Finite Group Databasebeta

← #82024

Subgroups — #82024

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#5766C211yesmaximal, characteristic, complemented
2#5757C211yesmaximal, characteristic, complemented
3C2×C3×SD32C211yesmaximal, characteristic, complemented
4#5179C211yesmaximal, characteristic, complemented
5#5147C211yesmaximal, characteristic, complemented
6#4916C212yesmaximal, complemented
7#4915C212yesmaximal, complemented
8#4914C212yesmaximal, complemented
9D3×SD32C212yesmaximal, complemented
10#4904C211yesmaximal, characteristic, complemented
11#4901C211yesmaximal, characteristic, complemented
12#3416—31nomaximal, Sylow
13#1007—21no—
14#1004—21no—
15C2×C3×Q16C2211yescharacteristic, complemented
16C2×C3×D8C2211yescharacteristic
17#942—21no—
18#932—21no—
19#919C2212yescomplemented
20#918—22no—
21D3×Q16C2212yescomplemented
22#912C2212yes—
23#911—22no—
24D3×D8C2212yes—
25#904C2211yescharacteristic
26#904C2211yescharacteristic, complemented
27#901C2211yescharacteristic
28C3×SD32C2212yescomplemented
29C3×SD32C2212yescomplemented
30C2×C3×C16C2211yescharacteristic, complemented
31#829C2212yescomplemented
32#828C2212yescomplemented
33#827C2212yescomplemented
34#826C2212yescomplemented
35#812C2211yescharacteristic, complemented
36Dic24C2211yescharacteristic, complemented
37#800C2211yescharacteristic, complemented
38#800C2211yescharacteristic, complemented
39D48C2211yescharacteristic, complemented
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#508—31no—
50C2×SD32—32no—
51C2×SD32D311yescharacteristic, complemented
52#504—31no—
53C2×C3×Q8—21no—
54C2×C3×D4—21no—
55C2×#67—42no—
56#239—22no—
57#239—41no—
58D3×Q8—22no—
59D3×Q8—41no—
60#237—22no—
61#237—41no—
62D3×D4—22no—
63D3×D4—41no—
64#235—21no—
65#235—21no—
66#235—42no—
67#235D411yescharacteristic
68C3×Q16—21no—
69C3×Q16C2312yes—
70C3×D8—21no—
71C3×D8C2312yes—
72C2×C3×C8C2311yescharacteristic
73#216—22no—
74#215—22no—
75#214—22no—
76#213—22no—
77#208D411yescharacteristic
78Dic12C2311yescharacteristic
79Dic12C2311yescharacteristic
80D24C2311yescharacteristic
81D24C2311yescharacteristic
82#204—21no—
83#204—21no—
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#135—31no—
98#135—31no—
99C2×Q16—32no—
100C2×Q16C2×D311yescharacteristic, complemented
101C2×D8—32no—
102C2×D8C2×D311yescharacteristic
103#131—31no—
104Q32—32no—
105Q32—31no—
106SD32—32no—
107SD32—32no—
108SD32—31no—
109SD32—31no—
110SD32C2×D312yescomplemented
111SD32C2×D312yescomplemented
112D16—32no—
113D16—31no—
114M32—31no—
115M32—32no—
116C2×C16—32no—
117C2×C16C2×D311yescharacteristic, complemented
118C23×C3—41no—
119C22×D3—42no—
120C3×Q8—22no—
121C3×Q8—41no—
122C3×D4—22no—
123C3×D4—41no—
124C2×C3×C4—41no—
125C2×C3×C4C2×D411yescharacteristic
126#67—41no—
127#67—41no—
128#67—42no—
129#67—42no—
130#67D812yes—
131C2×Dic3—42no—
132D12—21no—
133D12—21no—
134D12—42no—
135D12D811yescharacteristic
