Finite Group Databasebeta

← #10691246

Subgroups — #10691246

Order 768.

516 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2627C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic
3#71690C212yesmaximal
4C2×#4528C211yesmaximal, characteristic
5#67625C212yesmaximal
6#9202—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C3×#400C412yescomplemented
10C3×#398—22no—
11C3×#395C412yescomplemented
12C3×#385—21no—
13C3×#381—21no—
14C3×#380C2211yescharacteristic
15#5212C2212yes—
16#5211—21no—
17C4×#217—21no—
18C2×#830C2212yes—
19#4980—24no—
20#4538—24no—
21#4528—22no—
22#2627D311yescharacteristic
23#2313—31no—
24#1520—32no—
25C2×#324—31no—
26#1289—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—21no—
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×#217—22no—
44C2×C4⋊Dic3—22no—
45C2×C4×Dic3—22no—
46C22×C3⋊C8—22no—
47C3×#95—22no—
48C3×#95—22no—
49C3×#95—22no—
50C3×#95C2×C412yes—
51C3⋊#95—22no—
52C3⋊#95—24no—
53C3⋊#95D414yes—
54#830—22no—
55#830—22no—
56C2×C22⋊Q8C2×D311yescharacteristic
57C22×C4⋊C4C2×D311yescharacteristic
58C2×C22⋊C8—32no—
59#400Dic312yescomplemented
60#398—22no—
61#395Dic312yescomplemented
62#387—64no—
63#385—21no—
64#381—21no—
65#381—61no—
66#380—32no—
67#380C2×D311yescharacteristic
68C4×C22⋊C4—61no—
69#328—64no—
70#324—62no—
71C24×C3C2×D411yescharacteristic
72C2×C3×Q8—22no—
73C2×C3×Q8—81no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—22no—
82C22×C3×C4—21no—
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—81no—
97C3×C4⋊C4—82no—
98C3×C22⋊C4—41no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—42no—
101#217—42no—
102#217—42no—
103C4⋊Dic3—82no—
104C4×Dic3—82no—
105C2×C3⋊C8—42no—
106C2×C3⋊C8—82no—
107C22×Q8C2×Dic311yescharacteristic
108C23×C4#6711yescharacteristic
109C23×C4#6711yescharacteristic
110C23×C4C22×D311yescharacteristic
111C22×C8—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—21no—
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—
127C22⋊C8—62no—
128C22⋊C8—62no—
129#95—22no—
130#95—22no—
131#95—22no—
132#95—34no—
133#95—62no—
134#95—64no—
135#95C2×Dic312yes—
136C23×C3—41no—
137C23×C3—41no—
138C23×C3C2×C22⋊C411yescharacteristic
139C23×C3#12011yescharacteristic
140C23×C3#12311yescharacteristic
141C23×C3#13512yes—
142C23×C3#13712yes—
143C3×Q8—82no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—42no—
149C2×C3×C4—42no—
150C2×C3×C4—41no—
151C2×C3×C4—41no—
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—42no—
159C2×C3×C4—42no—
160C2×C3×C4—41no—
161C2×C3×C4—41no—
162C2×C3×C4—42no—
163C2×C3×C4—81no—
164C2×C3×C4C4×D412yes—
165C2×C3×C4#12112yes—
166C2×Dic3—44no—
167C2×Dic3—44no—
168C2×Dic3—44no—
169C3⋊C8—82no—
170C24C2×#6711yescharacteristic
171C2×Q8—22no—
172C2×Q8—81no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—64no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4#21712yes—
186C22×C4#21712yes—
187C22×C4D3×D412yes—
188C22×C4#23712yes—
189C22×C4C22×Dic311yescharacteristic
190C22×C4C2×#6711yescharacteristic
191C2×C8—122no—
192C2×C8—242no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—81no—
198C4⋊C4—82no—
199C4⋊C4—242no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—42no—
203C22⋊C4—122no—
204C22⋊C4—122no—
205C42—242no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3#38611yescharacteristic
215C22×C3#41611yescharacteristic
216C22×C3#41911yescharacteristic
217C22×C3#45212yes—
218C22×C3#48512yes—
219C3×C4—22no—
220C3×C4—22no—
221C3×C4—42no—
222C3×C4—42no—
223C3×C4—82no—
224C3×C4—81no—
225C3×C4—81no—
226Dic3—84no—
227C23—41no—
228C23—41no—
229C23#93511yescharacteristic
230C23#93711yescharacteristic
231C23#94212yes—
232C23#95012yes—
233C23C2×#21711yescharacteristic
234Q8—82no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—81no—
255C2×C4—124no—
256C2×C4—124no—
257C2×C4—124no—
258C2×C4D4×Dic312yes—
259C2×C4#93912yes—
260C8—242no—
261C2×C3—22no—
262C2×C3—21no—
263C2×C3—21no—
264C2×C3—22no—
265C2×C3#187311yescharacteristic
266C2×C3#191211yescharacteristic
267C2×C3#191311yescharacteristic
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—42no—
275C22—42no—
276C22#504512yes—
277C22#504712yes—
278C22#522011yescharacteristic
279C22#522511yescharacteristic
280C22#523311yescharacteristic
281C4—22no—
282C4—22no—
283C4—42no—
284C4—42no—
285C4—82no—
286C4—81no—
287C4—81no—
288C4—244no—
289C3#920211yescharacteristic, complemented, Sylow
290C2—21no—
291C2—22no—
292C2—21no—
293C2—22no—
294C2#7169311yescharacteristic
295C2#7184811yescharacteristic
296C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2627C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2627D3
hypocenterC3#9202
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
9. C3×#400C4semidirect
11. C3×#395C4semidirect
59. #400Dic3semidirect
61. #395Dic3semidirect
289. C3#9202semidirect

Maximal subgroups

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

subgroupindex
C3×#26272
#821552
#716902
C2×#45282
#676252
#92023