Finite Group Databasebeta

← #10693021

Subgroups — #10693021

Order 768.

518 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2313C211yesmaximal, characteristic
2#82176C211yesmaximal, characteristic
3#71552C211yesmaximal, characteristic
4#69442C212yesmaximal
5#67616C212yesmaximal
6#9482—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#926C2211yescharacteristic
9C3×#387—24no—
10C3×#381—21no—
11C3×#380—21no—
12C3×C4×C22⋊C4C2212yes—
13#5211—21no—
14#5211C2211yescharacteristic
15#5210—21no—
16#5207C2212yes—
17#4986C412yescomplemented
18#4981C412yescomplemented
19#4978—22no—
20#4528—22no—
21#4523—24no—
22#2484—31no—
23#2313D311yescharacteristic
24#1826—32no—
25#1463—31no—
26#1284—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4C2×C412yes—
34C2×C3×C4⋊C4D412yes—
35C2×C3×C22⋊C4—22no—
36C2×C3×C42D414yes—
37C2×#217—21no—
38C2×#217—21no—
39#926—22no—
40#926—22no—
41C2×C4⋊Dic3C2×C411yescharacteristic
42C2×C4⋊Dic3C2×C411yescharacteristic
43C2×Dic3⋊C4—22no—
44C2×C4×Dic3—21no—
45C2×#208—22no—
46C3×#95—22no—
47C3×#95—24no—
48C3×#95—22no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51C3⋊#95—22no—
52C3⋊#95—21no—
53C3⋊#95C2×C412yes—
54#830—22no—
55#830—22no—
56C2×#117—31no—
57C22×C4⋊C4C2×D311yescharacteristic
58#407—32no—
59#400—32no—
60#398—32no—
61#387—24no—
62#387—62no—
63#385—61no—
64#381—21no—
65#381—31no—
66#381—61no—
67#380—21no—
68C4×C22⋊C4C2×D312yes—
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—24no—
77C22×C3×C4—42no—
78C22×C3×C4—42no—
79C22×C3×C4—42no—
80C22×C3×C4C22⋊C412yes—
81C22×C3×C4C22⋊C412yes—
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D412yes—
85C22×C3×C4#4112yes—
86C22×Dic3—22no—
87C22×Dic3—21no—
88C22×Dic3—21no—
89C3×C4⋊C4—24no—
90C3×C4⋊C4—24no—
91C3×C4⋊C4—82no—
92C3×C22⋊C4—44no—
93C3×C42—44no—
94#217—42no—
95#217—41no—
96#217—41no—
97C4⋊Dic3—22no—
98C4⋊Dic3—22no—
99Dic3⋊C4—44no—
100C4×Dic3—42no—
101#208—44no—
102C2×C3⋊C8—42no—
103C23×C4#6711yescharacteristic
104C23×C4#6711yescharacteristic
105C23×C4C22×D311yescharacteristic
106C2×M16—62no—
107#117—62no—
108#117—62no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—62no—
115C2×C4⋊C4C4×D312yes—
116C2×C4⋊C4D1212yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—61no—
119C2×C22⋊C4—61no—
120C2×C42—61no—
121C2×C42#6714yes—
122C22⋊C8—62no—
123C22⋊C8—62no—
124#95—22no—
125#95—24no—
126#95—22no—
127#95—32no—
128#95—62no—
129#95—62no—
130#95—62no—
131#95—61no—
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—44no—
151C2×C3×C4—44no—
152C2×C3×C4—44no—
153C2×C3×C4—81no—
154C2×C3×C4C4×D412yes—
155C2×C3×C4#12112yes—
156C2×Dic3—42no—
157C2×Dic3—42no—
158C2×Dic3—42no—
159C2×Dic3—41no—
160C2×Dic3—41no—
161C2×Dic3—42no—
162C2×Dic3—42no—
163C3⋊C8—82no—
164C24C2×#6711yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—24no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—62no—
174C22×C4—61no—
175C22×C4—61no—
176C22×C4(C2×D3)⋊C412yes—
177C22×C4(C2×D3)⋊C412yes—
178C22×C4C2×C4×D311yescharacteristic
179C22×C4C2×D1211yescharacteristic
180C22×C4#23512yes—
181C22×C4C2×#6712yes—
182M16—124no—
183C2×C8—122no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—62no—
187C4⋊C4—62no—
188C4⋊C4—82no—
189C4⋊C4—124no—
190C22⋊C4—44no—
191C22⋊C4—122no—
192C22⋊C4—121no—
193C22⋊C4—121no—
194C42—44no—
195C42—122no—
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—84no—
212Dic3—82no—
213Dic3—81no—
214Dic3—81no—
215C23—22no—
216C23—22no—
217C23—41no—
218C23C2×(C2×D3)⋊C411yescharacteristic
219C23#92911yescharacteristic
220C23#95311yescharacteristic
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—81no—
237C2×C4—122no—
238C2×C4—122no—
239C2×C4—122no—
240C2×C4—121no—
241C2×C4—121no—
242C2×C4—122no—
243C2×C4—122no—
244C2×C4C4×#6712yes—
245C2×C4#93012yes—
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#500512yes—
262C22#521311yescharacteristic
263C22#524312yes—
264C4—22no—
265C4—22no—
266C4—42no—
267C4—42no—
268C4—84no—
269C4—242no—
270C4—241no—
271C4—241no—
272C3#948211yescharacteristic, complemented, Sylow
273C2—22no—
274C2—22no—
275C2—41no—
276C2#7159111yescharacteristic
277C2#7165511yescharacteristic
278C2#7166311yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2313C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2313D3
hypocenterC3#9482
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
17. #4986C4semidirect
18. #4981C4semidirect
272. C3#9482semidirect

Maximal subgroups

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

subgroupindex
C3×#23132
#821762
#715522
#694422
#676162
#94823