Finite Group Databasebeta

← C23×D3×C3⋊Dic3 #13181653

Subgroups — C23×D3×C3⋊Dic3 #13181653

Order 1728.

9116 conjugacy classes of subgroups; only the 3004 normal subgroups are shown.

normal subgroups only

This group has 9116 conjugacy classes of subgroups — too many to draw. Only its 3004 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×C32×D3C211yesmaximal, characteristic
2C23×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C22×D3×C3⋊Dic3C2128yesmaximal, direct product factor
5C24×C3⋊Dic3—31nomaximal
6C23×D3×Dic3—34nomaximal
7C24×C33C2211yescharacteristic
8C23×C32×D3C4116yescomplemented
9C23×C32×D3C22114yes—
10C22×C3⋊(C3⋊Dic3)C22114yescomplemented
11C22×C3×C3⋊Dic3C22114yescomplemented
12C2×D3×C3⋊Dic3C221112yesdirect product factor
13C25×C32—31no—
14C24×C3×D3D314yes—
15C23×C3⋊Dic3—328no—
16C23×C3⋊Dic3—31no—
17C23×C3⋊Dic3—34no—
18C23×C3⋊Dic3—64no—
19C23×C3⋊Dic3D311yescharacteristic, direct product factor
20C23×C3×Dic3—34no—
21C22×D3×Dic3—3112no—
22C23×C33C2×C418yescomplemented
23C23×C33C2317yes—
24C22×C32×D3C2×C41112yescomplemented
25C22×C32×D3C23128yes—
26C2×C3⋊(C3⋊Dic3)C23128yescomplemented
27C2×C3×C3⋊Dic3C23128yescomplemented
28D3×C3⋊Dic3C23164yesdirect product factor
29C24×Dic3—94no—
30C23×C4×D3—91no—
31C24×C32—24no—
32C24×C32—316no—
33C24×C32—314no—
34C24×C32C2×D311yescharacteristic
35C24×C32C2×D314yes—
36C23×C3×D3Dic3164yescomplemented
37C23×C3×D3C2×D3156yes—
38C22×C3⋊Dic3—314no—
39C22×C3⋊Dic3—356no—
40C22×C3⋊Dic3—3112no—
41C22×C3⋊Dic3—656no—
42C22×C3⋊Dic3C2×D3114yesdirect product factor
43C22×C3×Dic3—356no—
44C2×D3×Dic3—3448no—
45C22×C33C22×C4128yescomplemented
46C22×C33C2417yes—
47C2×C32×D3C22×C41112yescomplemented
48C2×C32×D3C2418yes—
49C3⋊(C3⋊Dic3)C2418yescomplemented
50C3×C3⋊Dic3C2418yescomplemented
51C25×C3—34no—
52C24×D3C3⋊D311yescharacteristic
53C23×C3×C4—91no—
54C23×Dic3—34no—
55C23×Dic3—9112no—
56C23×Dic3—94no—
57C23×Dic3—91no—
58C23×Dic3—184no—
59C22×C4×D3—928no—
60C23×C32—232no—
61C23×C32—228no—
62C23×C32—3112no—
63C23×C32—328no—
64C23×C32C4×D318yescomplemented
65C23×C32C2×Dic3132yescomplemented
66C23×C32C22×D3128yes—
67C23×C32C22×D317yes—
68C22×C3×D3C2×Dic31448yescomplemented
69C22×C3×D3C22×D31112yes—
70C2×C3⋊Dic3—328no—
71C2×C3⋊Dic3—364no—
72C2×C3⋊Dic3—3112no—
73C2×C3⋊Dic3—6112no—
74C2×C3⋊Dic3C22×D3128yesdirect product factor
75C2×C3×Dic3—3112no—
76D3×Dic3—3256no—
77C24×C4—271noSylow
78C2×C33C23×C4114yescomplemented
79C2×C33C2511yescharacteristic
80C32×D3C23×C4116yescomplemented
81C24×C3—24no—
82C24×C3—364no—
83C24×C3—356no—
84C24×C3D3214yes—
85C24×C3C2×C3⋊D311yescharacteristic
86C23×D3C3⋊Dic3116yesdirect product factor
87C23×D3C2×C3⋊D3114yes—
88C22×C3×C4—914no—
89C22×Dic3—356no—
90C22×Dic3—9448no—
91C22×Dic3—956no—
92C22×Dic3—914no—
93C22×Dic3—1856no—
94C2×C4×D3—9112no—
95C22×C32—228no—
96C22×C32—2112no—
97C22×C32—3112no—
98C22×C32—38no—
99C22×C32C2×C4×D3128yescomplemented
100C22×C32C22×Dic31112yescomplemented
101C22×C32C23×D3128yes—
102C22×C32C23×D317yes—
103C2×C3×D3C22×Dic31448yescomplemented
104C2×C3×D3C23×D3132yes—
105C3⋊Dic3—38no—
106C3⋊Dic3—332no—
107C3⋊Dic3—632no—
108C3⋊Dic3C23×D318yesdirect product factor
109C3×Dic3—332no—
110C25—31no—
111C23×C4—91no—
112C23×C4—2728no—
113C23×C4—271no—
114C33C24×C411yescharacteristic, complemented, Sylow
115C23×C3—232no—
116C23×C3—228no—
117C23×C3—3112no—
118C23×C3—3448no—
119C23×C3D3×Dic3132yescomplemented
120C23×C3C2×C3⋊Dic318yescomplemented
121C23×C3C2×D32128yes—
122C23×C3C22×C3⋊D317yes—
123C22×D3C2×C3⋊Dic31112yesdirect product factor
