8—36no—
60#385—23no—
61#380C2×D313yes—
62#379—21no—
63C24×C3—26no—
64C24×C3—26no—
65C24×C3Q1614yescomplemented
66C24×C3C2×D413yes—
67C2×C3×Q8—26no—
68C22×C3×C4—232no—
69C22×C3×C4—26no—
70C22×C3×C4—212no—
71C22×C3×C4—212no—
72C22×C3×C4—23no—
73C22×C3×C4—21no—
74C22×C3×C4C2×D4112yes—
75C22×C3×C4C2×D413yes—
76C22×C3×C4C2411yescharacteristic
77C22×Dic3—264no—
78C22×Dic3—248no—
79C22×Dic3—224no—
80C22×Dic3—24no—
81C2×Dic6—2192no—
82C2×Dic6—248no—
83C2×Dic6—296no—
84C2×Dic6—224no—
85C2×Dic6—212no—
86C2×Dic6—296no—
87C2×Dic6C2×D4124yes—
88C3×C4⋊C4—26no—
89C3×C4⋊C4—46no—
90C3×C22⋊C4—46no—
91C3×C22⋊C4—46no—
92#216—416no—
93C2×C3⋊C8—212no—
94C25#6711yescharacteristic
95C22×Q8—324no—
96C22×Q8—34no—
97C22×Q8—648no—
98C22×Q8—6128no—
99C22×Q8—624no—
100C22×Q8—648no—
101C22×Q8—612no—
102C22×Q8—66no—
103C22×Q8C22×D311yescharacteristic
104C23×C4—212no—
105C23×C4—624no—
106C23×C4—66no—
107C23×C4#6713yes—
108C23×C4C22×D313yes—
109C2×Q16—624no—
110C22×C8—34no—
111C22⋊Q8—212no—
112C2×C4⋊C4—23no—
113C2×C4⋊C4C22×D313yes—
114C2×C22⋊C4—23no—
115C2×C22⋊C4—23no—
116#103—624no—
117C22⋊C8—612no—
118#95—23no—
119C23×C3—28no—
120C23×C3—224no—
121C23×C3—26no—
122C23×C3—26no—
123C23×C3—224no—
124C23×C3#12016yes—
125C23×C3C2×Q1616yes—
126C23×C3#13716yes—
127C23×C3C22×D411yescharacteristic
128C3×Q8—44no—
129C2×C3×C4—224no—
130C2×C3×C4—26no—
131C2×C3×C4—224no—
132C2×C3×C4—43no—
133C2×C3×C4—43no—
134C2×C3×C4—46no—
135C2×C3×C4#12016yes—
136C2×C3×C4C22×D416yes—
137C2×Dic3—296no—
138C2×Dic3—224no—
139Dic6—296no—
140Dic6—224no—
141Dic6#120116yes—
142C3⋊C8—44no—
143C24—26no—
144C24—26no—
145C24#21614yescomplemented
146C24C2×#6713yes—
147C2×Q8—26no—
148C2×Q8—324no—
149C2×Q8—696no—
150C2×Q8—6192no—
151C2×Q8—648no—
152C2×Q8—696no—
153C2×Q8—624no—
154C2×Q8—612no—
155C22×C4—232no—
156C22×C4—26no—
157C22×C4—212no—
158C22×C4—212no—
159C22×C4—23no—
160C22×C4—21no—
161C22×C4—664no—
162C22×C4—648no—
163C22×C4—624no—
164C22×C4—64no—
165C22×C4D3×D4112yes—
166C22×C4C2×#6713yes—
167C22×C4C23×D311yescharacteristic
168Q16—1216no—
169C2×C8—612no—
170C4⋊C4—26no—
171C4⋊C4—46no—
172C22⋊C4—46no—
173C22⋊C4—46no—
174C22×C3—232no—
175C22×C3—26no—
176C22×C3—212no—
177C22×C3—26no—
178C22×C3—212no—
179C22×C3#451112yescomplemented
180C22×C3C2×#12013yes—
181C22×C3C22×Q1611yescharacteristic
182C22×C3C2×#13713yes—
183C3×C4—26no—
184C3×C4—81no—
185C3×C4—83no—
186C3×C4C2×#12014yes—
187Dic3—216no—
188C23—28no—
189C23—224no—
190C23—26no—
191C23—224no—
192C23—26no—
193C23#93716yes—
194C23C2×#21616yes—
195C23#95116yes—
196C23C22×#6711yescharacteristic
197Q8—316no—
198Q8—44no—
199Q8—696no—
200Q8—624no—
201C2×C4—224no—
202C2×C4—26no—
203C2×C4—224no—
204C2×C4—43no—
205C2×C4—43no—
206C2×C4—46no—
207C2×C4—696no—
208C2×C4—624no—
209C2×C4#93716yes—
210C2×C4C2×D3×D416yes—
211C8—124no—
212C2×C3—212no—
213C2×C3#284611yescharacteristic
214C2×C3C2×#45113yes—
215C2×C3#300413yes—
216C22—232no—
217C22—212no—
218C22—26no—
219C22—26no—
220C22—212no—
221C22#5050112yescomplemented
222C22C2×#93713yes—
223C22C22×#21611yescharacteristic
224C22C2×#95113yes—
225C4—26no—
226C4—616no—
227C4—81no—
228C4—83no—
229C4D3×#12014yes—
230C3#3288211yescharacteristic, complemented, Sylow
231C2—212no—
232C2#8261211yescharacteristic
233C2C2×#505013yes—
234C2#8293313yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2848C2
socleC23×C3C22×D4
prime radicalC25×C3D4
hypercenter#2848D3
hypocenterC3#32882
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
3. C2×#5050C2semidirect
5. #69525C2semidirect
13. C22×#216C22semidirect
19. C2×#810C22semidirect
65. C24×C3Q16semidirect
145. C24#216semidirect
179. C22×C3#451semidirect
221. C22#5050semidirect
230. C3#32882semidirect

Maximal subgroups

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

subgroupindex
C24×Dic62
C3×#28482
C2×#50502
#721262
#695252
#328823