Finite Group Databasebeta

← #10944242

Subgroups — #10944242

Order 768.

1630 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 1630 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
2C3×#2848C211yesmaximal, characteristic
3#84031C211yesmaximal, characteristic
4C2×#5051C213yesmaximal, complemented
5#71696C213yesmaximal, complemented
6#69514C213yesmaximal, complemented
7#69512C213yesmaximal, complemented
8#32931—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C2×C3×C22⋊Q8C2213yes—
11C22×#216C2211yescharacteristic, complemented
12C2×#924C2213yes—
13C23×C3⋊C8—212no—
14C23×C3⋊C8C2213yes—
15C23×C3⋊C8C2213yes—
16C3×#385—23no—
17C3×#380C2213yes—
18C3×#379—21no—
19#5247—21no—
20C2×#835C2213yescomplemented
21#5212C2213yes—
22#5210—23no—
23#5051—212no—
24C2×#810C2213yescomplemented
25C2×#807C2213yescomplemented
26#4529C2216yescomplemented
27C24×C8—31no—
28C2×#467—33no—
29#2848—31no—
30#2848D311yescharacteristic
31#1892—33no—
32#1892—33no—
33#1808—33no—
34C25×C3D411yescharacteristic
35C22×C3×Q8C2311yescharacteristic
36C23×C3×C4—26no—
37C23×C3×C4—26no—
38C23×C3×C4D413yes—
39C23×C3×C4C2313yes—
40C22×Dic6C2311yescharacteristic
41C3×C22⋊Q8—26no—
42C3×C22⋊Q8—26no—
43C2×C3×C4⋊C4—23no—
44C2×C3×C4⋊C4C2313yes—
45C2×C3×C22⋊C4—23no—
46C2×C3×C22⋊C4—23no—
47C2×#217—23no—
48C2×#217—23no—
49C2×#216—26no—
50C2×C4⋊Dic3C2313yes—
51#924—26no—
52#924—26no—
53C2×Dic3⋊C4—23no—
54C22×C3⋊C8—232no—
55C22×C3⋊C8—26no—
56C22×C3⋊C8—212no—
57C22×C3⋊C8—212no—
58C22×C3⋊C8D4112yes—
59C22×C3⋊C8C2313yes—
60C22×C3⋊C8C2311yescharacteristic
61C3×#95—23no—
62#835—26no—
63C3⋊#95—23no—
64#810—26no—
65#807—26no—
66C24×C4C2×D311yescharacteristic
67C22×Q16—31no—
68C23×C8—33no—
69C23×C8—33no—
70C23×C8—612no—
71C2×C22⋊Q8—33no—
72C2×C22⋊Q8C2×D313yes—
73#467—612no—
74C2×#107—33no—
75C2×#103—33no—
76C2×#103—33no—
77#385—23no—
78#385—63no—
79#380—33no—
80#380C2×D313yes—
81#379—21no—
82#379—61no—
83#340—36no—
84C24×C3—26no—
85C24×C3—26no—
86C24×C3Q1614yescomplemented
87C24×C3C2×D413yes—
88C2×C3×Q8—23no—
89C2×C3×Q8—23no—
90C22×C3×C4—28no—
91C22×C3×C4—224no—
92C22×C3×C4—26no—
93C22×C3×C4—212no—
94C22×C3×C4—26no—
95C22×C3×C4—26no—
96C22×C3×C4—23no—
97C22×C3×C4—21no—
98C22×C3×C4C2×D416yes—
99C22×C3×C4C2×D413yes—
100C22×C3×C4#4116yes—
101C22×C3×C4C2411yescharacteristic
102C22×Dic3—23no—
103C22×Dic3—21no—
104C2×Dic6—23no—
105C2×Dic6—23no—
106C3×C4⋊C4—23no—
107C3×C4⋊C4—23no—
108C3×C4⋊C4—46no—
109C3×C22⋊C4—46no—
110C3×C22⋊C4—46no—
111#217—46no—
112#217—46no—
113#216—44no—
114C4⋊Dic3—23no—
115C4⋊Dic3—23no—
116Dic3⋊C4—46no—
117C2×C3⋊C8—224no—
118C2×C3⋊C8—26no—
119C2×C3⋊C8—224no—
120C2×C3⋊C8C2×D416yes—
121C2×C3⋊C8C2×D416yes—
122C25#6711yescharacteristic
123C22×Q8—31no—
124C22×Q8C22×D311yescharacteristic
125C23×C4—26no—
126C23×C4—26no—
127C23×C4#6713yes—
128C23×C4C22×D313yes—
129C2×Q16—66no—
130C22×C8—31no—
131C22×C8—312no—
132C22×C8—33no—
133C22×C8—612no—
134C22×C8—612no—
135C22×C8—632no—
136C22×C8—66no—
137C22⋊Q8—26no—
138C22⋊Q8—26no—
139C22⋊Q8—66no—
140C22⋊Q8—66no—
141C2×C4⋊C4—23no—
142C2×C4⋊C4—33no—
143C2×C4⋊C4—63no—
144C2×C4⋊C4C22×D313yes—
145C2×C22⋊C4—23no—
146C2×C22⋊C4—23no—
147C2×C22⋊C4—63no—
148C2×C22⋊C4—63no—
149#107—66no—
150#103—66no—
