Finite Group Databasebeta

← C23×#6289 #13181655

Subgroups — C23×#6289 #13181655

Order 1728.

11164 conjugacy classes of subgroups; only the 1724 normal subgroups are shown.

normal subgroups only

This group has 11164 conjugacy classes of subgroups — too many to draw. Only its 1724 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
1C24×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C23×C32×Dic3C211yesmaximal, characteristic, complemented
4C22×#6289C2128yesmaximal, direct product factor
5C23×C4×C3⋊D3—31nomaximal
6C23×#624—34nomaximal
7C24×C33C2211yescharacteristic
8C23×C3⋊(C3⋊D3)C4116yescomplemented
9C23×C3⋊(C3⋊D3)C22114yes—
10C22×C3×C3⋊Dic3C22114yescomplemented
11C22×C32×Dic3C22114yescomplemented
12C2×#6289C221112yesdirect product factor
13C24×C3⋊D3—31no—
14C24×C3⋊D3—34no—
15C24×C3⋊D3—64no—
16C23×C32×C4—31no—
17C23×C3⋊Dic3D311yescharacteristic, complemented
18C22×C4×C3⋊D3—328no—
19C23×C3×Dic3—34no—
20C23×C3×Dic3D314yescomplemented
21C22×#624—3112no—
22C23×C33C2×C418yescomplemented
23C23×C33C2317yes—
24C22×C3⋊(C3⋊D3)C2×C41112yescomplemented
25C22×C3⋊(C3⋊D3)C23128yes—
26C2×C3×C3⋊Dic3C23128yescomplemented
27C2×C32×Dic3C23128yescomplemented
28#6289C23164yesdirect product factor
29C23×C4×D3—94no—
30C23×C4×D3—91no—
31C24×C32—24no—
32C24×C32C2×D314yes—
33C24×C32C2×D311yescharacteristic
34C23×C3⋊D3—316no—
35C23×C3⋊D3—364no—
36C23×C3⋊D3—314no—
37C23×C3⋊D3—356no—
38C23×C3⋊D3—664no—
39C23×C3⋊D3—656no—
40C22×C32×C4—314no—
41C22×C3⋊Dic3C2×D3114yescomplemented
42C2×C4×C3⋊D3—3112no—
43C22×C3×Dic3—356no—
44C22×C3×Dic3C2×D3156yescomplemented
45C2×#624—3448no—
46C22×C33C22×C4128yescomplemented
47C22×C33C2417yes—
48C2×C3⋊(C3⋊D3)C22×C41112yescomplemented
49C2×C3⋊(C3⋊D3)C2418yes—
50C3×C3⋊Dic3C2418yescomplemented
51C32×Dic3C2418yescomplemented
52C24×D3—94no—
53C24×D3—91no—
54C24×D3—184no—
55C23×C3×C4—34no—
56C23×C3×C4—91no—
57C23×Dic3—34no—
58C23×Dic3C3⋊D311yescharacteristic, complemented
59C22×C4×D3—9112no—
60C22×C4×D3—928no—
61C23×C32—232no—
62C23×C32—228no—
63C23×C32C4×D3132yescomplemented
64C23×C32C4×D318yescomplemented
65C23×C32C22×D3128yes—
66C23×C32C22×D317yes—
67C22×C3⋊D3—3112no—
68C22×C3⋊D3—328no—
69C22×C3⋊D3—3448no—
70C22×C3⋊D3—3112no—
71C22×C3⋊D3—6448no—
72C22×C3⋊D3—6112no—
73C2×C32×C4—328no—
74C2×C3⋊Dic3C22×D3128yescomplemented
75C4×C3⋊D3—364no—
76C2×C3×Dic3—3112no—
77C2×C3×Dic3C22×D31112yescomplemented
78#624—3256no—
79C24×C4—271noSylow
80C2×C33C23×C4114yescomplemented
81C2×C33C2511yescharacteristic
82C3⋊(C3⋊D3)C23×C4116yescomplemented
83C24×C3—24no—
84C24×C3D3214yes—
85C24×C3C2×C3⋊D311yescharacteristic
86C23×D3—964no—
87C23×D3—916no—
88C23×D3—956no—
89C23×D3—914no—
90C23×D3—1864no—
91C23×D3—1856no—
92C22×C3×C4—356no—
93C22×C3×C4—914no—
94C22×Dic3—356no—
95C22×Dic3C2×C3⋊D3114yescomplemented
96C2×C4×D3—9448no—
97C2×C4×D3—9112no—
98C22×C32—228no—
99C22×C32—2112no—
100C22×C32C2×C4×D31112yescomplemented
101C22×C32C2×C4×D3128yescomplemented
102C22×C32C23×D3128yes—
103C22×C32C23×D317yes—
104C2×C3⋊D3—3112no—
105C2×C3⋊D3—38no—
106C2×C3⋊D3—3448no—
107C2×C3⋊D3—332no—
108C2×C3⋊D3—6448no—
109C2×C3⋊D3—632no—
110C32×C4—38no—
111C3⋊Dic3C23×D318yescomplemented
112C3×Dic3—332no—
113C3×Dic3C23×D3132yescomplemented
114C25—271no—
115C23×C4—31no—
116C23×C4—91no—
117C23×C4—2728no—
118C33C24×C411yescharacteristic, complemented, Sylow
119C23×C3—232no—
120C23×C3—228no—
121C23×C3#624132yescomplemented
122C23×C3C4×C3⋊D318yescomplemented
123C23×C3C2×D32128yes—
124C23×C3C22×C3⋊D317yes—
125C22×D3—9112no—
