Finite Group Databasebeta

← #10944237

Subgroups — #10944237

Order 768.

1786 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 1786 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, complemented
3#84032C211yesmaximal, characteristic, complemented
4C2×#5038C213yesmaximal, complemented
5#71681C213yesmaximal, complemented
6#69515C213yesmaximal, complemented
7#69512C213yesmaximal, complemented
8#32930—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C2×C3×#121C2213yes—
11C2×C3×#120—21no—
12C2×#953—21no—
13C22×#213C2211yescharacteristic, complemented
14C2×#930C2213yes—
15C23×C3⋊C8—212no—
16C23×C3⋊C8C2213yes—
17C23×C3⋊C8C2213yes—
18C3×#386—23no—
19C3×#380C2213yes—
20C2×#832C2213yescomplemented
21#5213—23no—
22#5212C2213yes—
23#5038—212no—
24C2×#809C2213yescomplemented
25C2×#807C2213yescomplemented
26#4529C2216yescomplemented
27C24×C8—31no—
28C2×#466—33no—
29#2847—31no—
30#2847D311yescharacteristic, complemented
31#1893—33no—
32#1893—33no—
33#1808—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×#67—23no—
42C22×D12C2311yescharacteristic
43C3×#121—26no—
44C3×#121—26no—
45C3×#120—44no—
46C2×C3×C4⋊C4C2313yes—
47C2×C3×C22⋊C4—23no—
48C2×C3×C22⋊C4—23no—
49#953—44no—
50C2×#217—23no—
51C2×#213—26no—
52#930—26no—
53#930—26no—
54C2×(C2×D3)⋊C4—23no—
55C2×C4⋊Dic3C2313yes—
56C22×C3⋊C8—232no—
57C22×C3⋊C8—26no—
58C22×C3⋊C8—212no—
59C22×C3⋊C8—212no—
60C22×C3⋊C8D4112yescomplemented
61C22×C3⋊C8C2313yes—
62C22×C3⋊C8C2311yescharacteristic
63C3×#95—23no—
64#832—26no—
65C3⋊#95—23no—
66#809—26no—
67#807—26no—
68C24×C4C2×D311yescharacteristic
69C22×D8—31no—
70C23×C8—33no—
71C23×C8—33no—
72C23×C8—612no—
73C2×#121—33no—
74C2×#121C2×D313yes—
75C2×#120—21no—
76C2×#120—61no—
77#466—612no—
78C2×#107—33no—
79C2×#102—33no—
80C2×#102—33no—
81#386—23no—
82#386—63no—
83#380—33no—
84#380C2×D313yes—
85#340—36no—
86C24×C3—26no—
87C24×C3—26no—
88C24×C3—21no—
89C24×C3D814yescomplemented
90C24×C3C2×D413yes—
91C23×D3—21no—
92C2×C3×D4—23no—
93C2×C3×D4—23no—
94C2×C3×D4—46no—
95C2×C3×D4—46no—
96C2×C3×D4—46no—
97C22×C3×C4—28no—
98C22×C3×C4—224no—
99C22×C3×C4—26no—
100C22×C3×C4—212no—
101C22×C3×C4—26no—
102C22×C3×C4—26no—
103C22×C3×C4—23no—
104C22×C3×C4C2×D416yes—
105C22×C3×C4C2×D413yes—
106C22×C3×C4#4116yes—
107C22×C3×C4C2411yescharacteristic
108C2×#67—46no—
109C2×#67—46no—
110C2×#67—46no—
111C22×Dic3—23no—
112C2×D12—23no—
113C2×D12—23no—
114C3×C4⋊C4—23no—
115C3×C4⋊C4—23no—
116C3×C22⋊C4—46no—
117C3×C22⋊C4—46no—
118#217—46no—
119#213—44no—
120(C2×D3)⋊C4—46no—
121C4⋊Dic3—23no—
122C4⋊Dic3—23no—
123C2×C3⋊C8—224no—
124C2×C3⋊C8—26no—
125C2×C3⋊C8—224no—
126C2×C3⋊C8C2×D416yes—
127C2×C3⋊C8C2×D416yes—
128C25#6711yescharacteristic
129C22×D4—23no—
130C22×D4—31no—
131C22×D4—63no—
132C22×D4C22×D311yescharacteristic
133C23×C4—26no—
134C23×C4—26no—
135C23×C4#6713yes—
136C23×C4C22×D313yes—
137C2×D8—66no—
138C22×C8—31no—
139C22×C8—312no—
140C22×C8—33no—
141C22×C8—612no—
142C22×C8—612no—
143C22×C8—632no—
144C22×C8—66no—
145#121—26no—
146#121—26no—
147#121—66no—
148#121—66no—
149#120—44no—
150#120—124no—
151C2×C4⋊C4—33no—
152C2×C4⋊C4C22×D313yes—
153C2×C22⋊C4—23no—
154C2×C22⋊C4—23no—
155C2×C22⋊C4—63no—
156C2×C22⋊C4—63no—
157#107—66no—
158#102—66no—
159#102—66no—
160#95—23no—
161#95—63no—
162C23×C3—28no—
163C23×C3—224no—
164C23×C3—26no—
165C23×C3—26no—
166C23×C3—224no—
167C23×C3—43no—
168C23×C3—41no—
169C23×C3—43no—
170C23×C3#12116yes—
171C23×C3C2×D816yes—
172C23×C3#13516yes—
173C23×C3C22×D411yescharacteristic
174C22×D3—43no—
