Finite Group Databasebeta

← #10944234

Subgroups — #10944234

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×#2848C211yesmaximal, characteristic, complemented
3#84032C211yesmaximal, characteristic
4C2×#5048C213yesmaximal, complemented
5#71696C213yesmaximal, complemented
6#69515C213yesmaximal, complemented
7#69511C213yesmaximal, complemented
8#32929—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C2×C3×C22⋊Q8C2213yes—
11C2×#953—21no—
12C22×#215C2211yescharacteristic, complemented
13C2×#930C2213yes—
14C23×C3⋊C8—212no—
15C23×C3⋊C8C2213yes—
16C23×C3⋊C8C2213yes—
17C3×#385—23no—
18C3×#380C2213yes—
19C3×#379—21no—
20C2×#835C2213yescomplemented
21#5213—23no—
22#5212C2213yes—
23#5048—212no—
24C2×#809C2213yescomplemented
25C2×#808C2213yescomplemented
26#4529C2216yescomplemented
27C24×C8—31no—
28C2×#465—33no—
29#2848D311yescharacteristic, complemented
30#2847—31no—
31#1893—33no—
32#1892—33no—
33#1807—33no—
34C25×C3D411yescharacteristic
35C22×C3×Q8C2311yescharacteristic
36C23×C3×C4—26no—
37C23×C3×C4—26no—
38C23×C3×C4D413yes—
39C23×C3×C4C2313yes—
40C22×#67—23no—
41C22×D12C2311yescharacteristic
42C3×C22⋊Q8—26no—
43C3×C22⋊Q8—26no—
44C2×C3×C4⋊C4—23no—
45C2×C3×C4⋊C4C2313yes—
46C2×C3×C22⋊C4—23no—
47C2×C3×C22⋊C4—23no—
48#953—44no—
49C2×#217—23no—
50C2×#215—26no—
51#930—26no—
52#930—26no—
53C2×(C2×D3)⋊C4—23no—
54C2×C4⋊Dic3C2313yes—
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#835—26no—
64C3⋊#95—23no—
65#809—26no—
66#808—26no—
67C24×C4C2×D311yescharacteristic
68C22×SD16—31no—
69C23×C8—33no—
70C23×C8—33no—
71C23×C8—612no—
72C2×C22⋊Q8C2×D313yes—
73C2×#121—33no—
74C2×#120—61no—
75#465—612no—
76C2×#106—33no—
77C2×#103—33no—
78C2×#102—33no—
79#386—63no—
80#385—23no—
81#380—33no—
82#380C2×D313yes—
83#379—21no—
84#340—36no—
85C24×C3—26no—
86C24×C3—26no—
87C24×C3SD1614yescomplemented
88C24×C3C2×D413yes—
89C23×D3—21no—
90C2×C3×Q8—23no—
91C2×C3×Q8—23no—
92C22×C3×C4—28no—
93C22×C3×C4—224no—
94C22×C3×C4—26no—
95C22×C3×C4—212no—
96C22×C3×C4—26no—
97C22×C3×C4—26no—
98C22×C3×C4—23no—
99C22×C3×C4—21no—
100C22×C3×C4C2×D416yes—
101C22×C3×C4C2×D413yes—
102C22×C3×C4#4116yes—
103C22×C3×C4C2411yescharacteristic
104C2×#67—46no—
105C2×#67—46no—
106C2×#67—46no—
107C22×Dic3—23no—
108C2×D12—23no—
109C2×D12—23no—
110C3×C4⋊C4—23no—
111C3×C4⋊C4—23no—
112C3×C4⋊C4—46no—
113C3×C22⋊C4—46no—
114C3×C22⋊C4—46no—
115#217—46no—
116#215—44no—
117(C2×D3)⋊C4—46no—
118C4⋊Dic3—23no—
119C4⋊Dic3—23no—
120C2×C3⋊C8—224no—
121C2×C3⋊C8—26no—
122C2×C3⋊C8—224no—
123C2×C3⋊C8C2×D416yes—
124C2×C3⋊C8C2×D416yes—
125C25#6711yescharacteristic
126C22×Q8C22×D311yescharacteristic
127C22×D4—31no—
128C22×D4—63no—
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—26no—
142C22⋊Q8—26no—
143#121—66no—
144#121—66no—
145#120—124no—
146C2×C4⋊C4—23no—
147C2×C4⋊C4—33no—
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#12116yes—
164C23×C3C2×SD1616yes—
165C23×C3#13516yes—
166C23×C3C22×D411yescharacteristic
167C22×D3—43no—
168C22×D3—41no—
169C22×D3—43no—
