Finite Group Databasebeta

← #82032

Subgroups — #82032

Order 384.

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

normal subgroups only

This group has 360 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#5771C211yesmaximal, characteristic, complemented
2#5769C211yesmaximal, characteristic, complemented
3C3×#508C211yesmaximal, characteristic, complemented
4#5195C211yesmaximal, characteristic, complemented
5#5193C211yesmaximal, characteristic, complemented
6#4918C212yesmaximal, complemented
7#4916C212yesmaximal, complemented
8D3×SD32C212yesmaximal, complemented
9#4911C212yesmaximal, complemented
10C2×#800C211yesmaximal, characteristic, complemented
11#4901C211yesmaximal, characteristic, complemented
12#3416—31nomaximal, Sylow
13#1010—21no—
14#1009—21no—
15C3×#135C2211yescharacteristic
16C3×#135C2211yescharacteristic, complemented
17#951—21no—
18#949—21no—
19#919C2212yes—
20D3×Q16C2212yescomplemented
21#915—22no—
22#914—22no—
23#912C2212yescomplemented
24D3×D8C2212yes—
25C2×Dic12C2211yescharacteristic, complemented
26C2×D24C2211yescharacteristic
27#901C2211yescharacteristic
28C3×Q32C2211yescharacteristic, complemented
29C3×SD32C2211yescharacteristic, complemented
30C3×SD32C2211yescharacteristic, complemented
31C3×D16C2211yescharacteristic, complemented
32C2×C3×C16C2211yescharacteristic, complemented
33#829C2212yescomplemented
34#828C2212yescomplemented
35#827C2212yescomplemented
36#826C2212yescomplemented
37#812C2211yescharacteristic, complemented
38#800C2212yescomplemented
39#800C2212yescomplemented
40#798C2212yescomplemented
41D3×C16C2212yescomplemented
42#578—31no—
43#576—31no—
44#510—31no—
45#510—32no—
46#509—31no—
47#509—32no—
48#508—32no—
49#508D311yescharacteristic, complemented
50C2×SD32—32no—
51C2×SD32—31no—
52#504—31no—
53C3×#41—21no—
54C3×#41—21no—
55#239—22no—
56D3×Q8—22no—
57#237—22no—
58#237—42no—
59D3×D4—22no—
60D3×D4—42no—
61#235—41no—
62#235—41no—
63#235D411yescharacteristic
64C2×D12—21no—
65C2×D12—41no—
66C2×Dic6—21no—
67C2×Dic6—41no—
68C3×Q16C2311yescharacteristic
69C3×Q16C2311yescharacteristic
70C3×SD16—21no—
71C3×SD16—21no—
72C3×D8C2311yescharacteristic
73C3×D8C2311yescharacteristic
74C2×C3×C8C2311yescharacteristic
75#216—22no—
76#215—22no—
77#214—22no—
78#213—22no—
79#208D411yescharacteristic
80Dic12—21no—
81Dic12C2312yes—
82D24—21no—
83D24C2312yes—
84ΓL(1,𝔽25)D412yes—
85D3×C8C2312yes—
86C3×C16C2311yescharacteristic
87C3×C16C2311yescharacteristic
88C3⋊C16C2312yes—
892−1+4—61no—
90H(5,𝔽2)—61no—
91#137—62no—
92#137—61no—
93AGL(1,ℤ/8)—62no—
94AGL(1,ℤ/8)—61no—
95#135—32no—
96#135—32no—
97#135C2×D311yescharacteristic
98#135C2×D311yescharacteristic, complemented
99C2×Q16—32no—
100C2×Q16—31no—
101C2×D8—32no—
102C2×D8—31no—
103#131—31no—
104Q32—32no—
105Q32C2×D311yescharacteristic, complemented
106SD32—32no—
107SD32—32no—
108SD32—32no—
109SD32—32no—
110SD32C2×D311yescharacteristic, complemented
111SD32C2×D311yescharacteristic, complemented
112D16—32no—
113D16C2×D311yescharacteristic, complemented
114M32—31no—
115M32—32no—
116C2×C16—32no—
117C2×C16C2×D311yescharacteristic, complemented
118C22×D3—41no—
119C22×D3—42no—
120C3×Q8—21no—
121C3×Q8—21no—
122C3×D4—21no—
123C3×D4—21no—
124C3×D4—41no—
125C3×D4—41no—
126C2×C3×C4—41no—
127C2×C3×C4—41no—
128C2×C3×C4C2×D411yescharacteristic
129#67—42no—
130#67—42no—
131#67D812yes—
132C2×Dic3—41no—
133C2×Dic3—42no—