175C22×D3—41no—
176C22×D3—43no—
177C3×D4—41no—
178C3×D4—43no—
179C3×D4—812no—
180C2×C3×C4—224no—
181C2×C3×C4—26no—
182C2×C3×C4—26no—
183C2×C3×C4—26no—
184C2×C3×C4—212no—
185C2×C3×C4—43no—
186C2×C3×C4—43no—
187C2×C3×C4—43no—
188C2×C3×C4#12116yes—
189C2×C3×C4C22×D413yes—
190C2×C3×C4C2×#4113yes—
191#67—812no—
192C2×Dic3—43no—
193C2×Dic3—43no—
194C2×Dic3—43no—
195D12—41no—
196D12—43no—
197C3⋊C8—26no—
198C3⋊C8#12014yescomplemented
199C24—26no—
200C24—26no—
201C24—21no—
202C24—61no—
203C24#21314yescomplemented
204C24C2×#6713yes—
205C2×D4—23no—
206C2×D4—23no—
207C2×D4—46no—
208C2×D4—46no—
209C2×D4—46no—
210C2×D4—63no—
211C2×D4—63no—
212C2×D4—126no—
213C2×D4—126no—
214C2×D4—126no—
215C22×C4—28no—
216C22×C4—224no—
217C22×C4—26no—
218C22×C4—212no—
219C22×C4—26no—
220C22×C4—26no—
221C22×C4—23no—
222C22×C4—63no—
223C22×C4D3×D416yes—
224C22×C4#23716yes—
225C22×C4C2×#6713yes—
226C22×C4C23×D311yescharacteristic
227D8—124no—
228C2×C8—36no—
229C2×C8—36no—
230C2×C8—624no—
231C2×C8—624no—
232C2×C8—66no—
233C4⋊C4—23no—
234C4⋊C4—23no—
235C4⋊C4—63no—
236C4⋊C4—63no—
237C22⋊C4—46no—
238C22⋊C4—46no—
239C22⋊C4—126no—
240C22⋊C4—126no—
241C22×C3—232no—
242C22×C3—26no—
243C22×C3—212no—
244C22×C3—26no—
245C22×C3—212no—
246C22×C3—81no—
247C22×C3—83no—
248C22×C3—83no—
249C22×C3#466112yescomplemented
250C22×C3C2×#12113yes—
251C22×C3C22×D811yescharacteristic
252C22×C3C2×#13513yes—
253C2×D3—81no—
254C2×D3—83no—
255C2×D3—83no—
256C3×C4—26no—
257C3×C4—83no—
258C3×C4C2×#12011yescharacteristic
259C3×C4#52513yes—
260Dic3—83no—
261C23—28no—
262C23—224no—
263C23—26no—
264C23—224no—
265C23—26no—
266C23—41no—
267C23—43no—
268C23—43no—
269C23—121no—
270C23—123no—
271C23—123no—
272C23C2×#21316yes—
273C23#93916yes—
274C23#95016yes—
275C23C22×#6711yescharacteristic
276D4—43no—
277D4—41no—
278D4—812no—
279D4—123no—
280D4—121no—
281D4—2412no—
282C2×C4—224no—
283C2×C4—26no—
284C2×C4—26no—
285C2×C4—212no—
286C2×C4—26no—
287C2×C4—43no—
288C2×C4—43no—
289C2×C4—43no—
290C2×C4—123no—
291C2×C4—123no—
292C2×C4—123no—
293C2×C4#93916yes—
294C2×C4C2×D3×D413yes—
295C2×C4C2×#23713yes—
296C8—34no—
297C8—66no—
298C2×C3—212no—
299C2×C3—161no—
300C2×C3#284711yescharacteristic
301C2×C3C2×#46613yes—
302C2×C3#305013yes—
303D3—161no—
304C22—232no—
305C22—26no—
306C22—212no—
307C22—26no—
308C22—212no—
309C22—81no—
310C22—83no—
311C22—83no—
312C22—241no—
313C22—243no—
314C22—243no—
315C22#5038112yescomplemented
316C22C22×#21311yescharacteristic
317C22C2×#93913yes—
318C22C2×#95013yes—
319C4—26no—
320C4—83no—
321C4—243no—
322C4#558613yes—
323C4D3×#12011yescharacteristic
324C3#3293011yescharacteristic, complemented, Sylow
325C2—212no—
326C2—161no—
327C2—481no—
328C2#8260111yescharacteristic
329C2C2×#503813yes—
330C2#8292213yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2847C2
socleC23×C3C22×D4
hypercenter#2847D3
hypocenterC3#32930
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
2. C3×#2847C2semidirect
3. #84032C2semidirect
4. C2×#5038C2semidirect
5. #71681C2semidirect
6. #69515C2semidirect
7. #69512C2semidirect
13. C22×#213C22semidirect
20. C2×#832C22semidirect
24. C2×#809C22semidirect
25. C2×#807C22semidirect
26. #4529C22semidirect
30. #2847D3semidirect
60. C22×C3⋊C8D4semidirect
89. C24×C3D8semidirect
198. C3⋊C8#120semidirect
203. C24#213semidirect
249. C22×C3#466semidirect
315. C22#5038semidirect
324. C3#32930semidirect

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
#840322
C2×#50382
#716812
#695152
#695122
#329303