Finite Group Databasebeta

← #82036

Subgroups — #82036

Order 384.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#5769C211yesmaximal, characteristic, complemented
2#5757C211yesmaximal, characteristic, complemented
3C3×#509C211yesmaximal, characteristic, complemented
4#5194C211yesmaximal, characteristic, complemented
5C2×#826C211yesmaximal, characteristic, complemented
6#4916C212yesmaximal, complemented
7#4914C212yesmaximal, complemented
8#4911C212yesmaximal, complemented
9D3×D16C212yesmaximal, complemented
10#4908C211yesmaximal, characteristic, complemented
11#4907C211yesmaximal, characteristic, complemented
12#3415—31nomaximal, Sylow
13#1009—21no—
14#1004—21no—
15C3×#135C2211yescharacteristic, complemented
16C2×C3×D8C2211yescharacteristic, complemented
17#949—21no—
18#932—21no—
19#919C2212yescomplemented
20#914—22no—
21#912C2212yescomplemented
22#911—22no—
23D3×D8C2212yescomplemented
24D3×D8C2212yescomplemented
25#904C2211yescharacteristic, complemented
26C2×D24C2211yescharacteristic, complemented
27#901C2211yescharacteristic
28C3×SD32C2212yescomplemented
29C3×D16C2212yescomplemented
30C3×M32C2211yescharacteristic, complemented
31#829C2211yescharacteristic, complemented
32#828C2211yescharacteristic, complemented
33#827C2211yescharacteristic, complemented
34#826C2212yescomplemented
35#826C2212yescomplemented
36#826C2211yescharacteristic, complemented
37C2×C3⋊C16C2211yescharacteristic, complemented
38#800C2212yescomplemented
39D48C2212yescomplemented
40#798C2212yescomplemented
41D3×C16C2212yescomplemented
42#576—31no—
43#576—31no—
44#509—32no—
45#509—32no—
46#509—31no—
47#509D311yescharacteristic, complemented
48#508—31no—
49#508—32no—
50C2×D16—31no—
51C2×D16—32no—
52#504—31no—
53C3×#41—21no—
54C2×C3×D4—21no—
55C2×#67—42no—
56#239—22no—
57#237—22no—
58#237—41no—
59D3×D4—22no—
60D3×D4—22no—
61D3×D4—42no—
62D3×D4—41no—
63#235—21no—
64#235—41no—
65#235D411yescharacteristic
66C2×D12—21no—
67C2×D12—41no—
68C3×Q16C2311yescharacteristic
69C3×SD16—21no—
70C3×D8—21no—
71C3×D8C2312yes—
72C3×D8C2311yescharacteristic
73C2×C3×C8C2311yescharacteristic
74#215—22no—
75#214—22no—
76#213—22no—
77#213—22no—
78#208D411yescharacteristic
79Dic12C2311yescharacteristic
80D24—21no—
81D24C2311yescharacteristic
82D24C2312yes—
83#204—21no—
84ΓL(1,𝔽25)D412yes—
85D3×C8C2312yes—
86C3×C16C2312yes—
87C3⋊C16C2311yescharacteristic
88C3⋊C16C2311yescharacteristic
89H(5,𝔽2)—61no—
90H(5,𝔽2)—61no—
91AGL(1,ℤ/8)—62no—
92AGL(1,ℤ/8)—62no—
93AGL(1,ℤ/8)—61no—
94AGL(1,ℤ/8)—61no—
95#135—32no—
96#135—32no—
97#135—31no—
98#135C2×D311yescharacteristic, complemented
99C2×D8—32no—
100C2×D8—32no—
101C2×D8—31no—
102C2×D8C2×D311yescharacteristic, complemented
103#131—31no—
104Q32—31no—
105SD32—31no—
106SD32—31no—
107SD32—32no—
108SD32C2×D312yescomplemented
109D16—32no—
110D16—31no—
111D16—32no—
112D16—32no—
113D16C2×D312yescomplemented
114M32—32no—
115M32C2×D311yescharacteristic, complemented
116C2×C16—31no—
117C2×C16—32no—
118C23×C3—41no—
119C22×D3—41no—
120C22×D3—42no—
121C22×D3—42no—
122C3×Q8—21no—
123C3×D4—21no—
124C3×D4—22no—
125C3×D4—41no—
126C3×D4—41no—
127C2×C3×C4—41no—
128C2×C3×C4C2×D411yescharacteristic
129#67—41no—
130#67—42no—
131#67—42no—
132#67—42no—
133#67D812yes—
134C2×Dic3—42no—
135D12—21no—
136D12—22no—
137D12—42no—
138D12—41no—
139D12D811yescharacteristic
