Finite Group Databasebeta

← #82037

Subgroups — #82037

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#5757C211yesmaximal, characteristic, complemented
3C3×#509C211yesmaximal, characteristic, complemented
4#5194C211yesmaximal, characteristic, complemented
5C2×#827C211yesmaximal, characteristic, complemented
6#4915C212yesmaximal, complemented
7D3×SD32C212yesmaximal, complemented
8#4912C212yesmaximal, complemented
9#4911C212yesmaximal, complemented
10#4909C211yesmaximal, characteristic, complemented
11#4907C211yesmaximal, characteristic, complemented
12#3416—31nomaximal, Sylow
13#1010—21no—
14#1004—21no—
15C3×#135C2211yescharacteristic, complemented
16C2×C3×D8C2211yescharacteristic
17#951—21no—
18#932—21no—
19D3×Q16C2212yescomplemented
20#915—22no—
21#912C2212yes—
22#912C2212yescomplemented
23#911—22no—
24D3×D8C2212yes—
25C2×Dic12C2211yescharacteristic, complemented
26#904C2211yescharacteristic
27#901C2211yescharacteristic
28C3×SD32C2212yescomplemented
29C3×D16C2212yescomplemented
30C3×M32C2211yescharacteristic, complemented
31#829C2211yescharacteristic, complemented
32#828C2211yescharacteristic, complemented
33#827C2212yescomplemented
34#827C2212yescomplemented
35#827C2211yescharacteristic, complemented
36#826C2211yescharacteristic, complemented
37C2×C3⋊C16C2211yescharacteristic, complemented
38Dic24C2212yescomplemented
39#800C2212yescomplemented
40#798C2212yescomplemented
41D3×C16C2212yescomplemented
42#578—31no—
43#576—31no—
44#510—32no—
45#510—31no—
46#509—32no—
47#509D311yescharacteristic, complemented
48#508—31no—
49#508—32no—
50C2×SD32—31no—
51C2×SD32—32no—
52#504—31no—
53C3×#41—21no—
54C2×C3×D4—21no—
55C2×#67—42no—
56D3×Q8—22no—
57#237—22no—
58#237—22no—
59#237—42no—
60#237—41no—
61D3×D4—22no—
62D3×D4—41no—
63#235—21no—
64#235—41no—
65#235D411yescharacteristic
66C2×Dic6—21no—
67C2×Dic6—41no—
68C3×Q16C2311yescharacteristic
69C3×SD16—21no—
70C3×D8—21no—
71C3×D8C2311yescharacteristic
72C3×D8C2312yes—
73C2×C3×C8C2311yescharacteristic
74#216—22no—
75#214—22no—
76#214—22no—
77#213—22no—
78#208D411yescharacteristic
79Dic12—21no—
80Dic12C2312yes—
81Dic12C2311yescharacteristic
82D24C2311yescharacteristic
83#204—21no—
84ΓL(1,𝔽25)D412yes—
85D3×C8C2312yes—
86C3×C16C2312yes—
87C3⋊C16C2311yescharacteristic
88C3⋊C16C2311yescharacteristic
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#135—31no—
98#135C2×D311yescharacteristic, complemented
99C2×Q16—32no—
100C2×Q16—31no—
101C2×D8—32no—
102C2×D8C2×D311yescharacteristic
103#131—31no—
104Q32—31no—
105Q32—32no—
106SD32—31no—
107SD32—32no—
108SD32—32no—
109SD32—31no—
110SD32—32no—
111SD32C2×D312yescomplemented
112D16—31no—
113D16C2×D312yescomplemented
114M32—32no—
115M32C2×D311yescharacteristic, complemented
116C2×C16—31no—
117C2×C16—32no—
118C23×C3—41no—
119C22×D3—42no—
120C3×Q8—21no—
121C3×D4—22no—
122C3×D4—21no—
123C3×D4—41no—
124C3×D4—41no—
125C2×C3×C4—41no—
126C2×C3×C4C2×D411yescharacteristic
127#67—41no—
128#67—42no—
129#67—42no—
130#67—42no—
131#67D812yes—
132C2×Dic3—41no—
133C2×Dic3—42no—