136C4×D3—42no—
137C4×D3—41no—
138C4×D3—41no—
139C4×D3—42no—
140C4×D3C2×D412yes—
141Dic6—21no—
142Dic6—21no—
143Dic6—42no—
144Dic6D811yescharacteristic
145C3×C8C2×D411yescharacteristic
146C3×C8C2411yescharacteristic
147C3⋊C8C2×D412yes—
148#41—31no—
149#41—62no—
150#41—62no—
151#41—61no—
152#41—61no—
153#41—122no—
154#41—121no—
155#41—121no—
156C2×Q8—21no—
157C2×Q8—62no—
158C2×Q8—121no—
159C2×D4—21no—
160C2×D4—62no—
161C2×D4—121no—
162C2×D4—122no—
163Q16—21no—
164Q16—31no—
165Q16—31no—
166Q16—62no—
167Q16C22×D312yes—
168SD16—62no—
169SD16—62no—
170SD16—61no—
171SD16—61no—
172D8—21no—
173D8—31no—
174D8—31no—
175D8—62no—
176D8C22×D312yes—
177M16—31no—
178M16—32no—
179C2×C8—32no—
180C2×C8C22×D311yescharacteristic
181C16—32no—
182C16C22×D311yescharacteristic
183C16C22×D311yescharacteristic
184C22×C3—42no—
185C22×C3—161no—
186C22×C3C2×D811yescharacteristic
187C2×D3—41no—
188C2×D3—41no—
189C2×D3—162no—
190C2×D3C2×D812yes—
191C3×C4—42no—
192C3×C4C2×D811yescharacteristic
193C3×C4C22×D411yescharacteristic
194Dic3—41no—
195Dic3—41no—
196Dic3C2×D812yes—
197C23—41no—
198C23—122no—
199Q8—22no—
200Q8—31no—
201Q8—41no—
202Q8—61no—
203Q8—61no—
204Q8—122no—
205D4—22no—
206D4—32no—
207D4—31no—
208D4—41no—
209D4—61no—
210D4—61no—
211D4—122no—
212D4—122no—
213D4—122no—
214D4—121no—
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×C3C22×D811yescharacteristic
230D3—22no—
231D3—81no—
232D3—81no—
233C22—32no—
234C22—42no—
235C22—121no—
236C22—121no—
237C22—161no—
238C22—482no—
239C22D3×D811yescharacteristic
240C4—32no—
241C4—42no—
242C4—121no—
243C4—121no—
244C4D3×D811yescharacteristic
245C4C2×D3×D411yescharacteristic
246C3#341611yescharacteristic, complemented, Sylow
247C2—21no—
248C2—62no—
249C2—82no—
250C2—241no—
251C2—241no—
252C2C2×D3×D811yescharacteristic
subgroupG/H
centerC2C2×D3×D8
commutator, cosocleC3×C8C24
FrattiniC8C23×D3
FittingC2×C3×SD32C2
socleC2×C3C22×D8
hypercenterC2×SD32D3
hypocenterC3#3416
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
1. #5766C2semidirect
2. #5757C2semidirect
3. C2×C3×SD32C2semidirect
4. #5179C2semidirect
5. #5147C2semidirect
6. #4916C2semidirect
7. #4915C2semidirect
8. #4914C2semidirect
9. D3×SD32C2semidirect
10. #4904C2semidirect
11. #4901C2semidirect
15. C2×C3×Q16C22semidirect
19. #919C22semidirect
21. D3×Q16C22semidirect
26. #904C22semidirect
28. C3×SD32C22semidirect
29. C3×SD32C22semidirect
30. C2×C3×C16C22semidirect
31. #829C22semidirect
32. #828C22semidirect
33. #827C22semidirect
34. #826C22semidirect
35. #812C22semidirect
36. Dic24C22semidirect
37. #800C22semidirect
38. #800C22semidirect
39. D48C22semidirect
40. #798C22semidirect
41. D3×C16C22semidirect
51. C2×SD32D3semidirect
100. C2×Q16C2×D3semidirect
110. SD32C2×D3semidirect
111. SD32C2×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
#57662
#57572
C2×C3×SD322
#51792
#51472
#49162
#49152
#49142
D3×SD322
#49042
#49012
#34163