Finite Group Databasebeta

← #10693022

Subgroups — #10693022

Order 768.

518 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2484C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic
3#71590C212yesmaximal
4#71552C211yesmaximal, characteristic
5#67628C212yesmaximal
6#9482—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C3×#400C412yescomplemented
10C3×#398C412yescomplemented
11C3×#387—22no—
12C3×#385—21no—
13C3×#381—21no—
14C3×#381C2211yescharacteristic
15#5212—21no—
16#5211—21no—
17C4×#217C2212yes—
18#5207C2212yes—
19#4980—24no—
20#4532—24no—
21#4528—22no—
22#2484D311yescharacteristic
23#2313—31no—
24#1826—32no—
25#1463—31no—
26#1284—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C3×#117—22no—
31C3×#117—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4C2×C411yescharacteristic
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C4⋊C4D412yes—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C42—21no—
42C2×#217—22no—
43C2×C4⋊Dic3—22no—
44C2×C4×Dic3D414yes—
45C2×#208—22no—
46C3×#95—22no—
47C3×#95—22no—
48C3×#95—22no—
49C3×#95—21no—
50C3×#95C2×C412yes—
51C3⋊#95—24no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54#830—22no—
55#830—22no—
56C2×#117C2×D311yescharacteristic
57C22×C4⋊C4C2×D311yescharacteristic
58#407—32no—
59#400Dic312yescomplemented
60#398Dic312yescomplemented
61#387—22no—
62#387—64no—
63#385—21no—
64#381—21no—
65#381—61no—
66#381C2×D311yescharacteristic
67#380—61no—
68C4×C22⋊C4—32no—
69#339—64no—
70#324—62no—
71C24×C3C2×D411yescharacteristic
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—22no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—42no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4C22⋊C412yes—
83C22×C3×C4C22⋊C412yes—
84C22×C3×C4C22×C411yescharacteristic
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4#4112yes—
88C22×Dic3—24no—
89C3×C4⋊C4—24no—
90C3×C4⋊C4—24no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—44no—
94C3×C22⋊C4—42no—
95C3×C22⋊C4—41no—
96C3×C22⋊C4—41no—
97C3×C42—42no—
98#217—44no—
99C4⋊Dic3—82no—
100C4×Dic3—44no—
101#208—44no—
102C2×C3⋊C8—42no—
103C23×C4#6711yescharacteristic
104C23×C4#6711yescharacteristic
105C23×C4C22×D311yescharacteristic
106C2×M16—62no—
107#117—22no—
108#117—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—62no—
113C2×C4⋊C4C2×Dic311yescharacteristic
114C2×C4⋊C4C2×Dic311yescharacteristic
115C2×C4⋊C4C2×Dic312yes—
116C2×C4⋊C4#6712yes—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—62no—
120C2×C42—21no—
121C2×C42—34no—
122C22⋊C8—62no—
123C22⋊C8—62no—
124#95—22no—
125#95—22no—
126#95—22no—
127#95—21no—
128#95—62no—
129#95—64no—
130#95—62no—
131#95C2×Dic312yes—
132C23×C3—22no—
133C23×C3—22no—
134C23×C3—41no—
135C23×C3C2×C22⋊C411yescharacteristic
136C23×C3#12011yescharacteristic
137C23×C3#12311yescharacteristic
138C2×C3×C4—24no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—41no—
144C2×C3×C4—41no—
145C2×C3×C4—42no—
146C2×C3×C4—42no—
147C2×C3×C4—42no—
148C2×C3×C4—42no—
149C2×C3×C4—42no—
150C2×C3×C4—42no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—81no—
158C2×C3×C4C4×D412yes—
159C2×C3×C4#12112yes—
160C2×Dic3—44no—
161C2×Dic3—44no—
162C2×Dic3—44no—
163C3⋊C8—82no—
164C24C2×#6711yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—64no—
176C22×C4#21712yes—
177C22×C4#21712yes—
178C22×C4D3×D412yes—
179C22×C4#23712yes—
180C22×C4C22×Dic311yescharacteristic
181C22×C4C2×#6711yescharacteristic
182M16—124no—
183C2×C8—122no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—44no—
189C4⋊C4—242no—
190C22⋊C4—42no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C22⋊C4—124no—
194C42—42no—
195C42—124no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—44no—
204C22×C3#38611yescharacteristic
205C22×C3#42112yes—
206C22×C3#45812yes—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4—42no—
210C3×C4—42no—
211C3×C4—82no—
212C3×C4—81no—
213C3×C4—81no—
214Dic3—84no—
215C23—22no—
216C23—22no—
217C23—41no—
218C23#93511yescharacteristic
219C23#93711yescharacteristic
220C23C2×#21711yescharacteristic
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—81no—
241C2×C4—124no—
242C2×C4—124no—
243C2×C4—124no—
244C2×C4D4×Dic312yes—
245C2×C4#93912yes—
246C8—242no—
247C2×C3—22no—
248C2×C3—22no—
249C2×C3—41no—
250C2×C3#189011yescharacteristic
251C2×C3#189711yescharacteristic
252C2×C3#191211yescharacteristic
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—44no—
261C22#505312yes—
262C22#522011yescharacteristic
263C22#523612yes—
264C4—22no—
265C4—22no—
266C4—42no—
267C4—42no—
268C4—82no—
269C4—81no—
270C4—81no—
271C4—244no—
272C3#948211yescharacteristic, complemented, Sylow
273C2—22no—
274C2—22no—
275C2—41no—
276C2#7170411yescharacteristic
277C2#7171211yescharacteristic
278C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2484C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2484D3
hypocenterC3#9482
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
9. C3×#400C4semidirect
10. C3×#398C4semidirect
59. #400Dic3semidirect
60. #398Dic3semidirect
272. C3#9482semidirect

Maximal subgroups

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

subgroupindex
C3×#24842
#821552
#715902
#715522
#676282
#94823