124C22×D3C22×C3⋊D3128yes—
125C2×C3×C4—928no—
126C2×Dic3—3112no—
127C2×Dic3—9256no—
128C2×Dic3—9112no—
129C2×Dic3—928no—
130C2×Dic3—18112no—
131C4×D3—964no—
132C2×C32—256no—
133C2×C32—24no—
134C2×C32—316no—
135C2×C32C22×C4×D3114yescomplemented
136C2×C32C23×Dic3156yescomplemented
137C2×C32C24×D314yes—
138C2×C32C24×D311yescharacteristic
139C3×D3C23×Dic3164yescomplemented
140C24—316no—
141C24—314no—
142C24D3×C3⋊D311yescharacteristic
143C22×C4—914no—
144C22×C4—27112no—
145C22×C4—2714no—
146C22×C3—228no—
147C22×C3—2112no—
148C22×C3—3448no—
149C22×C3—332no—
150C22×C3C2×D3×Dic31112yescomplemented
151C22×C3C22×C3⋊Dic3128yescomplemented
152C22×C3C22×D32128yes—
153C22×C3C23×C3⋊D317yes—
154C2×D3C22×C3⋊Dic31112yesdirect product factor
155C2×D3C23×C3⋊D318yes—
156C3×C4—98no—
157Dic3—332no—
158Dic3—932no—
159Dic3—98no—
160Dic3—1832no—
161C32—24no—
162C32C23×C4×D311yescharacteristic, complemented
163C32C24×Dic314yescomplemented
164C23—328no—
165C23—3112no—
166C23D3×C3⋊Dic318yesdirect product factor
167C23C2×D3×C3⋊D317yes—
168C2×C4—928no—
169C2×C4—2764no—
170C2×C4—2728no—
171C2×C3—256no—
172C2×C3—24no—
173C2×C3—364no—
174C2×C3C22×D3×Dic3156yescomplemented
175C2×C3C23×C3⋊Dic3114yescomplemented
176C2×C3C23×D3214yes—
177C2×C3C24×C3⋊D311yescharacteristic
178D3C23×C3⋊Dic3116yesdirect product factor
179C22—3112no—
180C22—38no—
181C22C2×D3×C3⋊Dic3128yesdirect product factor
182C22C22×D3×C3⋊D317yes—
183C4—98no—
184C4—278no—
185C3—24no—
186C3C23×D3×Dic314yescomplemented
187C3C24×C3⋊Dic311yescharacteristic, complemented
188C2—316no—
189C2C22×D3×C3⋊Dic3114yesdirect product factor
190C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC24D3×C3⋊D3
commutator, hypocenterC33C24×C4
FrattiniC2C23×D3×C3⋊D3
Fitting, socleC24×C33C22
prime radicalC24×C32×D3C2
cosocleC2×C33C25
LCSG ▷ C33
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×C3⋊(C3⋊Dic3)C2semidirect
3. C23×C3×C3⋊Dic3C2semidirect
4. C22×D3×C3⋊Dic3C2direct (N × H)
8. C23×C32×D3C4semidirect
10. C22×C3⋊(C3⋊Dic3)C22semidirect
11. C22×C3×C3⋊Dic3C22semidirect
12. C2×D3×C3⋊Dic3C22direct (N × H)
19. C23×C3⋊Dic3D3direct (N × H)
22. C23×C33C2×C4semidirect
24. C22×C32×D3C2×C4semidirect
26. C2×C3⋊(C3⋊Dic3)C23semidirect
27. C2×C3×C3⋊Dic3C23semidirect
28. D3×C3⋊Dic3C23direct (N × H)
36. C23×C3×D3Dic3semidirect
42. C22×C3⋊Dic3C2×D3direct (N × H)
45. C22×C33C22×C4semidirect
47. C2×C32×D3C22×C4semidirect
49. C3⋊(C3⋊Dic3)C24semidirect
50. C3×C3⋊Dic3C24semidirect
64. C23×C32C4×D3semidirect
65. C23×C32C2×Dic3semidirect
68. C22×C3×D3C2×Dic3semidirect
74. C2×C3⋊Dic3C22×D3direct (N × H)
78. C2×C33C23×C4semidirect
80. C32×D3C23×C4semidirect
86. C23×D3C3⋊Dic3direct (N × H)
99. C22×C32C2×C4×D3semidirect
100. C22×C32C22×Dic3semidirect
103. C2×C3×D3C22×Dic3semidirect
108. C3⋊Dic3C23×D3direct (N × H)
114. C33C24×C4semidirect
119. C23×C3D3×Dic3semidirect
120. C23×C3C2×C3⋊Dic3semidirect
123. C22×D3C2×C3⋊Dic3direct (N × H)
135. C2×C32C22×C4×D3semidirect
136. C2×C32C23×Dic3semidirect
139. C3×D3C23×Dic3semidirect
150. C22×C3C2×D3×Dic3semidirect
151. C22×C3C22×C3⋊Dic3semidirect
154. C2×D3C22×C3⋊Dic3direct (N × H)
162. C32C23×C4×D3semidirect
163. C32C24×Dic3semidirect
166. C23D3×C3⋊Dic3direct (N × H)
174. C2×C3C22×D3×Dic3semidirect
175. C2×C3C23×C3⋊Dic3semidirect
178. D3C23×C3⋊Dic3direct (N × H)
181. C22C2×D3×C3⋊Dic3direct (N × H)
186. C3C23×D3×Dic3semidirect
187. C3C24×C3⋊Dic3semidirect
189. C2C22×D3×C3⋊Dic3direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C32×D32
C23×C3⋊(C3⋊Dic3)2
C23×C3×C3⋊Dic32
C22×D3×C3⋊Dic32
C24×C3⋊Dic33
C23×D3×Dic33