Finite Group Databasebeta

← C23×(C3⋊S4)⋊C3 #13180081

Subgroups — C23×(C3⋊S4)⋊C3 #13180081

Order 1728.

2848 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 2848 conjugacy classes of subgroups — too many to draw. Only its 230 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×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×(C3⋊S4)⋊C3C2114yesmaximal, direct product factor
3C23×C3⋊S4C311yesmaximal, characteristic, complemented
4C23×C3×#67—31nomaximal
5C23×(C3⋊D3)⋊C3—41nomaximal
6C22×C22⋊H(3,𝔽3)C2217yescomplemented
7C2×(C3⋊S4)⋊C3C22128yesdirect product factor
8C25×C32D311yescharacteristic, complemented
9C24×C3×D3—31no—
10C23×C3×A4—21no—
11C23×C3×A4C2×C311yescharacteristic, complemented
12C22×C3⋊S4C2×C3114yescomplemented
13C22×C3×#67—314no—
14C22×C3×#67—314no—
15C23×C3×Dic3—31no—
16C23×H(3,𝔽3)—41no—
17C22×(C3⋊D3)⋊C3—414no—
18C2×C22⋊H(3,𝔽3)C2317yescomplemented
19(C3⋊S4)⋊C3C2318yesdirect product factor
20C23×S4—91no—
21C23×C3×D4—91no—
22C23×#67—31no—
23C24×C32—31no—
24C24×C32—37no—
25C24×C32C2×D317yescomplemented
26C23×C3⋊D3—41no—
27C23×C3×D3—314no—
28C23×C3×D3—67no—
29C23×C3×D3—61no—
30C22×C3×A4—27no—
31C22×C3×A4C22×C317yescomplemented
32C2×C3⋊S4C22×C3128yescomplemented
33C2×C3×#67—384no—
34C2×C3×#67—328no—
35C22×C3×Dic3—314no—
36C22×H(3,𝔽3)—47no—
37C2×(C3⋊D3)⋊C3—428no—
38C22⋊H(3,𝔽3)C2411yescharacteristic, complemented
39C25×C3—31no—
40C25×C3—91no—
41C25×C3C3×D311yescharacteristic
42C24×D3—31no—
43C23×A4—31no—
44C23×A4—61no—
45C22×S4—914no—
46C22×C3×D4—914no—
47C22×C3×D4—914no—
48C23×C3×C4—91no—
49C22×#67—314no—
50C22×#67—314no—
51C23×Dic3—31no—
52C23×C32—37no—
53C23×C32—37no—
54C23×C32—321no—
55C23×C32—41no—
56C23×C32—67no—
57C23×C32—81no—
58C23×C32S411yescharacteristic, complemented
59C23×C32C22×D317yescomplemented
60C22×C3⋊D3—414no—
61C22×C3×D3—328no—
62C22×C3×D3—642no—
63C22×C3×D3—614no—
64C2×C3×A4—27no—
65C2×C3×A4C23×C317yescomplemented
66C3⋊S4C23×C318yescomplemented
67C3×#67—356no—
68C3×#67—38no—
69C2×C3×Dic3—328no—
70C23×D4—91noSylow
71C2×H(3,𝔽3)—47no—
72(C3⋊D3)⋊C3—48no—
73C24×C3—31no—
74C24×C3—31no—
75C24×C3—37no—
76C24×C3—37no—
77C24×C3—37no—
78C24×C3—61no—
79C24×C3—67no—
80C24×C3—914no—
81C24×C3—187no—
82C24×C3—181no—
83C24×C3C2×C3×D317yes—
84C23×D3—314no—
85C23×D3—67no—
86C23×D3—61no—
87C23×D3—361no—
88C22×A4—37no—
89C22×A4—67no—
90C2×S4—928no—
91C2×C3×D4—984no—
92C2×C3×D4—928no—
93C22×C3×C4—914no—
94C2×#67—384no—
95C2×#67—328no—
96C22×Dic3—314no—
97C22×C32—37no—
98C22×C32—37no—
99C22×C32—321no—
100C22×C32—47no—
101C22×C32—67no—
102C22×C32—87no—
103C22×C32C2×S417yescomplemented
104C22×C32C23×D311yescharacteristic, complemented
105C2×C3⋊D3—428no—
106C2×C3×D3—38no—
107C2×C3×D3—628no—
108C2×C3×D3—628no—
109C3×A4—21no—
110C3×A4C24×C311yescharacteristic, complemented
111C3×Dic3—38no—
112C25—91no—
113C25(C3⋊D3)⋊C311yescharacteristic, complemented
114C22×D4—914no—
115C22×D4—914no—
116C23×C4—91no—
117H(3,𝔽3)—41noSylow
118C23×C3—37no—
119C23×C3—37no—
120C23×C3—37no—
121C23×C3—37no—
122C23×C3—321no—
123C23×C3—321no—
124C23×C3—31no—
125C23×C3—37no—
126C23×C3—67no—
127C23×C3—67no—
128C23×C3—621no—
129C23×C3—67no—
130C23×C3—621no—
131C23×C3—928no—
132C23×C3—121no—