134C2×Dic3—42no—
135D12—21no—
136D12D811yescharacteristic
137C4×D3—42no—
138C4×D3—41no—
139C4×D3C2×D412yes—
140Dic6—21no—
141Dic6—22no—
142Dic6—41no—
143Dic6—42no—
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—122no—
155#41—121no—
156C2×Q8—62no—
157C2×Q8—61no—
158C2×Q8—121no—
159C2×D4—21no—
160C2×D4—62no—
161C2×D4—121no—
162C2×D4—122no—
163Q16—32no—
164Q16—31no—
165Q16—62no—
166Q16—61no—
167Q16C22×D311yescharacteristic
168SD16—21no—
169SD16—62no—
170SD16—62no—
171SD16—61no—
172D8—21no—
173D8—31no—
174D8—62no—
175D8C22×D311yescharacteristic
176D8C22×D312yes—
177M16—31no—
178M16—32no—
179C2×C8—32no—
180C2×C8C22×D311yescharacteristic
181C16—31no—
182C16—31no—
183C16C22×D312yes—
184C22×C3—41no—
185C22×C3—42no—
186C22×C3—161no—
187C22×C3C2×D811yescharacteristic
188C2×D3—41no—
189C2×D3—162no—
190C2×D3C2×D812yes—
191C3×C4—41no—
192C3×C4C2×D811yescharacteristic
193C3×C4C22×D411yescharacteristic
194Dic3—42no—
195Dic3—41no—
196Dic3C2×D812yes—
197C23—41no—
198C23—122no—
199Q8—21no—
200Q8—31no—
201Q8—61no—
202Q8—62no—
203Q8—122no—
204Q8—121no—
205D4—22no—
206D4—21no—
207D4—32no—
208D4—31no—
209D4—41no—
210D4—41no—
211D4—61no—
212D4—122no—
213D4—122no—
214D4—122no—
215D4—121no—
216C2×C4—32no—
217C2×C4—41no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4—122no—
221C2×C4—121no—
222C2×C4—121no—
223C2×C4D3×D411yescharacteristic
224C8—32no—
225C8D3×D411yescharacteristic
226C8C23×D311yescharacteristic
227C2×C3—21no—
228C2×C3—82no—
229C2×C3—81no—
230C2×C3C22×D811yescharacteristic
231D3—22no—
232D3—81no—
233C22—32no—
234C22—41no—
235C22—42no—
236C22—121no—
237C22—161no—
238C22—482no—
239C22D3×D811yescharacteristic
240C4—32no—
241C4—41no—
242C4—121no—
243C4—122no—
244C4D3×D811yescharacteristic
245C4C2×D3×D411yescharacteristic
246C3#341611yescharacteristic, complemented, Sylow
247C2—21no—
248C2—62no—
249C2—82no—
250C2—81no—
251C2—241no—
252C2C2×D3×D811yescharacteristic
subgroupG/H
centerC2C2×D3×D8
commutator, cosocleC3×C8C24
FrattiniC8C23×D3
FittingC3×#509C2
socleC2×C3C22×D8
hypercenter#509D3
hypocenterC3#3416
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. #5771C2semidirect
2. #5757C2semidirect
3. C3×#509C2semidirect
4. #5194C2semidirect
5. C2×#827C2semidirect
6. #4915C2semidirect
7. D3×SD32C2semidirect
8. #4912C2semidirect
9. #4911C2semidirect
10. #4909C2semidirect
11. #4907C2semidirect
15. C3×#135C22semidirect
19. D3×Q16C22semidirect
22. #912C22semidirect
25. C2×Dic12C22semidirect
28. C3×SD32C22semidirect
29. C3×D16C22semidirect
30. C3×M32C22semidirect
31. #829C22semidirect
32. #828C22semidirect
33. #827C22semidirect
34. #827C22semidirect
35. #827C22semidirect
36. #826C22semidirect
37. C2×C3⋊C16C22semidirect
38. Dic24C22semidirect
39. #800C22semidirect
40. #798C22semidirect
41. D3×C16C22semidirect
47. #509D3semidirect
98. #135C2×D3semidirect
111. SD32C2×D3semidirect
113. D16C2×D3semidirect
115. M32C2×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
#57572
C3×#5092
#51942
C2×#8272
#49152
D3×SD322
#49122
#49112
#49092
#49072
#34163