151#103—66no—
152#95—23no—
153#95—63no—
154C23×C3—28no—
155C23×C3—224no—
156C23×C3—26no—
157C23×C3—26no—
158C23×C3—224no—
159C23×C3#12116yes—
160C23×C3C2×Q1616yes—
161C23×C3#13516yes—
162C23×C3C22×D411yescharacteristic
163C3×Q8—41no—
164C3×Q8—43no—
165C2×C3×C4—224no—
166C2×C3×C4—26no—
167C2×C3×C4—26no—
168C2×C3×C4—26no—
169C2×C3×C4—212no—
170C2×C3×C4—43no—
171C2×C3×C4—43no—
172C2×C3×C4—46no—
173C2×C3×C4#12116yes—
174C2×C3×C4C22×D413yes—
175C2×C3×C4C2×#4113yes—
176C2×Dic3—43no—
177C2×Dic3—43no—
178C2×Dic3—46no—
179Dic6—41no—
180Dic6—43no—
181C3⋊C8—26no—
182C3⋊C8#12014yes—
183C24—26no—
184C24—26no—
185C24#21614yescomplemented
186C24C2×#6713yes—
187C2×Q8—23no—
188C2×Q8—23no—
189C2×Q8—63no—
190C2×Q8—63no—
191C22×C4—28no—
192C22×C4—224no—
193C22×C4—26no—
194C22×C4—212no—
195C22×C4—26no—
196C22×C4—26no—
197C22×C4—23no—
198C22×C4—21no—
199C22×C4—63no—
200C22×C4—61no—
201C22×C4D3×D416yes—
202C22×C4#23716yes—
203C22×C4C2×#6713yes—
204C22×C4C23×D311yescharacteristic
205Q16—124no—
206C2×C8—36no—
207C2×C8—36no—
208C2×C8—624no—
209C2×C8—624no—
210C2×C8—66no—
211C4⋊C4—23no—
212C4⋊C4—23no—
213C4⋊C4—46no—
214C4⋊C4—63no—
215C4⋊C4—63no—
216C4⋊C4—126no—
217C22⋊C4—46no—
218C22⋊C4—46no—
219C22⋊C4—126no—
220C22⋊C4—126no—
221C22×C3—232no—
222C22×C3—26no—
223C22×C3—212no—
224C22×C3—26no—
225C22×C3—212no—
226C22×C3#467112yescomplemented
227C22×C3C2×#12113yes—
228C22×C3C22×Q1611yescharacteristic
229C22×C3C2×#13513yes—
230C3×C4—26no—
231C3×C4—81no—
232C3×C4—83no—
233C3×C4C2×#12011yescharacteristic
234C3×C4#52513yes—
235Dic3—81no—
236Dic3—83no—
237C23—28no—
238C23—224no—
239C23—26no—
240C23—224no—
241C23—26no—
242C23#93916yes—
243C23C2×#21616yes—
244C23#95016yes—
245C23C22×#6711yescharacteristic
246Q8—43no—
247Q8—41no—
248Q8—123no—
249Q8—121no—
250C2×C4—224no—
251C2×C4—26no—
252C2×C4—26no—
253C2×C4—212no—
254C2×C4—26no—
255C2×C4—43no—
256C2×C4—43no—
257C2×C4—46no—
258C2×C4—123no—
259C2×C4—123no—
260C2×C4—126no—
261C2×C4#93916yes—
262C2×C4C2×D3×D413yes—
263C2×C4C2×#23713yes—
264C8—34no—
265C8—66no—
266C2×C3—212no—
267C2×C3#284711yescharacteristic
268C2×C3C2×#46713yes—
269C2×C3#305013yes—
270C22—232no—
271C22—212no—
272C22—26no—
273C22—26no—
274C22—212no—
275C22#5051112yescomplemented
276C22C2×#93913yes—
277C22C22×#21611yescharacteristic
278C22C2×#95013yes—
279C4—26no—
280C4—81no—
281C4—83no—
282C4—241no—
283C4—243no—
284C4#558613yes—
285C4D3×#12011yescharacteristic
286C3#3293111yescharacteristic, complemented, Sylow
287C2—212no—
288C2#8260111yescharacteristic
289C2C2×#505113yes—
290C2#8292213yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2848C2
socleC23×C3C22×D4
prime radicalC25×C3D4
hypercenter#2848D3
hypocenterC3#32931
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
4. C2×#5051C2semidirect
5. #71696C2semidirect
6. #69514C2semidirect
7. #69512C2semidirect
11. C22×#216C22semidirect
20. C2×#835C22semidirect
24. C2×#810C22semidirect
25. C2×#807C22semidirect
26. #4529C22semidirect
86. C24×C3Q16semidirect
185. C24#216semidirect
226. C22×C3#467semidirect
275. C22#5051semidirect
286. C3#32931semidirect

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
#840312
C2×#50512
#716962
#695142
#695122
#329313