Finite Group Databasebeta

← #10691243

Subgroups — #10691243

Order 768.

516 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#82183C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic
3C2×C3×#324C211yesmaximal, characteristic
4#69946C212yesmaximal
5#67676C212yesmaximal
6#9202—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#924C2211yescharacteristic
9C2×C3×C22⋊C8C2212yes—
10#5212C2211yescharacteristic
11#5212C2212yes—
12#5211—21no—
13#5211—21no—
14#5210—21no—
15C4×#217—21no—
16#4986C412yescomplemented
17#4982—22no—
18#4980—24no—
19#4979C412yescomplemented
20C3×#324—22no—
21#4467—24no—
22#2627—31no—
23#2313—31no—
24#1520—32no—
25C2×#324D311yescharacteristic
26#1289—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×Dic6C2×C411yescharacteristic
31C22×C3×C8—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4C2×C412yes—
35C2×C3×C4⋊C4D412yes—
36C2×#217—22no—
37C2×#217—21no—
38C2×#217—21no—
39C2×C4⋊Dic3—22no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3C2×C411yescharacteristic
42#924—22no—
43#924—22no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—22no—
46C2×C4×Dic3—22no—
47C3×C22⋊C8—22no—
48C3×C22⋊C8—22no—
49C3⋊#95—22no—
50C3⋊#95—24no—
51C3⋊#95—22no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95C2×C412yes—
55C3⋊#95D414yes—
56C2×C22⋊Q8—31no—
57C22×C4⋊C4C2×D311yescharacteristic
58C2×C22⋊C8C2×D312yes—
59#400—32no—
60#398—62no—
61#395—32no—
62#387—64no—
63#385—61no—
64#381—61no—
65#381—61no—
66#380—31no—
67#380—32no—
68C4×C22⋊C4—61no—
69#328—64no—
70#324—22no—
71C24×C3C2×D411yescharacteristic
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—21no—
78C22×C3×C4—81no—
79C22×C3×C4—81no—
80C22×C3×C4C22⋊C412yes—
81C22×C3×C4C22⋊C412yes—
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D412yes—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4#4112yes—
86C22×Dic3—21no—
87C22×Dic3—22no—
88C22×Dic3—21no—
89C22×Dic3—24no—
90C2×Dic6—22no—
91C2×Dic6—81no—
92C2×C3×C8—42no—
93C2×C3×C8—82no—
94C3×C4⋊C4—24no—
95C3×C4⋊C4—24no—
96#217—41no—
97#217—41no—
98#217—42no—
99#217—42no—
100#217—42no—
101C4⋊Dic3—22no—
102C4⋊Dic3—82no—
103C4⋊Dic3—81no—
104Dic3⋊C4—42no—
105Dic3⋊C4—82no—
106C4×Dic3—82no—
107C22×Q8—31no—
108C23×C4D1211yescharacteristic
109C23×C4D1211yescharacteristic
110C23×C4C22×D311yescharacteristic
111C22×C8—22no—
112C22⋊Q8—62no—
113C22⋊Q8—62no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4—61no—
121C2×C4⋊C4C4×D312yes—
122C2×C4⋊C4#6712yes—
123C2×C22⋊C4—61no—
124C2×C22⋊C4—62no—
125C2×C22⋊C4—61no—
126C2×C42—62no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129#95—32no—
130#95—34no—
131#95—62no—
132#95—64no—
133#95—62no—
134#95—62no—
135#95—62no—
136C23×C3—41no—
137C23×C3—41no—
138C23×C3C2×C22⋊C411yescharacteristic
139C23×C3#12011yescharacteristic
140C23×C3#12311yescharacteristic
141C23×C3#13512yes—
142C23×C3#13712yes—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—42no—
148C2×C3×C4—41no—
149C2×C3×C4—42no—
150C2×C3×C4—41no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—81no—
156C2×C3×C4C4×D412yes—
157C2×C3×C4#12112yes—
158C2×Dic3—44no—
159C2×Dic3—42no—
160C2×Dic3—42no—
161C2×Dic3—42no—
162C2×Dic3—44no—
163C2×Dic3—42no—
164C2×Dic3—41no—
165C2×Dic3—41no—
166C2×Dic3—42no—
167C2×Dic3—44no—
168Dic6—82no—
169C3×C8—82no—
170C24C2×D1211yescharacteristic
171C2×Q8—62no—
172C2×Q8—241no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—64no—
180C22×C4—62no—
181C22×C4—61no—
182C22×C4—61no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4(C2×D3)⋊C412yes—
186C22×C4(C2×D3)⋊C412yes—
187C22×C4C2×C4×D311yescharacteristic
188C22×C4D3×D412yes—
189C22×C4#23712yes—
190C22×C4C2×#6711yescharacteristic
191C2×C8—42no—
192C2×C8—82no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—62no—
196C4⋊C4—122no—
197C4⋊C4—242no—
198C4⋊C4—242no—
199C4⋊C4—241no—
200C22⋊C4—121no—
201C22⋊C4—121no—
202C22⋊C4—122no—
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—
223Dic3—82no—
224Dic3—84no—
225Dic3—81no—
226Dic3—81no—
227C23—41no—
228C23—41no—
229C23#88211yescharacteristic
230C23#88611yescharacteristic
231C23#90412yes—
232C23#90912yes—
233C23C2×(C2×D3)⋊C411yescharacteristic
234Q8—242no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—124no—
249C2×C4—122no—
250C2×C4—122no—
251C2×C4—124no—
252C2×C4—122no—
253C2×C4—122no—
254C2×C4—121no—
255C2×C4—121no—
256C2×C4—124no—
257C2×C4—122no—
258C2×C4#67⋊C412yes—
259C2×C4#93912yes—
260C8—82no—
261C2×C3—21no—
262C2×C3—22no—
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#472312yes—
277C22#473312yes—
278C22#495811yescharacteristic
279C22#511311yescharacteristic
280C22#512011yescharacteristic
281C4—22no—
282C4—22no—
283C4—42no—
284C4—42no—
285C4—244no—
286C4—242no—
287C4—241no—
288C4—241no—
289C3#920211yescharacteristic, complemented, Sylow
290C2—22no—
291C2—21no—
292C2—21no—
293C2—22no—
294C2#6993111yescharacteristic
295C2#7002411yescharacteristic
296C2#7010511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#324D3
hypocenterC3#9202
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
16. #4986C4semidirect
19. #4979C4semidirect
289. C3#9202semidirect

Maximal subgroups

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

subgroupindex
#821832
#821552
C2×C3×#3242
#699462
#676762
#92023