Finite Group Databasebeta

← #10944230

Subgroups — #10944230

Order 768.

1708 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 1708 conjugacy classes of subgroups — too many to draw. Only its 287 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
1C24×C3⋊C8C211yesmaximal, characteristic, complemented
2C3×#2847C211yesmaximal, characteristic
3#84031C211yesmaximal, characteristic, complemented
4C2×#5041C213yesmaximal, complemented
5#71681C213yesmaximal, complemented
6#69514C213yesmaximal, complemented
7#69511C213yesmaximal, complemented
8#32929—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C2×C3×#121C2213yes—
11C2×C3×#120—21no—
12C22×#214C2211yescharacteristic, complemented
13C2×#924C2213yes—
14C23×C3⋊C8—212no—
15C23×C3⋊C8C2213yes—
16C23×C3⋊C8C2213yes—
17C3×#386—23no—
18C3×#380C2213yes—
19#5247—21no—
20C2×#832C2213yescomplemented
21#5212C2213yes—
22#5210—23no—
23#5041—212no—
24C2×#810C2213yescomplemented
25C2×#808C2213yescomplemented
26#4529C2216yescomplemented
27C24×C8—31no—
28C2×#465—33no—
29#2848—31no—
30#2847D311yescharacteristic
31#1893—33no—
32#1892—33no—
33#1807—33no—
34C25×C3D411yescharacteristic
35C22×C3×D4—23no—
36C22×C3×D4C2311yescharacteristic
37C23×C3×C4—26no—
38C23×C3×C4—26no—
39C23×C3×C4D413yes—
40C23×C3×C4C2313yes—
41C22×Dic6C2311yescharacteristic
42C3×#121—26no—
43C3×#121—26no—
44C3×#120—44no—
45C2×C3×C4⋊C4C2313yes—
46C2×C3×C22⋊C4—23no—
47C2×C3×C22⋊C4—23no—
48C2×#217—23no—
49C2×#217—23no—
50C2×#214—26no—
51C2×C4⋊Dic3C2313yes—
52#924—26no—
53#924—26no—
54C2×Dic3⋊C4—23no—
55C22×C3⋊C8—232no—
56C22×C3⋊C8—26no—
57C22×C3⋊C8—212no—
58C22×C3⋊C8—212no—
59C22×C3⋊C8D4112yescomplemented
60C22×C3⋊C8C2313yes—
61C22×C3⋊C8C2311yescharacteristic
62C3×#95—23no—
63#832—26no—
64C3⋊#95—23no—
65#810—26no—
66#808—26no—
67C24×C4C2×D311yescharacteristic
68C22×SD16—31no—
69C23×C8—33no—
70C23×C8—33no—
71C23×C8—612no—
72C2×C22⋊Q8—33no—
73C2×#121C2×D313yes—
74C2×#120—21no—
75#465—612no—
76C2×#106—33no—
77C2×#103—33no—
78C2×#102—33no—
79#386—23no—
80#385—63no—
81#380—33no—
82#380C2×D313yes—
83#379—61no—
84#340—36no—
85C24×C3—26no—
86C24×C3—26no—
87C24×C3—21no—
88C24×C3SD1614yescomplemented
89C24×C3C2×D413yes—
90C2×C3×D4—23no—
91C2×C3×D4—23no—
92C2×C3×D4—46no—
93C2×C3×D4—46no—
94C2×C3×D4—46no—
95C22×C3×C4—28no—
96C22×C3×C4—224no—
97C22×C3×C4—26no—
98C22×C3×C4—212no—
99C22×C3×C4—26no—
100C22×C3×C4—26no—
101C22×C3×C4—23no—
102C22×C3×C4C2×D416yes—
103C22×C3×C4C2×D413yes—
104C22×C3×C4#4116yes—
105C22×C3×C4C2411yescharacteristic
106C22×Dic3—23no—
107C22×Dic3—21no—
108C2×Dic6—23no—
109C2×Dic6—23no—
110C3×C4⋊C4—23no—
111C3×C4⋊C4—23no—
112C3×C22⋊C4—46no—
113C3×C22⋊C4—46no—
114#217—46no—
115#217—46no—
116#214—44no—
117C4⋊Dic3—23no—
118C4⋊Dic3—23no—
119Dic3⋊C4—46no—
120C2×C3⋊C8—224no—
121C2×C3⋊C8—26no—
122C2×C3⋊C8—224no—
123C2×C3⋊C8C2×D416yes—
124C2×C3⋊C8C2×D416yes—
125C25#6711yescharacteristic
126C22×Q8—31no—
127C22×D4—23no—
128C22×D4C22×D311yescharacteristic
129C23×C4—26no—
130C23×C4—26no—
131C23×C4#6713yes—
132C23×C4C22×D313yes—
133C2×SD16—66no—
134C22×C8—31no—
135C22×C8—312no—
136C22×C8—33no—
137C22×C8—612no—
138C22×C8—612no—
139C22×C8—632no—
140C22×C8—66no—
141C22⋊Q8—66no—
142C22⋊Q8—66no—
143#121—26no—
144#121—26no—
145#120—44no—
146C2×C4⋊C4—33no—
147C2×C4⋊C4—63no—
148C2×C4⋊C4C22×D313yes—
149C2×C22⋊C4—23no—
150C2×C22⋊C4—23no—
151C2×C22⋊C4—63no—
152C2×C22⋊C4—63no—
153#106—66no—
154#103—66no—
155#102—66no—
156#95—23no—
157#95—63no—
158C23×C3—28no—
159C23×C3—224no—
160C23×C3—26no—