140C4×D3—41no—
141C4×D3—42no—
142C4×D3C2×D412yes—
143Dic6—21no—
144Dic6D811yescharacteristic
145C3×C8C2×D411yescharacteristic
146C3×C8C2411yescharacteristic
147C3⋊C8C2×D412yes—
148#41—21no—
149#41—31no—
150#41—62no—
151#41—62no—
152#41—61no—
153#41—121no—
154#41—121no—
155C2×D4—21no—
156C2×D4—62no—
157C2×D4—62no—
158C2×D4—61no—
159C2×D4—121no—
160C2×D4—122no—
161C2×D4—121no—
162C2×D4—122no—
163Q16—31no—
164Q16C22×D311yescharacteristic
165SD16—21no—
166SD16—62no—
167SD16—62no—
168SD16—61no—
169D8—21no—
170D8—32no—
171D8—31no—
172D8—62no—
173D8—62no—
174D8—61no—
175D8C22×D312yes—
176D8C22×D311yescharacteristic
177M16—31no—
178M16—32no—
179C2×C8—32no—
180C2×C8C22×D311yescharacteristic
181C16—31no—
182C16—31no—
183C16C22×D312yes—
184C22×C3—42no—
185C22×C3—41no—
186C22×C3—161no—
187C22×C3C2×D811yescharacteristic
188C2×D3—42no—
189C2×D3—41no—
190C2×D3—162no—
191C2×D3—162no—
192C2×D3—161no—
193C2×D3C2×D812yes—
194C3×C4—41no—
195C3×C4C2×D811yescharacteristic
196C3×C4C22×D411yescharacteristic
197Dic3—41no—
198Dic3C2×D812yes—
199C23—41no—
200C23—122no—
201C23—122no—
202C23—121no—
203Q8—21no—
204Q8—31no—
205Q8—61no—
206D4—22no—
207D4—21no—
208D4—32no—
209D4—31no—
210D4—41no—
211D4—41no—
212D4—61no—
213D4—62no—
214D4—122no—
215D4—122no—
216D4—122no—
217D4—122no—
218D4—121no—
219D4—121no—
220C2×C4—32no—
221C2×C4—41no—
222C2×C4—122no—
223C2×C4—122no—
224C2×C4—121no—
225C2×C4D3×D411yescharacteristic
226C8—32no—
227C8D3×D411yescharacteristic
228C8C23×D311yescharacteristic
229C2×C3—21no—
230C2×C3—81no—
231C2×C3—82no—
232C2×C3C22×D811yescharacteristic
233D3—22no—
234D3—82no—
235D3—81no—
236C22—32no—
237C22—41no—
238C22—42no—
239C22—122no—
240C22—121no—
241C22—161no—
242C22—481no—
243C22—482no—
244C22—482no—
245C22D3×D811yescharacteristic
246C4—32no—
247C4—41no—
248C4—121no—
249C4D3×D811yescharacteristic
250C4C2×D3×D411yescharacteristic
251C3#341511yescharacteristic, complemented, Sylow
252C2—21no—
253C2—62no—
254C2—82no—
255C2—81no—
256C2—242no—
257C2—241no—
258C2C2×D3×D811yescharacteristic
subgroupG/H
centerC2C2×D3×D8
commutator, cosocleC3×C8C24
FrattiniC8C23×D3
FittingC3×#509C2
socleC2×C3C22×D8
hypercenter#509D3
hypocenterC3#3415
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #5769C2semidirect
2. #5757C2semidirect
3. C3×#509C2semidirect
4. #5194C2semidirect
5. C2×#826C2semidirect
6. #4916C2semidirect
7. #4914C2semidirect
8. #4911C2semidirect
9. D3×D16C2semidirect
10. #4908C2semidirect
11. #4907C2semidirect
15. C3×#135C22semidirect
16. C2×C3×D8C22semidirect
19. #919C22semidirect
21. #912C22semidirect
23. D3×D8C22semidirect
24. D3×D8C22semidirect
25. #904C22semidirect
26. C2×D24C22semidirect
28. C3×SD32C22semidirect
29. C3×D16C22semidirect
30. C3×M32C22semidirect
31. #829C22semidirect
32. #828C22semidirect
33. #827C22semidirect
34. #826C22semidirect
35. #826C22semidirect
36. #826C22semidirect
37. C2×C3⋊C16C22semidirect
38. #800C22semidirect
39. D48C22semidirect
40. #798C22semidirect
41. D3×C16C22semidirect
47. #509D3semidirect
98. #135C2×D3semidirect
102. C2×D8C2×D3semidirect
108. SD32C2×D3semidirect
113. D16C2×D3semidirect
115. M32C2×D3semidirect
251. C3#3415semidirect

Maximal subgroups

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

subgroupindex
#57692
#57572
C3×#5092
#51942
C2×#8262
#49162
#49142
#49112
D3×D162
#49082
#49072
#34153