Finite Group Databasebeta

← C23×C32×S4 #13181853

Subgroups — C23×C32×S4 #13181853

Order 1728.

4462 conjugacy classes of subgroups; only the 594 normal subgroups are shown.

normal subgroups only

This group has 4462 conjugacy classes of subgroups — too many to draw. Only its 594 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
1C23×C32×A4C211yesmaximal, characteristic, complemented
2C22×C32×S4C2114yesmaximal, direct product factor
3C23×C3×S4C314yesmaximal, direct product factor
4C23×C32×D4—31nomaximal
5C23×C32×D3—41nomaximal
6C22×C32×A4C2217yescomplemented
7C2×C32×S4C22128yesdirect product factor
8C25×C32—31no—
9C25×C32D311yescharacteristic, complemented
10C23×C3×A4—24no—
11C23×C3×A4C2×C314yescomplemented
12C22×C3×S4C2×C3156yesdirect product factor
13C22×C32×D4—314no—
14C22×C32×D4—314no—
15C23×C32×C4—31no—
16C23×C33—41no—
17C22×C32×D3—414no—
18C2×C32×A4C2317yescomplemented
19C32×S4C2318yesdirect product factor
20C23×S4C3211yescharacteristic, direct product factor
21C23×C3×D4—34no—
22C24×C32—31no—
23C24×C32—37no—
24C24×C32—314no—
25C24×C32—67no—
26C24×C32—61no—
27C24×C32C2×D317yescomplemented
28C23×C3×D3—44no—
29C22×C3×A4—228no—
30C22×C3×A4C22×C3128yescomplemented
31C2×C3×S4C22×C31112yesdirect product factor
32C2×C32×D4—384no—
33C2×C32×D4—328no—
34C22×C32×C4—314no—
35C22×C33—47no—
36C2×C32×D3—428no—
37C32×A4C2411yescharacteristic, complemented
38C25×C3—34no—
39C25×C3C3×D314yescomplemented
40C23×A4—24no—
41C23×A4C2×C3211yescharacteristic, complemented
42C22×S4C2×C32114yesdirect product factor
43C22×C3×D4—356no—
44C22×C3×D4—356no—
45C23×C3×C4—34no—
46C23×C32—37no—
47C23×C32—37no—
48C23×C32—321no—
49C23×C32—328no—
50C23×C32—44no—
51C23×C32—67no—
52C23×C32—642no—
53C23×C32—614no—
54C23×C32—84no—
55C23×C32S411yescharacteristic, direct product factor
56C23×C32C22×D317yescomplemented
57C22×C3×D3—456no—
58C2×C3×A4—228no—
59C2×C3×A4C23×C3128yescomplemented
60C3×S4C23×C3132yesdirect product factor
61C32×D4—356no—
62C32×D4—38no—
63C2×C32×C4—328no—
64C23×D4—31noSylow
65C2×C33—47no—
66C32×D3—48no—
67C24×C3—34no—
68C24×C3—328no—
69C24×C3—356no—
70C24×C3—628no—
71C24×C3—64no—
72C24×C3C2×C3×D3128yescomplemented
73C23×D3—41no—
74C22×A4—228no—
75C22×A4C22×C3217yescomplemented
76C2×S4C22×C32128yesdirect product factor
77C2×C3×D4—3336no—
78C2×C3×D4—3112no—
79C22×C3×C4—356no—
80C22×C32—37no—
81C22×C32—37no—
82C22×C32—321no—
83C22×C32—38no—
84C22×C32—428no—
85C22×C32—67no—
86C22×C32—628no—
87C22×C32—628no—
88C22×C32—828no—
89C22×C32C2×S417yesdirect product factor
90C22×C32C23×D311yescharacteristic, complemented
91C2×C3×D3—4112no—
92C3×A4—24no—
93C3×A4C24×C314yescomplemented
94C32×C4—38no—
95C25—31no—
96C25C32×D311yescharacteristic, complemented
97C22×D4—314no—
98C22×D4—314no—
99C23×C4—31no—
100C33—41noSylow
101C23×C3—328no—
102C23×C3—328no—
103C23×C3—384no—
104C23×C3—3112no—
105C23×C3—41no—
106C23×C3—628no—
107C23×C3—6168no—
108C23×C3—656no—
109C23×C3—84no—
110C23×C3C3×S414yesdirect product factor
111C23×C3C22×C3×D3128yescomplemented
112C22×D3—414no—
113C2×A4—228no—
114C2×A4C23×C3217yescomplemented
115S4C23×C3218yesdirect product factor
116C3×D4—3224no—
117C3×D4—332no—
118C2×C3×C4—3112no—
119C2×C32—31no—
120C2×C32—37no—
121C2×C32—428no—
122C2×C32—68no—
123C2×C32—828no—
124C2×C32C22×S417yesdirect product factor
125C3×D3—432no—
126C24—31no—
127C24—37no—
128C24—314no—
129C24—67no—
130C24—61no—
131C24C2×C32×D317yescomplemented
132C2×D4—384no—
133C2×D4—328no—
134C22×C4—314no—
135C22×C3—328no—
136C22×C3—328no—
137C22×C3—384no—
138C22×C3—332no—
139C22×C3—47no—
140C22×C3—628no—
141C22×C3—6112no—
142C22×C3—6112no—
143C22×C3—828no—
144C22×C3C2×C3×S4128yesdirect product factor
145C22×C3C23×C3×D314yescomplemented
146C2×D3—428no—
147A4—24no—
148A4C24×C3211yescharacteristic, complemented
149C3×C4—332no—
150C32—44no—
151C32—84no—
152C32C23×S411yescharacteristic, direct product factor
153C23—37no—
154C23—37no—
155C23—321no—
156C23—328no—
157C23—67no—
158C23—642no—
159C23—614no—
160C23C32×S411yescharacteristic, direct product factor
161C23C22×C32×D317yescomplemented
162D4—356no—
163D4—38no—
164C2×C4—328no—
165C2×C3—34no—
166C2×C3—328no—
167C2×C3—47no—
168C2×C3—632no—
169C2×C3—828no—
170C2×C3C22×C3×S4128yesdirect product factor
171D3—48no—
172C22—37no—
173C22—37no—
174C22—321no—
175C22—38no—
176C22—67no—
177C22—628no—
178C22—628no—
179C22C2×C32×S417yesdirect product factor
180C22C23×C32×D311yescharacteristic, complemented
181C4—38no—
182C3—41no—
183C3—84no—
184C3C23×C3×S414yesdirect product factor
185C2—31no—
186C2—37no—
187C2—68no—
188C2C22×C32×S417yesdirect product factor
subgroupG/H
center, hypercenterC23×C32S4
commutator, cosocle, hypocenterA4C24×C32
FrattiniC1G
Fitting, socleC25×C32D3
LCSG ▷ A4
UCSC1 ◁ C23×C32

