Finite Group Databasebeta

← #10692917

Subgroups — #10692917

Order 768.

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

normal subgroups only

This group has 528 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×#2502C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic
3#71689C212yesmaximal
4#71562C211yesmaximal, characteristic
5#67616C212yesmaximal
6#9461—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C3×#398—22no—
10C3×#391C412yescomplemented
11C3×#385—21no—
12C3×#385—21no—
13C3×#384C412yescomplemented
14C3×C4×C22⋊C4C2211yescharacteristic
15#5212—21no—
16#5211C2212yes—
17C4×#217—21no—
18#5207C2212yes—
19#4980—24no—
20#4538—24no—
21#4530—22no—
22#2502D311yescharacteristic
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⋊C4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4C2×C412yes—
40C2×C3×C4⋊C4D412yes—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C42C2×C412yes—
44C2×#217—22no—
45C2×C4⋊Dic3—22no—
46C2×C4×Dic3—22no—
47C2×#208—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#398—22no—
61#391Dic312yescomplemented
62#387—64no—
63#385—21no—
64#385—21no—
65#384Dic312yescomplemented
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—22no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—42no—
83C22×C3×C4—81no—
84C22×C3×C4—81no—
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×C4⋊C4—82no—
97C3×C4⋊C4—82no—
98C3×C22⋊C4—24no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—41no—
101C3×C42—42no—
102#217—44no—
103C4⋊Dic3—82no—
104C4×Dic3—82no—
105#208—44no—
106C2×C3⋊C8—42no—
107C22×Q8C2×Dic311yescharacteristic
108C23×C4#6711yescharacteristic
109C23×C4#6711yescharacteristic
110C23×C4C22×D311yescharacteristic
111C2×M16—62no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4C2×Dic311yescharacteristic
121C2×C4⋊C4C2×Dic312yes—
122C2×C4⋊C4#6712yes—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—62no—
126C2×C42—62no—
127C2×C42C2×Dic312yes—
128C22⋊C8—62no—
129C22⋊C8—62no—
130#95—22no—
131#95—22no—
132#95—21no—
133#95—34no—
134#95—62no—
135#95—62no—
136#95—62no—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3—41no—
140C23×C3C2×C22⋊C411yescharacteristic
141C23×C3#12011yescharacteristic
142C23×C3#12311yescharacteristic
143C3×Q8—82no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—24no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
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—42no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4—42no—
161C2×C3×C4—81no—
162C2×C3×C4—81no—
163C2×C3×C4C4×D412yes—
164C2×C3×C4#12112yes—
165C2×Dic3—44no—
166C2×Dic3—44no—
167C2×Dic3—44no—
168C3⋊C8—82no—
169C24C2×#6711yescharacteristic
170C2×Q8—22no—
171C2×Q8—42no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—42no—
180C22×C4—64no—
181C22×C4—81no—
182C22×C4—81no—
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—42no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4—242no—
198C22⋊C4—24no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—124no—
202C42—42no—
203C42—242no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—44no—
212C22×C3#38611yescharacteristic
213C22×C3#41012yes—
214C22×C3#45612yes—
215C3×C4—22no—
216C3×C4—22no—
217C3×C4—44no—
218C3×C4—42no—
219C3×C4—42no—
220C3×C4—81no—
221C3×C4—81no—
222Dic3—84no—
223C23—22no—
224C23—22no—
225C23—41no—
226C23#93511yescharacteristic
227C23#93711yescharacteristic
228C23C2×#21711yescharacteristic
229Q8—82no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—124no—
250C2×C4—124no—
251C2×C4—124no—
252C2×C4D4×Dic312yes—
253C2×C4#93912yes—
254C8—242no—
255C2×C3—22no—
256C2×C3—22no—
257C2×C3—41no—
258C2×C3#188611yescharacteristic
259C2×C3#189911yescharacteristic
260C2×C3#191111yescharacteristic
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—44no—
269C22#509512yes—
270C22#522011yescharacteristic
271C22#524012yes—
272C4—22no—
273C4—22no—
274C4—44no—
275C4—42no—
276C4—42no—
277C4—81no—
278C4—81no—
279C4—244no—
280C3#946111yescharacteristic, complemented, Sylow
281C2—22no—
282C2—22no—
283C2—41no—
284C2#7170511yescharacteristic
285C2#7178411yescharacteristic
286C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2502C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2502D3
hypocenterC3#9461
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
10. C3×#391C4semidirect
13. C3×#384C4semidirect
61. #391Dic3semidirect
65. #384Dic3semidirect
280. C3#9461semidirect

Maximal subgroups

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

subgroupindex
C3×#25022
#821552
#716892
#715622
#676162
#94613