Finite Group Databasebeta

← #10693002

Subgroups — #10693002

Order 768.

610 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 115 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#82144C211yesmaximal, characteristic
3#71590C212yesmaximal
4#71552C211yesmaximal, characteristic
5#67628C212yesmaximal
6#9478—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C2×#926—21no—
10C3×#400C412yescomplemented
11C3×#398C412yescomplemented
12C3×#387—22no—
13C3×#385—21no—
14C3×#381—21no—
15C3×#381C2211yescharacteristic
16C4×#217—21no—
17C4×#217C2212yes—
18#5207C2212yes—
19Dic3×C4⋊C4—24no—
20#4532—24no—
21#4528—22no—
22#2484D311yescharacteristic
23#2274—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—
43#926—28no—
44C2×C4⋊Dic3—22no—
45C2×C4×Dic3—22no—
46C2×C4×Dic3—22no—
47C2×C4×Dic3D414yes—
48C2×#208—22no—
49C3×#95—22no—
50C3×#95—22no—
51C3×#95—22no—
52C3×#95—21no—
53C3×#95C2×C412yes—
54C3⋊#95—22no—
55C3⋊#95—22no—
56#830—22no—
57#830—22no—
58C2×#117—61no—
59C2×#117C2×D311yescharacteristic
60C22×C4⋊C4C2×D311yescharacteristic
61#407—32no—
62#400Dic312yescomplemented
63#398Dic312yescomplemented
64#387—22no—
65#385—21no—
66#381—21no—
67#381C2×D311yescharacteristic
68C4×C4⋊C4—64no—
69C4×C22⋊C4—32no—
70C4×C22⋊C4—61no—
71#339—64no—
72#324—62no—
73C24×C3C2×D411yescharacteristic
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—24no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—22no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—42no—
83C22×C3×C4—42no—
84C22×C3×C4C22⋊C412yes—
85C22×C3×C4C22⋊C412yes—
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C2×D412yes—
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4#4112yes—
90C22×Dic3—24no—
91C3×C4⋊C4—24no—
92C3×C4⋊C4—24no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—44no—
96C3×C22⋊C4—42no—
97C3×C22⋊C4—41no—
98C3×C22⋊C4—41no—
99C3×C42—42no—
100#217—28no—
101C4⋊Dic3—28no—
102C4×Dic3—28no—
103C4×Dic3—44no—
104C4×Dic3—82no—
105#208—44no—
106C2×C3⋊C8—42no—
107C23×C4#6711yescharacteristic
108C23×C4#6711yescharacteristic
109C23×C4C22×D311yescharacteristic
110C2×M16—62no—
111#117—22no—
112#117—22no—
113#117—68no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4C2×Dic311yescharacteristic
119C2×C4⋊C4C2×Dic311yescharacteristic
120C2×C4⋊C4C2×Dic312yes—
121C2×C4⋊C4#6712yes—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—62no—
125C2×C42—21no—
126C2×C42—34no—
127C2×C42—62no—
128C2×C42—62no—
129C22⋊C8—62no—
130C22⋊C8—62no—
131#95—22no—
132#95—22no—
133#95—22no—
134#95—21no—
135#95—62no—
136#95—62no—
137#95C2×Dic312yes—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—41no—
141C23×C3C2×C22⋊C411yescharacteristic
142C23×C3#12011yescharacteristic
143C23×C3#12311yescharacteristic
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—24no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—24no—
150C2×C3×C4—22no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—41no—
160C2×C3×C4—41no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—81no—
164C2×C3×C4#10414yes—
165C2×C3×C4C4×D412yes—
166C2×C3×C4#12112yes—
167C2×Dic3—28no—
168C2×Dic3—44no—
169C2×Dic3—44no—
170C3⋊C8—82no—
171C24C2×#6711yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—24no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—64no—
183C22×C4#21712yes—
184C22×C4#21712yes—
185C22×C4D3×D412yes—
186C22×C4#23712yes—
187C22×C4C22×Dic311yescharacteristic
188C22×C4C2×#6711yescharacteristic
189M16—124no—
190C2×C8—122no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—44no—
196C4⋊C4—68no—
197C22⋊C4—42no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—68no—
201C42—42no—
202C42—68no—
203C42—124no—
204C42—242no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—44no—
213C22×C3#38611yescharacteristic
214C22×C3C2×#10412yes—
215C22×C3#45812yes—
216C3×C4—22no—
217C3×C4—22no—
218C3×C4—24no—
219C3×C4—42no—
220C3×C4—82no—
221C3×C4—81no—
222C3×C4—81no—
223Dic3—48no—
224C23—22no—
225C23—22no—
226C23—41no—
227C23#93511yescharacteristic
228C23#93711yescharacteristic
229C23C2×#21711yescharacteristic
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—68no—
250C2×C4—81no—
251C2×C4—124no—
252C2×C4—124no—
253C2×C4#83714yes—
254C2×C4D4×Dic312yes—
255C2×C4#93912yes—
256C8—242no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—41no—
260C2×C3#189011yescharacteristic
261C2×C3#189611yescharacteristic
262C2×C3#191211yescharacteristic
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—44no—
271C22#505312yes—
272C22#522011yescharacteristic
273C22C2×#83712yes—
274C4—22no—
275C4—22no—
276C4—24no—
277C4—42no—
278C4—82no—
279C4—81no—
280C4—81no—
281C4—128no—
282C3#947811yescharacteristic, complemented, Sylow
283C2—22no—
284C2—22no—
285C2—41no—
286C2#7170411yescharacteristic
287C2#7171111yescharacteristic
288C2#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#9478
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
10. C3×#400C4semidirect
11. C3×#398C4semidirect
62. #400Dic3semidirect
63. #398Dic3semidirect
282. C3#9478semidirect

Maximal subgroups

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

subgroupindex
C3×#24842
#821442
#715902
#715522
#676282
#94783