133C23×C3—1842no—
134C23×C3—1814no—
135C23×C3—241no—
136C23×C3C3×S411yescharacteristic
137C23×C3C22×C3×D317yes—
138C22×D3—328no—
139C22×D3—642no—
140C22×D3—614no—
141C22×D3—3614no—
142C2×A4—37no—
143C2×A4—67no—
144S4—98no—
145C3×D4—956no—
146C3×D4—98no—
147C2×C3×C4—928no—
148#67—356no—
149#67—38no—
150C2×Dic3—328no—
151C2×C32—31no—
152C2×C32—37no—
153C2×C32—47no—
154C2×C32—87no—
155C2×C32C22×S417yescomplemented
156C3⋊D3—48no—
157C3×D3—68no—
158C24—31no—
159C24—37no—
160C24—914no—
161C24—187no—
162C24—181no—
163C24C2×(C3⋊D3)⋊C317yescomplemented
164C2×D4—984no—
165C2×D4—928no—
166C22×C4—914no—
167C22×C3—37no—
168C22×C3—37no—
169C22×C3—321no—
170C22×C3—37no—
171C22×C3—321no—
172C22×C3—37no—
173C22×C3—37no—
174C22×C3—31no—
175C22×C3—67no—
176C22×C3—67no—
177C22×C3—621no—
178C22×C3—67no—
179C22×C3—621no—
180C22×C3—98no—
181C22×C3—127no—
182C22×C3—1828no—
183C22×C3—1828no—
184C22×C3—247no—
185C22×C3C2×C3×S417yes—
186C22×C3C23×C3×D311yescharacteristic
187C2×D3—38no—
188C2×D3—628no—
189C2×D3—628no—
190C2×D3—3628no—
191A4—31no—
192A4—61no—
193C3×C4—98no—
194Dic3—38no—
195C32—41no—
196C32—81no—
197C32C23×S411yescharacteristic, complemented
198C23—37no—
199C23—37no—
200C23—321no—
201C23—67no—
202C23—928no—
203C23—1842no—
204C23—1814no—
205C23(C3⋊S4)⋊C311yescharacteristic, direct product factor
206C23C22×(C3⋊D3)⋊C317yescomplemented
207D4—956no—
208D4—98no—
209C2×C4—928no—
210C2×C3—31no—
211C2×C3—37no—
212C2×C3—31no—
213C2×C3—37no—
214C2×C3—37no—
215C2×C3—61no—
216C2×C3—67no—
217C2×C3—127no—
218C2×C3—188no—
219C2×C3—247no—
220C2×C3C22×C3×S417yes—
221D3—68no—
222D3—368no—
223C22—37no—
224C22—37no—
225C22—321no—
226C22—67no—
227C22—98no—
228C22—1828no—
229C22—1828no—
230C22C2×(C3⋊S4)⋊C317yesdirect product factor
231C22C23×(C3⋊D3)⋊C311yescharacteristic, complemented
232C4—98no—
233C3—31no—
234C3—121no—
235C3—241no—
236C3C23×C3×S411yescharacteristic
237C2—31no—
238C2—37no—
239C2—188no—
240C2C22×(C3⋊S4)⋊C317yesdirect product factor
subgroupG/H
center, hypercenterC23(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×A4C24×C3
FrattiniC3C23×C3×S4
FittingC25×C32D3
socleC25×C3C3×D3
LCSG ▷ C3×A4
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊H(3,𝔽3)C2semidirect
2. C22×(C3⋊S4)⋊C3C2direct (N × H)
3. C23×C3⋊S4C3semidirect
6. C22×C22⋊H(3,𝔽3)C22semidirect
7. C2×(C3⋊S4)⋊C3C22direct (N × H)
8. C25×C32D3semidirect
11. C23×C3×A4C2×C3semidirect
12. C22×C3⋊S4C2×C3semidirect
18. C2×C22⋊H(3,𝔽3)C23semidirect
19. (C3⋊S4)⋊C3C23direct (N × H)
25. C24×C32C2×D3semidirect
31. C22×C3×A4C22×C3semidirect
32. C2×C3⋊S4C22×C3semidirect
38. C22⋊H(3,𝔽3)C24semidirect
58. C23×C32S4semidirect
59. C23×C32C22×D3semidirect
65. C2×C3×A4C23×C3semidirect
66. C3⋊S4C23×C3semidirect
103. C22×C32C2×S4semidirect
104. C22×C32C23×D3semidirect
110. C3×A4C24×C3semidirect
113. C25(C3⋊D3)⋊C3semidirect
155. C2×C32C22×S4semidirect
163. C24C2×(C3⋊D3)⋊C3semidirect
197. C32C23×S4semidirect
205. C23(C3⋊S4)⋊C3direct (N × H)
206. C23C22×(C3⋊D3)⋊C3semidirect
230. C22C2×(C3⋊S4)⋊C3direct (N × H)
231. C22C23×(C3⋊D3)⋊C3semidirect
240. C2C22×(C3⋊S4)⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊H(3,𝔽3)2
C22×(C3⋊S4)⋊C32
C23×C3⋊S43
C23×C3×#673
C23×(C3⋊D3)⋊C34