Finite Group Databasebeta

← #10943973

Subgroups — #10943973

Order 768.

3706 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 3706 conjugacy classes of subgroups — too many to draw. Only its 315 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
1C2×C3×#585C211yesmaximal, characteristic
2C2×#5243C214yesmaximal, complemented
3#84090C213yesmaximal, complemented
4#72142C214yesmaximal, complemented
5#72139C213yesmaximal, complemented
6#32901—31nomaximal, Sylow
7C3×#585—216no—
8C2×C3×2−1+4—23no—
9C2×C3×H(5,𝔽2)—23no—
10C2×C3×H(5,𝔽2)C2214yes—
11C22×C3×#41—26no—
12C22×C3×#41C2213yes—
13C2×#935C2216yescomplemented
14C22×Dic3⋊C4C2211yescharacteristic, complemented
15#5243—216no—
16#5230—23no—
17#5222—23no—
18#5220C2216yescomplemented
19C2×#834C22112yescomplemented
20#5211C2213yescomplemented
21#5210—26no—
22C2×#585D311yescharacteristic
23C2×#458—34no—
24#2406—33no—
25#1790—34no—
26#1782—33no—
27C3×2−1+4—248no—
28C3×H(5,𝔽2)—248no—
29C3×H(5,𝔽2)—224no—
30C3×H(5,𝔽2)D4116yes—
31C2×C3×#41—24no—
32C2×C3×#41—212no—
33C2×C3×#41—248no—
34C2×C3×#41—248no—
35C2×C3×#41—224no—
36C2×C3×#41—26no—
37C2×C3×#41—212no—
38C2×C3×#41—26no—
39C22×C3×Q8—26no—
40C22×C3×Q8D413yes—
41C22×C3×D4—26no—
42C22×C3×D4—212no—
43C22×C3×D4D413yes—
44C22×C3×D4C2316yes—
45C23×C3×C4—24no—
46C23×C3×C4D416yes—
47C23×C3×C4C2311yescharacteristic
48C23×Dic3C2311yescharacteristic
49C23×Dic3C2311yescharacteristic
50C2×#217—26no—
51C2×#217—26no—
52C2×#217C2316yes—
53#935—224no—
54C2×Dic3⋊C4—23no—
55C2×Dic3⋊C4—26no—
56C2×Dic3⋊C4—23no—
57#834—224no—
58C3⋊#95—23no—
59C3⋊#95—26no—
60C3⋊#95—23no—
61#585—216no—
62C2×2−1+4—23no—
63C2×H(5,𝔽2)—23no—
64C2×H(5,𝔽2)C2×D314yes—
65C22×#41—26no—
66C22×#41C2×D313yes—
67C2×#123—36no—
68C22×C4⋊C4—31no—
69#458—616no—
70C2×#99—312no—
71#395—63no—
72#394—63no—
73#386—36no—
74#385—66no—
75#381—33no—
76C24×C3—24no—
77C24×C3C2×D416yes—
78C24×C3C2411yescharacteristic
79C3×#41—232no—
80C3×#41—248no—
81C3×#41—296no—
82C3×#41—248no—
83C3×#41—248no—
84C3×#41—248no—
85C2×C3×Q8—24no—
86C2×C3×Q8—248no—
87C2×C3×Q8—26no—
88C2×C3×Q8—212no—
89C2×C3×D4—212no—
90C2×C3×D4—248no—
91C2×C3×D4—248no—
92C2×C3×D4—248no—
93C2×C3×D4—212no—
94C2×C3×D4—212no—
95C2×C3×D4—212no—
96C2×C3×D4—26no—
97C2×C3×D4C2×D4124yes—
98C22×C3×C4—212no—
99C22×C3×C4—232no—
100C22×C3×C4—212no—
101C22×C3×C4—24no—
102C22×C3×C4—26no—
103C22×C3×C4—26no—
104C22×C3×C4—212no—
105C22×C3×C4C2×D413yes—
106C22×C3×C4C2×D416yes—
107C22×C3×C4C2×D413yes—
108C22×Dic3—23no—
109C22×Dic3—26no—
110C22×Dic3—23no—
111C22×Dic3—24no—
112C22×Dic3—42no—
113#217—212no—
114#217—412no—
115#217—412no—
116Dic3⋊C4—41no—
117Dic3⋊C4—43no—
118Dic3⋊C4—412no—
1192−1+4—248no—
120H(5,𝔽2)—248no—
121H(5,𝔽2)—224no—
122H(5,𝔽2)#67116yes—
123C2×#41—24no—
124C2×#41—212no—
125C2×#41—248no—
126C2×#41—248no—
127C2×#41—224no—
128C2×#41—26no—
129C2×#41—212no—
130C2×#41—26no—
131C22×Q8—26no—
132C22×Q8#6713yes—
133C22×D4—26no—
134C22×D4—212no—
135C22×D4#6713yes—
136C22×D4C22×D316yes—
137C23×C4—24no—
138C23×C4—31no—
139C23×C4—31no—
140C23×C4#6716yes—
141C23×C4C22×D311yescharacteristic
142#123—624no—
143C2×C4⋊C4—63no—
144C2×C4⋊C4—63no—
145C2×C4⋊C4—66no—
146C2×C22⋊C4—36no—
147C2×C22⋊C4—66no—
148C2×C22⋊C4—66no—
149#99—624no—
150#95—63no—
151#95—63no—
152#95—66no—
153C23×C3—216no—
154C23×C3—26no—
155C23×C3—26no—
156C23×C3—41no—
157C23×C3—41no—
