Finite Group Databasebeta

← #10693154

Subgroups — #10693154

Order 768.

524 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 103 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#82374C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3C3×#1465C211yesmaximal, characteristic
4#69973C212yesmaximal, complemented
5#67685C212yesmaximal, complemented
6#9509—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C22×C4⋊Dic3C2211yescharacteristic
9C3×#407C2212yes—
10#5220C412yes—
11#5212—21no—
12#5212C2211yescharacteristic
13#5207C2212yes—
14C2×#830—21no—
15#5011—24no—
16#4986—21no—
17#4949—21no—
18#4946C412yes—
19#4928—21no—
20#4926—21no—
21C3×#324—21no—
22C3×#323—21no—
23#4554—24no—
24#2973—31no—
25#2487—31no—
26#1821—32no—
27#1465D311yescharacteristic
28#1305—32no—
29C22×C3×D4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C23×Dic3C2×C412yes—
33C2×C3×M16—22no—
34C3×C4×D4—22no—
35C3×C4×D4—22no—
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C22⋊C4C2×C411yescharacteristic
38C2×C3×C22⋊C4D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×#217—22no—
41C2×#217C2×C412yes—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3—22no—
47C2×Dic3⋊C4—22no—
48C2×#208D414yes—
49C22×C3⋊C8—22no—
50C3×C22⋊C8—22no—
51C3×C22⋊C8—22no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95—21no—
55#830—22no—
56#830—22no—
57#830—22no—
58C4⋊(C3⋊C8)—22no—
59#803—22no—
60C2×C4×D4C2×D311yescharacteristic
61C22×C4⋊C4—31no—
62#435—64no—
63#432—61no—
64#407—32no—
65#407C2×D312yes—
66C2×C22⋊C8—61no—
67#400—61no—
68#399—61no—
69#389—61no—
70#386—32no—
71#385—32no—
72#380—31no—
73#380—61no—
74#343—64no—
75#324—21no—
76#323—21no—
77C24×C3C22⋊C411yescharacteristic
78C24×C3C22×C411yescharacteristic
79C2×C3×D4—22no—
80C2×C3×D4—22no—
81C2×C3×D4—41no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D412yes—
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4#4112yes—
93C22×Dic3—22no—
94C22×Dic3—22no—
95C22×Dic3—24no—
96C22×Dic3—82no—
97C3×M16—44no—
98C2×C3×C8—42no—
99C3×C4⋊C4—22no—
100C3×C22⋊C4—22no—
101C3×C22⋊C4—22no—
102C3×C42—21no—
103C3×C42—21no—
104#217—42no—
105#217—82no—
106C4⋊Dic3—42no—
107C4⋊Dic3—41no—
108C4⋊Dic3—42no—
109C4⋊Dic3—41no—
110Dic3⋊C4—82no—
111#208—44no—
112C2×C3⋊C8—24no—
113C2×C3⋊C8—82no—
114C22×D4C4×D311yescharacteristic
115C23×C4—32no—
116C23×C4D1211yescharacteristic
117C23×C4C22×D311yescharacteristic
118C2×M16—22no—
119C2×M16—34no—
120C22×C8—62no—
121C4×D4—22no—
122C4×D4—22no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4—62no—
126C2×C4⋊C4—62no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—62no—
129C2×C4⋊C4#6711yescharacteristic
130C2×C22⋊C4—32no—
131C2×C22⋊C4—62no—
132C2×C22⋊C4C4×D311yescharacteristic
133C2×C22⋊C4#6711yescharacteristic
134C2×C42D1211yescharacteristic
135C4⋊C8—62no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—62no—
139C22⋊C8—62no—
140C22⋊C8—62no—
141#97—62no—
142#95—62no—
143#95—62no—
144#95—61no—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—41no—
150C23×C3—41no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3#10112yes—
154C23×C3C2×C22⋊C411yescharacteristic
155C23×C3C4×D412yes—
156C3×D4—44no—
157C2×C3×C4—24no—
158C2×C3×C4—22no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—41no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—81no—
170C2×C3×C4#12011yescharacteristic
171C2×C3×C4#12112yes—
172C2×C3×C4#12311yescharacteristic
173C2×Dic3—42no—
174C2×Dic3—42no—
175C2×Dic3—42no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179C2×Dic3—84no—
180C3×C8—82no—
181C3⋊C8—44no—
182C24(C2×D3)⋊C411yescharacteristic
183C24C2×C4×D311yescharacteristic
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—41no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4—64no—
194C22×C4—242no—
195C22×C4(C2×D3)⋊C411yescharacteristic
196C22×C4(C2×D3)⋊C411yescharacteristic
197C22×C4(C2×D3)⋊C411yescharacteristic
198C22×C4C2×D1211yescharacteristic
199C22×C4D3×D412yes—
200C22×C4#23712yes—
201C22×C4C2×#6711yescharacteristic
202M16—44no—
203M16—124no—
204C2×C8—42no—
205C2×C8—64no—
206C2×C8—242no—
207C4⋊C4—22no—
208C4⋊C4—122no—
209C4⋊C4—121no—
210C4⋊C4—121no—
211C4⋊C4—122no—
212C4⋊C4—242no—
213C22⋊C4—22no—
214C22⋊C4—22no—
215C22⋊C4—122no—
216C22⋊C4—242no—
217C42—21no—
218C42—21no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—22no—
222C22×C3—22no—
223C22×C3—22no—
224C22×C3—22no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—42no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—81no—
231C22×C3#38611yescharacteristic
232C22×C3C2×#10111yescharacteristic
233C22×C3#41311yescharacteristic
234C3×C4—22no—
235C3×C4—21no—
236C3×C4—41no—
237C3×C4—41no—
238C3×C4—41no—
239C3×C4#45812yes—
240Dic3—82no—
241Dic3—82no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23#82412yes—
251C23#67⋊C412yes—
252C23C2×(C2×D3)⋊C411yescharacteristic
253D4—44no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—122no—
268C2×C4—122no—
269C2×C4—122no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4—122no—
273C2×C4—244no—
274C2×C4#88211yescharacteristic
275C2×C4#88611yescharacteristic
276C2×C4#93912yes—
277C8—82no—
278C8—124no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—42no—
282C2×C3—42no—
283C2×C3#189011yescharacteristic
284C2×C3#190111yescharacteristic
285C2×C3#191411yescharacteristic
286C22—21no—
287C22—21no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—81no—
298C22#495811yescharacteristic
299C22#513211yescharacteristic
300C22C2×#82411yescharacteristic
301C4—22no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—242no—
307C4—242no—
308C4#474212yes—
309C3#950911yescharacteristic, complemented, Sylow
310C2—21no—
311C2—21no—
312C2—42no—
313C2—42no—
314C2#6998611yescharacteristic
315C2#7002311yescharacteristic
316C2#7013711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1465C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC23×C3×C4C23
hypercenter#1465D3
hypocenterC3#9509
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
4. #69973C2semidirect
5. #67685C2semidirect
309. C3#9509semidirect

Maximal subgroups

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

subgroupindex
#823742
#823022
C3×#14652
#699732
#676852
#95093