126C22×D3—928no—
127C22×D3—9448no—
128C22×D3—9112no—
129C22×D3—18112no—
130C22×D3—18448no—
131C2×C3×C4—3112no—
132C2×C3×C4—928no—
133C2×Dic3—3112no—
134C2×Dic3C22×C3⋊D3128yescomplemented
135C4×D3—9256no—
136C4×D3—964no—
137C2×C32—256no—
138C2×C32—24no—
139C2×C32C22×C4×D3156yescomplemented
140C2×C32C22×C4×D3114yescomplemented
141C2×C32C24×D314yes—
142C2×C32C24×D311yescharacteristic
143C3⋊D3—316no—
144C3⋊D3—364no—
145C3⋊D3—664no—
146C24—2716no—
147C24—2714no—
148C24D3×C3⋊D311yescharacteristic
149C22×C4—314no—
150C22×C4—914no—
151C22×C4—27112no—
152C22×C3—228no—
153C22×C3—2112no—
154C22×C3C2×#6241112yescomplemented
155C22×C3C2×C4×C3⋊D3128yescomplemented
156C22×C3C22×D32128yes—
157C22×C3C23×C3⋊D317yes—
158C2×D3—9448no—
159C2×D3—9112no—
160C2×D3—932no—
161C2×D3—98no—
162C2×D3—18448no—
163C2×D3—1832no—
164C3×C4—332no—
165C3×C4—98no—
166Dic3—332no—
167Dic3C23×C3⋊D318yescomplemented
168C32—24no—
169C32C23×C4×D314yescomplemented
170C32C23×C4×D311yescharacteristic, complemented
171C23—2728no—
172C23—27112no—
173C23#628918yesdirect product factor
174C23C2×D3×C3⋊D317yes—
175C2×C4—328no—
176C2×C4—928no—
177C2×C4—2764no—
178C2×C3—256no—
179C2×C3—24no—
180C2×C3C22×#624156yescomplemented
181C2×C3C22×C4×C3⋊D3114yescomplemented
182C2×C3C23×D3214yes—
183C2×C3C24×C3⋊D311yescharacteristic
184D3—964no—
185D3—916no—
186D3—1864no—
187C22—27112no—
188C22—278no—
189C22C2×#6289128yesdirect product factor
190C22C22×D3×C3⋊D317yes—
191C4—38no—
192C4—98no—
193C3—24no—
194C3C23×#62414yescomplemented
195C3C23×C4×C3⋊D311yescharacteristic, complemented
196C2—2716no—
197C2C22×#6289114yesdirect product factor
198C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC24D3×C3⋊D3
commutator, hypocenterC33C24×C4
FrattiniC2C23×D3×C3⋊D3
Fitting, socleC24×C33C22
prime radicalC24×C3⋊(C3⋊D3)C2
cosocleC2×C33C25
LCSG ▷ C33
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×C3×C3⋊Dic3C2semidirect
3. C23×C32×Dic3C2semidirect
4. C22×#6289C2direct (N × H)
8. C23×C3⋊(C3⋊D3)C4semidirect
10. C22×C3×C3⋊Dic3C22semidirect
11. C22×C32×Dic3C22semidirect
12. C2×#6289C22direct (N × H)
17. C23×C3⋊Dic3D3semidirect
20. C23×C3×Dic3D3semidirect
22. C23×C33C2×C4semidirect
24. C22×C3⋊(C3⋊D3)C2×C4semidirect
26. C2×C3×C3⋊Dic3C23semidirect
27. C2×C32×Dic3C23semidirect
28. #6289C23direct (N × H)
41. C22×C3⋊Dic3C2×D3semidirect
44. C22×C3×Dic3C2×D3semidirect
46. C22×C33C22×C4semidirect
48. C2×C3⋊(C3⋊D3)C22×C4semidirect
50. C3×C3⋊Dic3C24semidirect
51. C32×Dic3C24semidirect
58. C23×Dic3C3⋊D3semidirect
63. C23×C32C4×D3semidirect
64. C23×C32C4×D3semidirect
74. C2×C3⋊Dic3C22×D3semidirect
77. C2×C3×Dic3C22×D3semidirect
80. C2×C33C23×C4semidirect
82. C3⋊(C3⋊D3)C23×C4semidirect
95. C22×Dic3C2×C3⋊D3semidirect
100. C22×C32C2×C4×D3semidirect
101. C22×C32C2×C4×D3semidirect
111. C3⋊Dic3C23×D3semidirect
113. C3×Dic3C23×D3semidirect
118. C33C24×C4semidirect
121. C23×C3#624semidirect
122. C23×C3C4×C3⋊D3semidirect
134. C2×Dic3C22×C3⋊D3semidirect
139. C2×C32C22×C4×D3semidirect
140. C2×C32C22×C4×D3semidirect
154. C22×C3C2×#624semidirect
155. C22×C3C2×C4×C3⋊D3semidirect
167. Dic3C23×C3⋊D3semidirect
169. C32C23×C4×D3semidirect
170. C32C23×C4×D3semidirect
173. C23#6289direct (N × H)
180. C2×C3C22×#624semidirect
181. C2×C3C22×C4×C3⋊D3semidirect
189. C22C2×#6289direct (N × H)
194. C3C23×#624semidirect
195. C3C23×C4×C3⋊D3semidirect
197. C2C22×#6289direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3⋊(C3⋊D3)2
C23×C3×C3⋊Dic32
C23×C32×Dic32
C22×#62892
C23×C4×C3⋊D33
C23×#6243