Finite Group Databasebeta

← #80302

Subgroups — #80302

Order 384.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#5837C212yesmaximal, complemented
2#5778C212yesmaximal, complemented
3C3×#456C211yesmaximal, characteristic, complemented
4#5203C212yesmaximal, complemented
5#5095C211yesmaximal, characteristic, complemented
6#5085C211yesmaximal, characteristic, complemented
7#4824C212yesmaximal, complemented
8#4822C212yesmaximal, complemented
9#4754C212yesmaximal, complemented
10#3291—31nomaximal, Sylow
11C3×2−1+4C2211yescharacteristic
12#1009C2212yescomplemented
13#1007C2211yescharacteristic
14C3×#137C2212yescomplemented
15C3×C4⋊Q8C2211yescharacteristic, complemented
16#950—22no—
17#950C2212yes—
18#949—22no—
19#949C2212yes—
20#948C2212yescomplemented
21#942—22no—
22#942C2212yescomplemented
23C2×#215—22no—
24C2×#215C2212yescomplemented
25C2×#215C2212yescomplemented
26#919—22no—
27#918—22no—
28#914—22no—
29D3×SD16—22no—
30#908C2212yes—
31#907C2212yescomplemented
32#874C2211yescharacteristic
33C3×#104C2212yescomplemented
34C3×#101C2211yescharacteristic, complemented
35#836C2211yescharacteristic, complemented
36#823C2212yescomplemented
37#822C2212yescomplemented
38#805C2212yescomplemented
39C4×C3⋊C8C2211yescharacteristic, complemented
40#577—32no—
41#577—32no—
42#492—31no—
43#471—32no—
44#471—32no—
45#456—31no—
46#456D311yescharacteristic, complemented
47#453—32no—
48#443—32no—
49C3×#41—22no—
50C3×#41—41no—
51C3×#41C2312yes—
52C2×C3×Q8—22no—
53C2×C3×Q8C2311yescharacteristic
54C2×C3×Q8C2311yescharacteristic
55#239—22no—
56#239—22no—
57D3×Q8—22no—
58D3×D4—22no—
59D3×D4—42no—
60#235—22no—
61#235—41no—
62#235C2312yes—
63C2×D12—22no—
64C2×D12—41no—
65C2×D12C2312yes—
66C3×Q16—22no—
67C3×SD16—22no—
68C3×M16C2312yes—
69C3×C4⋊C4—41no—
70C3×C42C2311yescharacteristic
71#216—22no—
72#216D412yescomplemented
73#215—22no—
74#215—22no—
75#215—22no—
76#215—22no—
77#215—22no—
78#215—22no—
79#215D412yescomplemented
80#214D412yescomplemented
81#213—22no—
82#213D412yescomplemented
83#208—22no—
84#208C2312yes—
85C2×C3⋊C8—22no—
86C2×C3⋊C8C2312yes—
87D24—22no—
88#204—22no—
89ΓL(1,𝔽25)—22no—
90D3×C8—22no—
912−1+4—31no—
922−1+4C2×D311yescharacteristic
93H(5,𝔽2)—32no—
94#137—32no—
95#137—62no—
96#137—62no—
97#137C2×D312yescomplemented
98AGL(1,ℤ/8)—32no—
99AGL(1,ℤ/8)—32no—
100AGL(1,ℤ/8)—62no—
101AGL(1,ℤ/8)—62no—
102#135—32no—
103#135—62no—
104#135—62no—
105C2×SD16—32no—
106C2×SD16—32no—
107C2×SD16—62no—
108C2×SD16—62no—
109#131—32no—
110#131—32no—
111C4⋊Q8C2×D311yescharacteristic, complemented
112#127—31no—
113#108—32no—
114#104—32no—
115#104C2×D312yescomplemented
116#101—31no—
117#101C2×D311yescharacteristic, complemented
118#100—32no—
119C4×C8—31no—
120C22×D3—22no—
121C22×D3—42no—
122C3×Q8—22no—
123C3×Q8—22no—
124C3×Q8—41no—
125C3×Q8C2×D412yes—
126C3×D4—22no—
127C3×D4—41no—
128C3×D4C2×D412yes—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—42no—
134C2×C3×C4C2411yescharacteristic
135#67—22no—
136#67—42no—
137D12—22no—
138D12—22no—
139D12—42no—
140D12—41no—