Complements

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

NH ≅ G/Ntype
1. C23×C32×A4C2semidirect
2. C22×C32×S4C2direct (N × H)
3. C23×C3×S4C3direct (N × H)
6. C22×C32×A4C22semidirect
7. C2×C32×S4C22direct (N × H)
9. C25×C32D3semidirect
11. C23×C3×A4C2×C3semidirect
12. C22×C3×S4C2×C3direct (N × H)
18. C2×C32×A4C23semidirect
19. C32×S4C23direct (N × H)
20. C23×S4C32direct (N × H)
27. C24×C32C2×D3semidirect
30. C22×C3×A4C22×C3semidirect
31. C2×C3×S4C22×C3direct (N × H)
37. C32×A4C24semidirect
39. C25×C3C3×D3semidirect
41. C23×A4C2×C32semidirect
42. C22×S4C2×C32direct (N × H)
55. C23×C32S4direct (N × H)
56. C23×C32C22×D3semidirect
59. C2×C3×A4C23×C3semidirect
60. C3×S4C23×C3direct (N × H)
72. C24×C3C2×C3×D3semidirect
75. C22×A4C22×C32semidirect
76. C2×S4C22×C32direct (N × H)
89. C22×C32C2×S4direct (N × H)
90. C22×C32C23×D3semidirect
93. C3×A4C24×C3semidirect
96. C25C32×D3semidirect
110. C23×C3C3×S4direct (N × H)
111. C23×C3C22×C3×D3semidirect
114. C2×A4C23×C32semidirect
115. S4C23×C32direct (N × H)
124. C2×C32C22×S4direct (N × H)
131. C24C2×C32×D3semidirect
144. C22×C3C2×C3×S4direct (N × H)
145. C22×C3C23×C3×D3semidirect
148. A4C24×C32semidirect
152. C32C23×S4direct (N × H)
160. C23C32×S4direct (N × H)
161. C23C22×C32×D3semidirect
170. C2×C3C22×C3×S4direct (N × H)
179. C22C2×C32×S4direct (N × H)
180. C22C23×C32×D3semidirect
184. C3C23×C3×S4direct (N × H)
188. C2C22×C32×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×A42
C22×C32×S42
C23×C3×S43
C23×C32×D43
C23×C32×D34