158C23×C3—46no—
159C23×C3—46no—
160C23×C3—84no—
161C23×C3—84no—
162C23×C3#120112yes—
163C23×C3C22×D413yes—
164C23×C3C22×D414yes—
165C3×Q8—216no—
166C3×Q8—224no—
167C3×D4—248no—
168C3×D4—224no—
169C3×D4—248no—
170C2×C3×C4—224no—
171C2×C3×C4—248no—
172C2×C3×C4—224no—
173C2×C3×C4—26no—
174C2×C3×C4—26no—
175C2×C3×C4—26no—
176C2×C3×C4—26no—
177C2×C3×C4#12011yescharacteristic
178C2×C3×C4#12013yes—
179C2×C3×C4#120112yes—
180C2×Dic3—26no—
181C2×Dic3—43no—
182C2×Dic3—41no—
183C2×Dic3—46no—
184C2×Dic3—46no—
185C2×Dic3—41no—
186C2×Dic3—46no—
187C2×Dic3—46no—
188C24—24no—
189C24C2×#6716yes—
190C24C23×D311yescharacteristic
191#41—232no—
192#41—248no—
193#41—296no—
194#41—248no—
195#41—248no—
196#41—248no—
197C2×Q8—24no—
198C2×Q8—248no—
199C2×Q8—26no—
200C2×Q8—212no—
201C2×D4—212no—
202C2×D4—248no—
203C2×D4—248no—
204C2×D4—248no—
205C2×D4—212no—
206C2×D4—212no—
207C2×D4—212no—
208C2×D4—26no—
209C2×D4C2×#67124yes—
210C22×C4—212no—
211C22×C4—232no—
212C22×C4—212no—
213C22×C4—24no—
214C22×C4—26no—
215C22×C4—26no—
216C22×C4—212no—
217C22×C4—63no—
218C22×C4—66no—
219C22×C4—63no—
220C22×C4—64no—
221C22×C4—122no—
222C22×C4C2×#6713yes—
223C22×C4C2×#6716yes—
224C22×C4C2×#6713yes—
225C4⋊C4—121no—
226C4⋊C4—123no—
227C4⋊C4—1212no—
228C22⋊C4—612no—
229C22⋊C4—1212no—
230C22⋊C4—1212no—
231C22×C3—23no—
232C22×C3—23no—
233C22×C3—212no—
234C22×C3—44no—
235C22×C3—412no—
236C22×C3—46no—
237C22×C3—46no—
238C22×C3—86no—
239C22×C3—816no—
240C22×C3C2×#12016yes—
241C22×C3C2×#12011yescharacteristic
242C3×C4—21no—
243C3×C4—212no—
244C3×C4—23no—
245Dic3—42no—
246Dic3—86no—
247Dic3—81no—
248C23—216no—
249C23—26no—
250C23—26no—
251C23—41no—
252C23—41no—
253C23—46no—
254C23—46no—
255C23—84no—
256C23—84no—
257C23#953112yes—
258C23C22×#6713yes—
259C23C22×#6714yes—
260Q8—216no—
261Q8—224no—
262D4—248no—
263D4—224no—
264D4—248no—
265C2×C4—224no—
266C2×C4—248no—
267C2×C4—224no—
268C2×C4—26no—
269C2×C4—26no—
270C2×C4—26no—
271C2×C4—26no—
272C2×C4—66no—
273C2×C4—123no—
274C2×C4—121no—
275C2×C4—126no—
276C2×C4—126no—
277C2×C4—126no—
278C2×C4—126no—
279C2×C4—121no—
280C2×C4#953112yes—
281C2×C4#95311yescharacteristic
282C2×C4#95313yes—
283C2×C3—26no—
284C2×C3—412no—
285C2×C3#284611yescharacteristic
286C2×C3#302511yescharacteristic
287C2×C3#302811yescharacteristic
288C22—23no—
289C22—23no—
290C22—212no—
291C22—44no—
292C22—412no—
293C22—46no—
294C22—46no—
295C22—86no—
296C22—816no—
297C22C2×#95316yes—
298C22C2×#95311yescharacteristic
299C4—21no—
300C4—212no—
301C4—23no—
302C4—122no—
303C4—246no—
304C4—241no—
305C3#3290111yescharacteristic, complemented, Sylow
306C2—26no—
307C2—412no—
308C2#8412411yescharacteristic
309C2#8412511yescharacteristic
310C2#8412811yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
Fitting, prime radicalC2×C3×#585C2
socleC22×C3C2×#120
hypercenterC2×#585D3
hypocenterC3#32901
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#585

Complements

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

NH ≅ G/Ntype
2. C2×#5243C2semidirect
3. #84090C2semidirect
4. #72142C2semidirect
5. #72139C2semidirect
13. C2×#935C22semidirect
14. C22×Dic3⋊C4C22semidirect
18. #5220C22semidirect
19. C2×#834C22semidirect
20. #5211C22semidirect
305. C3#32901semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5852
C2×#52432
#840902
#721422
#721392
#329013