134D12—22no—
135D12—42no—
136D12—41no—
137D12D811yescharacteristic
138C4×D3—42no—
139C4×D3—42no—
140C4×D3C2×D412yes—
141Dic6—22no—
142Dic6—41no—
143Dic6—42no—
144Dic6D811yescharacteristic
145C3×C8C2×D411yescharacteristic
146C3×C8C2411yescharacteristic
147C3⋊C8C2×D412yes—
148#41—21no—
149#41—21no—
150#41—31no—
151#41—62no—
152#41—62no—
153#41—121no—
154#41—121no—
155#41—122no—
156C2×Q8—62no—
157C2×Q8—61no—
158C2×Q8—121no—
159C2×D4—62no—
160C2×D4—61no—
161C2×D4—121no—
162C2×D4—122no—
163Q16—32no—
164Q16—62no—
165Q16—61no—
166Q16C22×D311yescharacteristic
167Q16C22×D311yescharacteristic
168SD16—21no—
169SD16—21no—
170SD16—62no—
171SD16—62no—
172D8—32no—
173D8—62no—
174D8—61no—
175D8C22×D311yescharacteristic
176D8C22×D311yescharacteristic
177M16—31no—
178M16—32no—
179C2×C8—32no—
180C2×C8C22×D311yescharacteristic
181C16—32no—
182C16C22×D311yescharacteristic
183C16C22×D311yescharacteristic
184C22×C3—41no—
185C22×C3—41no—
186C22×C3C2×D811yescharacteristic
187C2×D3—42no—
188C2×D3—162no—
189C2×D3—161no—
190C2×D3C2×D812yes—
191C3×C4—41no—
192C3×C4—41no—
193C3×C4C2×D811yescharacteristic
194C3×C4C22×D411yescharacteristic
195Dic3—42no—
196Dic3C2×D812yes—
197C23—122no—
198C23—121no—
199Q8—21no—
200Q8—21no—
201Q8—31no—
202Q8—62no—
203Q8—122no—
204Q8—121no—
205D4—21no—
206D4—21no—
207D4—32no—
208D4—31no—
209D4—41no—
210D4—41no—
211D4—62no—
212D4—122no—
213D4—122no—
214D4—122no—
215D4—121no—
216C2×C4—32no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—122no—
220C2×C4—122no—
221C2×C4—122no—
222C2×C4—121no—
223C2×C4D3×D411yescharacteristic
224C8—32no—
225C8D3×D411yescharacteristic
226C8C23×D311yescharacteristic
227C2×C3—21no—
228C2×C3—81no—
229C2×C3—81no—
230C2×C3C22×D811yescharacteristic
231D3—22no—
232D3—82no—
233C22—32no—
234C22—41no—
235C22—41no—
236C22—122no—
237C22—481no—
238C22—482no—
239C22D3×D811yescharacteristic
240C4—32no—
241C4—41no—
242C4—41no—
243C4—122no—
244C4D3×D811yescharacteristic
245C4C2×D3×D411yescharacteristic
246C3#341611yescharacteristic, complemented, Sylow
247C2—21no—
248C2—62no—
249C2—81no—
250C2—81no—
251C2—242no—
252C2C2×D3×D811yescharacteristic
subgroupG/H
centerC2C2×D3×D8
commutator, cosocleC3×C8C24
FrattiniC8C23×D3
FittingC3×#508C2
socleC2×C3C22×D8
hypercenter#508D3
hypocenterC3#3416
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #5771C2semidirect
2. #5769C2semidirect
3. C3×#508C2semidirect
4. #5195C2semidirect
5. #5193C2semidirect
6. #4918C2semidirect
7. #4916C2semidirect
8. D3×SD32C2semidirect
9. #4911C2semidirect
10. C2×#800C2semidirect
11. #4901C2semidirect
16. C3×#135C22semidirect
20. D3×Q16C22semidirect
23. #912C22semidirect
25. C2×Dic12C22semidirect
28. C3×Q32C22semidirect
29. C3×SD32C22semidirect
30. C3×SD32C22semidirect
31. C3×D16C22semidirect
32. C2×C3×C16C22semidirect
33. #829C22semidirect
34. #828C22semidirect
35. #827C22semidirect
36. #826C22semidirect
37. #812C22semidirect
38. #800C22semidirect
39. #800C22semidirect
40. #798C22semidirect
41. D3×C16C22semidirect
49. #508D3semidirect
98. #135C2×D3semidirect
105. Q32C2×D3semidirect
110. SD32C2×D3semidirect
111. SD32C2×D3semidirect
113. D16C2×D3semidirect
117. C2×C16C2×D3semidirect
246. C3#3416semidirect

Maximal subgroups

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

subgroupindex
#57712
#57692
C3×#5082
#51952
#51932
#49182
#49162
D3×SD322
#49112
C2×#8002
#49012
#34163