170C3×Q8—41no—
171C3×Q8—43no—
172C2×C3×C4—224no—
173C2×C3×C4—26no—
174C2×C3×C4—26no—
175C2×C3×C4—26no—
176C2×C3×C4—212no—
177C2×C3×C4—43no—
178C2×C3×C4—43no—
179C2×C3×C4—46no—
180C2×C3×C4#12116yes—
181C2×C3×C4C22×D413yes—
182C2×C3×C4C2×#4113yes—
183#67—812no—
184C2×Dic3—43no—
185C2×Dic3—43no—
186C2×Dic3—43no—
187D12—41no—
188D12—43no—
189C3⋊C8—26no—
190C3⋊C8#12014yescomplemented
191C24—26no—
192C24—26no—
193C24—61no—
194C24#21514yescomplemented
195C24C2×#6713yes—
196C2×Q8—23no—
197C2×Q8—23no—
198C2×D4—63no—
199C2×D4—63no—
200C2×D4—126no—
201C2×D4—126no—
202C2×D4—126no—
203C22×C4—28no—
204C22×C4—224no—
205C22×C4—26no—
206C22×C4—212no—
207C22×C4—26no—
208C22×C4—26no—
209C22×C4—23no—
210C22×C4—21no—
211C22×C4—63no—
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—46no—
225C4⋊C4—63no—
226C4⋊C4—63no—
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#465112yescomplemented
237C22×C3C2×#12113yes—
238C22×C3C22×SD1611yescharacteristic
239C22×C3C2×#13513yes—
240C2×D3—81no—
241C2×D3—83no—
242C2×D3—83no—
243C3×C4—26no—
244C3×C4—81no—
245C3×C4—83no—
246C3×C4C2×#12011yescharacteristic
247C3×C4#52513yes—
248Dic3—83no—
249C23—28no—
250C23—224no—
251C23—26no—
252C23—224no—
253C23—26no—
254C23—123no—
255C23—121no—
256C23—123no—
257C23#93916yes—
258C23C2×#21516yes—
259C23#95016yes—
260C23C22×#6711yescharacteristic
261Q8—43no—
262Q8—41no—
263D4—123no—
264D4—121no—
265D4—2412no—
266C2×C4—224no—
267C2×C4—26no—
268C2×C4—26no—
269C2×C4—212no—
270C2×C4—26no—
271C2×C4—43no—
272C2×C4—43no—
273C2×C4—46no—
274C2×C4—123no—
275C2×C4—123no—
276C2×C4—123no—
277C2×C4#93916yes—
278C2×C4C2×D3×D413yes—
279C2×C4C2×#23713yes—
280C8—34no—
281C8—66no—
282C2×C3—212no—
283C2×C3#284711yescharacteristic
284C2×C3C2×#46513yes—
285C2×C3#305013yes—
286D3—161no—
287C22—232no—
288C22—212no—
289C22—26no—
290C22—26no—
291C22—212no—
292C22—243no—
293C22—241no—
294C22—243no—
295C22#5048112yescomplemented
296C22C2×#93913yes—
297C22C22×#21511yescharacteristic
298C22C2×#95013yes—
299C4—26no—
300C4—81no—
301C4—83no—
302C4—243no—
303C4#558613yes—
304C4D3×#12011yescharacteristic
305C3#3292911yescharacteristic, complemented, Sylow
306C2—212no—
307C2—481no—
308C2#8260111yescharacteristic
309C2C2×#504813yes—
310C2#8292213yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2848C2
socleC23×C3C22×D4
prime radical#84032C2
hypercenter#2848D3
hypocenterC3#32929
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
1. C24×C3⋊C8C2semidirect
2. C3×#2848C2semidirect
4. C2×#5048C2semidirect
5. #71696C2semidirect
6. #69515C2semidirect
7. #69511C2semidirect
12. C22×#215C22semidirect
20. C2×#835C22semidirect
24. C2×#809C22semidirect
25. C2×#808C22semidirect
26. #4529C22semidirect
29. #2848D3semidirect
59. C22×C3⋊C8D4semidirect
87. C24×C3SD16semidirect
190. C3⋊C8#120semidirect
194. C24#215semidirect
236. C22×C3#465semidirect
295. C22#5048semidirect
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×#28482
#840322
C2×#50482
#716962
#695152
#695112
#329293