161C23×C3—26no—
162C23×C3—224no—
163C23×C3—43no—
164C23×C3—41no—
165C23×C3—43no—
166C23×C3#12116yes—
167C23×C3C2×SD1616yes—
168C23×C3#13516yes—
169C23×C3C22×D411yescharacteristic
170C3×D4—41no—
171C3×D4—43no—
172C3×D4—812no—
173C2×C3×C4—224no—
174C2×C3×C4—26no—
175C2×C3×C4—26no—
176C2×C3×C4—26no—
177C2×C3×C4—212no—
178C2×C3×C4—43no—
179C2×C3×C4—43no—
180C2×C3×C4—43no—
181C2×C3×C4#12116yes—
182C2×C3×C4C22×D413yes—
183C2×C3×C4C2×#4113yes—
184C2×Dic3—43no—
185C2×Dic3—43no—
186C2×Dic3—46no—
187Dic6—41no—
188Dic6—43no—
189C3⋊C8—26no—
190C3⋊C8#12014yescomplemented
191C24—26no—
192C24—26no—
193C24—21no—
194C24#21414yescomplemented
195C24C2×#6713yes—
196C2×Q8—63no—
197C2×Q8—63no—
198C2×D4—23no—
199C2×D4—23no—
200C2×D4—46no—
201C2×D4—46no—
202C2×D4—46no—
203C22×C4—28no—
204C22×C4—224no—
205C22×C4—26no—
206C22×C4—212no—
207C22×C4—26no—
208C22×C4—26no—
209C22×C4—23no—
210C22×C4—63no—
211C22×C4—61no—
212C22×C4D3×D416yes—
213C22×C4#23716yes—
214C22×C4C2×#6713yes—
215C22×C4C23×D311yescharacteristic
216SD16—124no—
217C2×C8—36no—
218C2×C8—36no—
219C2×C8—624no—
220C2×C8—624no—
221C2×C8—66no—
222C4⋊C4—23no—
223C4⋊C4—23no—
224C4⋊C4—63no—
225C4⋊C4—63no—
226C4⋊C4—126no—
227C22⋊C4—46no—
228C22⋊C4—46no—
229C22⋊C4—126no—
230C22⋊C4—126no—
231C22×C3—232no—
232C22×C3—26no—
233C22×C3—212no—
234C22×C3—26no—
235C22×C3—212no—
236C22×C3—81no—
237C22×C3—83no—
238C22×C3—83no—
239C22×C3#465112yescomplemented
240C22×C3C2×#12113yes—
241C22×C3C22×SD1611yescharacteristic
242C22×C3C2×#13513yes—
243C3×C4—26no—
244C3×C4—83no—
245C3×C4C2×#12011yescharacteristic
246C3×C4#52513yes—
247Dic3—81no—
248Dic3—83no—
249C23—28no—
250C23—224no—
251C23—26no—
252C23—224no—
253C23—26no—
254C23—41no—
255C23—43no—
256C23—43no—
257C23C2×#21416yes—
258C23#93916yes—
259C23#95016yes—
260C23C22×#6711yescharacteristic
261Q8—123no—
262Q8—121no—
263D4—43no—
264D4—41no—
265D4—812no—
266C2×C4—224no—
267C2×C4—26no—
268C2×C4—26no—
269C2×C4—212no—
270C2×C4—26no—
271C2×C4—43no—
272C2×C4—43no—
273C2×C4—43no—
274C2×C4—123no—
275C2×C4—123no—
276C2×C4—126no—
277C2×C4#93916yes—
278C2×C4C2×D3×D413yes—
279C2×C4C2×#23713yes—
280C8—34no—
281C8—66no—
282C2×C3—212no—
283C2×C3—161no—
284C2×C3#284711yescharacteristic
285C2×C3C2×#46513yes—
286C2×C3#305013yes—
287C22—232no—
288C22—26no—
289C22—212no—
290C22—26no—
291C22—212no—
292C22—81no—
293C22—83no—
294C22—83no—
295C22#5041112yescomplemented
296C22C22×#21411yescharacteristic
297C22C2×#93913yes—
298C22C2×#95013yes—
299C4—26no—
300C4—83no—
301C4—241no—
302C4—243no—
303C4#558613yes—
304C4D3×#12011yescharacteristic
305C3#3292911yescharacteristic, complemented, Sylow
306C2—212no—
307C2—161no—
308C2#8260111yescharacteristic
309C2C2×#504113yes—
310C2#8292213yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2847C2
socleC23×C3C22×D4
hypercenter#2847D3
hypocenterC3#32929
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
1. C24×C3⋊C8C2semidirect
3. #84031C2semidirect
4. C2×#5041C2semidirect
5. #71681C2semidirect
6. #69514C2semidirect
7. #69511C2semidirect
12. C22×#214C22semidirect
20. C2×#832C22semidirect
24. C2×#810C22semidirect
25. C2×#808C22semidirect
26. #4529C22semidirect
59. C22×C3⋊C8D4semidirect
88. C24×C3SD16semidirect
190. C3⋊C8#120semidirect
194. C24#214semidirect
239. C22×C3#465semidirect
295. C22#5041semidirect
305. C3#32929semidirect

Maximal subgroups

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

subgroupindex
C24×C3⋊C82
C3×#28472
#840312
C2×#50412
#716812
#695142
#695112
#329293