141D12—82no—
142D12C2×D412yes—
143C4×D3—22no—
144C4×D3—42no—
145C4×D3—42no—
146Dic6—41no—
147Dic6C2×D412yes—
148C3×C8—22no—
149C3⋊C8—22no—
150C3⋊C8C2×D412yes—
151C3⋊C8C2×D412yes—
152#41—22no—
153#41—32no—
154#41—41no—
155#41—62no—
156#41—62no—
157#41—62no—
158#41—121no—
159#41C22×D312yes—
160C2×Q8—22no—
161C2×Q8—62no—
162C2×Q8C22×D311yescharacteristic
163C2×Q8C22×D311yescharacteristic
164C2×D4—32no—
165C2×D4—62no—
166C2×D4—62no—
167C2×D4—121no—
168C2×D4—122no—
169Q16—22no—
170Q16—32no—
171Q16—62no—
172SD16—22no—
173SD16—32no—
174SD16—32no—
175SD16—62no—
176SD16—62no—
177SD16—62no—
178SD16—62no—
179SD16—62no—
180SD16—62no—
181SD16—62no—
182D8—32no—
183D8—62no—
184D8—62no—
185M16—32no—
186M16—62no—
187M16—62no—
188M16C22×D312yes—
189C2×C8—32no—
190C2×C8—62no—
191C2×C8—62no—
192C4⋊C4—41no—
193C42C22×D311yescharacteristic
194C22×C3—22no—
195C22×C3H(5,𝔽2)11yescharacteristic
196C2×D3—22no—
197C2×D3—42no—
198C2×D3—82no—
199C2×D3—162no—
200C3×C4—22no—
201C3×C4—22no—
202C3×C4—41no—
203C3×C4—41no—
204C3×C4C22×D412yes—
205Dic3—22no—
206C23—62no—
207C23—122no—
208Q8—22no—
209Q8—22no—
210Q8—32no—
211Q8—41no—
212Q8—121no—
213Q8D3×D412yes—
214D4—22no—
215D4—32no—
216D4—41no—
217D4—62no—
218D4—62no—
219D4—62no—
220D4—122no—
221D4—121no—
222D4—122no—
223D4—242no—
224D4D3×D412yes—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—42no—
230C2×C4—62no—
231C2×C4—122no—
232C2×C4—122no—
233C2×C4C23×D311yescharacteristic
234C8—22no—
235C8—32no—
236C8—32no—
237C8—62no—
238C2×C3—21no—
239C2×C3—42no—
240C2×C3D4211yescharacteristic
241D3—42no—
242D3—82no—
243C22—22no—
244C22—62no—
245C22—122no—
246C22—242no—
247C22—482no—
248C22#100411yescharacteristic
249C4—22no—
250C4—22no—
251C4—41no—
252C4—41no—
253C4—62no—
254C4C2×D3×D412yes—
255C3#329111yescharacteristic, complemented, Sylow
256C2—21no—
257C2—42no—
258C2—122no—
259C2—242no—
260C2#559011yescharacteristic
subgroupG/H
centerC2#5590
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#456C2
socleC2×C3D42
hypercenter#456D3
hypocenterC3#3291
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #5837C2semidirect
2. #5778C2semidirect
3. C3×#456C2semidirect
4. #5203C2semidirect
5. #5095C2semidirect
6. #5085C2semidirect
7. #4824C2semidirect
8. #4822C2semidirect
9. #4754C2semidirect
12. #1009C22semidirect
14. C3×#137C22semidirect
15. C3×C4⋊Q8C22semidirect
20. #948C22semidirect
22. #942C22semidirect
24. C2×#215C22semidirect
25. C2×#215C22semidirect
31. #907C22semidirect
33. C3×#104C22semidirect
34. C3×#101C22semidirect
35. #836C22semidirect
36. #823C22semidirect
37. #822C22semidirect
38. #805C22semidirect
39. C4×C3⋊C8C22semidirect
46. #456D3semidirect
72. #216D4semidirect
79. #215D4semidirect
80. #214D4semidirect
82. #213D4semidirect
97. #137C2×D3semidirect
111. C4⋊Q8C2×D3semidirect
115. #104C2×D3semidirect
117. #101C2×D3semidirect
255. C3#3291semidirect

Maximal subgroups

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

subgroupindex
#58372
#57782
C3×#4562
#52032
#50952
#50852
#48242
#48222
#47542
#32913