Finite Group Databasebeta

← #10692912

Subgroups — #10692912

Order 768.

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

normal subgroups only

This group has 512 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×#2503C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic
3#71689C212yesmaximal
4#71562C211yesmaximal, characteristic
5#67616C212yesmaximal
6#9460—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C3×#400—22no—
10C3×#391C412yescomplemented
11C3×#382C412yescomplemented
12C3×#381—21no—
13C3×#381—21no—
14C3×C4×C22⋊C4C2211yescharacteristic
15#5212—21no—
16#5211C2212yes—
17C4×#217—21no—
18#5207C2212yes—
19#4980—24no—
20#4538—24no—
21#4530—22no—
22#2503D311yescharacteristic
23#2313—31no—
24#1821—31no—
25#1511—32no—
26#1284—32no—
27C22×C3×Q8C2×C411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C3×C22⋊Q8—22no—
32C3×C22⋊Q8—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—21no—
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×C42C2×C412yes—
42C2×#217—22no—
43C2×C4⋊Dic3—22no—
44C2×C4×Dic3—22no—
45C2×#208—22no—
46C3×#95—22no—
47C3×#95—22no—
48C3×#95—22no—
49C3×#95—22no—
50C3×#95—21no—
51C3⋊#95—22no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95D414yes—
55#830—22no—
56#830—22no—
57C2×C22⋊Q8C2×D311yescharacteristic
58C22×C4⋊C4C2×D311yescharacteristic
59#407—32no—
60#400—22no—
61#391Dic312yescomplemented
62#387—64no—
63#382Dic312yescomplemented
64#381—21no—
65#381—21no—
66#381—32no—
67#380—61no—
68C4×C22⋊C4—61no—
69C4×C22⋊C4C2×D311yescharacteristic
70#343—62no—
71#328—64no—
72C24×C3C2×D411yescharacteristic
73C2×C3×Q8—22no—
74C2×C3×Q8—42no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
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×C4—42no—
85C22×C3×C4C22⋊C412yes—
86C22×C3×C4C22⋊C412yes—
87C22×C3×C4C22×C411yescharacteristic
88C22×C3×C4C2×D412yes—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4#4112yes—
91C22×Dic3—24no—
92C3×C4⋊C4—24no—
93C3×C4⋊C4—24no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—42no—
96C3×C22⋊C4—42no—
97C3×C22⋊C4—41no—
98C3×C22⋊C4—41no—
99C3×C42—42no—
100#217—44no—
101C4⋊Dic3—82no—
102C4×Dic3—82no—
103#208—44no—
104C2×C3⋊C8—42no—
105C22×Q8C2×Dic311yescharacteristic
106C23×C4#6711yescharacteristic
107C23×C4#6711yescharacteristic
108C23×C4C22×D311yescharacteristic
109C2×M16—62no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—62no—
116C2×C4⋊C4C2×Dic311yescharacteristic
117C2×C4⋊C4C2×Dic312yes—
118C2×C4⋊C4#6712yes—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—62no—
122C2×C42—62no—
123C2×C42C2×Dic312yes—
124C22⋊C8—62no—
125C22⋊C8—62no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—22no—
130#95—21no—
131#95—34no—
132#95—62no—
133#95—62no—
134#95—62no—
135C23×C3—22no—
136C23×C3—22no—
137C23×C3—41no—
138C23×C3C2×C22⋊C411yescharacteristic
139C23×C3#12011yescharacteristic
140C23×C3#12311yescharacteristic
141C3×Q8—82no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
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—42no—
154C2×C3×C4—42no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—81no—
160C2×C3×C4—81no—
161C2×C3×C4C4×D412yes—
162C2×C3×C4#12112yes—
163C2×Dic3—44no—
164C2×Dic3—44no—
165C2×Dic3—44no—
166C3⋊C8—82no—
167C24C2×#6711yescharacteristic
168C2×Q8—22no—
169C2×Q8—42no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—64no—
181C22×C4#21712yes—
182C22×C4#21712yes—
183C22×C4D3×D412yes—
184C22×C4#23712yes—
185C22×C4C22×Dic311yescharacteristic
186C22×C4C2×#6711yescharacteristic
187M16—124no—
188C2×C8—122no—
189C4⋊C4—24no—
190C4⋊C4—24no—
191C4⋊C4—22no—
192C4⋊C4—42no—
193C4⋊C4—242no—
194C22⋊C4—42no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—124no—
198C42—42no—
199C42—242no—
200C22×C3—21no—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—44no—
208C22×C3#38611yescharacteristic
209C22×C3#41012yes—
210C22×C3#45512yes—
211C3×C4—22no—
212C3×C4—22no—
213C3×C4—42no—
214C3×C4—42no—
215C3×C4—82no—
216C3×C4—81no—
217C3×C4—81no—
218Dic3—84no—
219C23—22no—
220C23—22no—
221C23—41no—
222C23#93511yescharacteristic
223C23#93711yescharacteristic
224C23C2×#21711yescharacteristic
225Q8—82no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—124no—
246C2×C4—124no—
247C2×C4—124no—
248C2×C4D4×Dic312yes—
249C2×C4#93912yes—
250C8—242no—
251C2×C3—22no—
252C2×C3—22no—
253C2×C3—41no—
254C2×C3#188511yescharacteristic
255C2×C3#189911yescharacteristic
256C2×C3#191211yescharacteristic
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22#506112yes—
266C22#522011yescharacteristic
267C22#524012yes—
268C4—22no—
269C4—22no—
270C4—42no—
271C4—42no—
272C4—82no—
273C4—81no—
274C4—81no—
275C4—244no—
276C3#946011yescharacteristic, complemented, Sylow
277C2—22no—
278C2—22no—
279C2—41no—
280C2#7170511yescharacteristic
281C2#7172311yescharacteristic
282C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2503C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2503D3
hypocenterC3#9460
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
10. C3×#391C4semidirect
11. C3×#382C4semidirect
61. #391Dic3semidirect
63. #382Dic3semidirect
276. C3#9460semidirect

Maximal subgroups

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

subgroupindex
C3×#25032
#821552
#716892
#715622
#676162
#94603