Finite Group Databasebeta

← C23×#6258 #13180084

Subgroups — C23×#6258 #13180084

Order 1728.

3320 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 3320 conjugacy classes of subgroups — too many to draw. Only its 211 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×#6258C2114yesmaximal, direct product factor
3C23×C3×S4—33nomaximal
4C23×C3×#67—31nomaximal
5C23×#272—41nomaximal
6C22×C22⋊H(3,𝔽3)C2217yescomplemented
7C2×#6258C22128yesdirect product factor
8C25×C32D311yescharacteristic, complemented
9C24×C3×D3—31no—
10C23×C3×A4D313yescomplemented
11C22×C3×S4—342no—
12C22×C3×#67—314no—
13C22×C3×#67—314no—
14C23×C3×Dic3—31no—
15C23×H(3,𝔽3)—41no—
16C22×#272—414no—
17C2×C22⋊H(3,𝔽3)C2317yescomplemented
18#6258C2318yesdirect product factor
19C23×S4—33no—
20C23×C3×D4—91no—
21C23×#67—31no—
22C24×C32—31no—
23C24×C32—37no—
24C24×C32C2×D317yescomplemented
25C23×C3×D3—314no—
26C23×C3×D3—67no—
27C23×C3×D3—61no—
28C23×C3×D3—123no—
29C22×C3×A4C2×D3121yescomplemented
30C2×C3×S4—384no—
31C2×C3×#67—384no—
32C2×C3×#67—328no—
33C22×C3×Dic3—314no—
34C22×H(3,𝔽3)—47no—
35C2×#272—428no—
36C22⋊H(3,𝔽3)C2411yescharacteristic, complemented
37C25×C3—31no—
38C25×C3—91no—
39C25×C3C3⋊D311yescharacteristic
40C24×D3—31no—
41C23×A4—33no—
42C22×S4—342no—
43C22×C3×D4—914no—
44C22×C3×D4—914no—
45C23×C3×C4—91no—
46C22×#67—314no—
47C22×#67—314no—
48C23×Dic3—31no—
49C23×C32—37no—
50C23×C32—37no—
51C23×C32—321no—
52C23×C32—43no—
53C23×C32—67no—
54C23×C32S411yescharacteristic, complemented
55C23×C32C22×D317yescomplemented
56C22×C3×D3—328no—
57C22×C3×D3—642no—
58C22×C3×D3—614no—
59C22×C3×D3—1242no—
60C2×C3×A4C22×D3121yescomplemented
61C3×S4—324no—
62C3×#67—356no—
63C3×#67—38no—
64C2×C3×Dic3—328no—
65C23×D4—91noSylow
66C2×H(3,𝔽3)—47no—
67#272—48no—
68C24×C3—31no—
69C24×C3—31no—
70C24×C3—37no—
71C24×C3—37no—
72C24×C3—37no—
73C24×C3—61no—
74C24×C3—67no—
75C24×C3—914no—
76C24×C3—187no—
77C24×C3—181no—
78C24×C3C2×C3⋊D317yes—
79C23×D3—314no—
80C23×D3—67no—
81C23×D3—61no—
82C23×D3—123no—
83C22×A4—321no—
84C2×S4—384no—
85C2×C3×D4—984no—
86C2×C3×D4—928no—
87C22×C3×C4—914no—
88C2×#67—384no—
89C2×#67—328no—
90C22×Dic3—314no—
91C22×C32—37no—
92C22×C32—37no—
93C22×C32—321no—
94C22×C32—421no—
95C22×C32—67no—
96C22×C32C2×S417yescomplemented
97C22×C32C23×D311yescharacteristic, complemented
98C2×C3×D3—38no—
99C2×C3×D3—628no—
100C2×C3×D3—628no—
101C2×C3×D3—1284no—
102C3×A4C23×D313yescomplemented
103C3×Dic3—38no—
104C25—91no—
105C25#27211yescharacteristic, complemented
106C22×D4—914no—
107C22×D4—914no—
108C23×C4—91no—
109H(3,𝔽3)—41noSylow
110C23×C3—37no—
111C23×C3—37no—
112C23×C3—37no—
113C23×C3—37no—
114C23×C3—321no—
115C23×C3—321no—
116C23×C3—31no—
117C23×C3—37no—
118C23×C3—67no—
119C23×C3—67no—
120C23×C3—621no—
121C23×C3—67no—
122C23×C3—621no—
123C23×C3—928no—
124C23×C3—123no—
125C23×C3—1842no—
126C23×C3—1814no—
127C23×C3C3⋊S411yescharacteristic
128C23×C3C22×C3⋊D317yes—
129C22×D3—328no—
130C22×D3—642no—
131C22×D3—614no—
132C22×D3—1242no—
133C2×A4—321no—
134S4—324no—
135C3×D4—956no—
136C3×D4—98no—
137C2×C3×C4—928no—
138#67—356no—
139#67—38no—
140C2×Dic3—328no—
141C2×C32—31no—
142C2×C32—37no—
143C2×C32—421no—
144C2×C32C22×S417yescomplemented
145C3×D3—68no—
146C3×D3—1224no—
147C24—31no—
148C24—37no—
149C24—914no—
150C24—187no—
151C24—181no—
152C24C2×#27217yescomplemented
153C2×D4—984no—
154C2×D4—928no—
155C22×C4—914no—
156C22×C3—37no—
157C22×C3—37no—
158C22×C3—321no—
159C22×C3—37no—
160C22×C3—321no—
161C22×C3—37no—
162C22×C3—37no—
163C22×C3—31no—
164C22×C3—67no—
165C22×C3—67no—
166C22×C3—621no—
167C22×C3—67no—
168C22×C3—621no—
169C22×C3—98no—
170C22×C3—1221no—
171C22×C3—1828no—
172C22×C3—1828no—
173C22×C3C2×C3⋊S417yes—
174C22×C3C23×C3⋊D311yescharacteristic
175C2×D3—38no—
176C2×D3—628no—
177C2×D3—628no—
178C2×D3—1284no—
179A4—33no—
180C3×C4—98no—
181Dic3—38no—
182C32—43no—
183C32C23×S411yescharacteristic, complemented
184C23—37no—
185C23—37no—
186C23—321no—
187C23—67no—
188C23—928no—
189C23—1842no—
190C23—1814no—
191C23#625811yescharacteristic, direct product factor
192C23C22×#27217yescomplemented
193D4—956no—
194D4—98no—
195C2×C4—928no—
196C2×C3—31no—
197C2×C3—37no—
198C2×C3—31no—
199C2×C3—37no—
200C2×C3—37no—
201C2×C3—61no—
202C2×C3—67no—
203C2×C3—1221no—
204C2×C3—188no—
205C2×C3C22×C3⋊S417yes—
206D3—68no—
207D3—1224no—
208C22—37no—
209C22—37no—
210C22—321no—
211C22—67no—
212C22—98no—
213C22—1828no—
214C22—1828no—
215C22C2×#625817yesdirect product factor
216C22C23×#27211yescharacteristic, complemented
217C4—98no—
218C3—31no—
219C3—123no—
220C3C23×C3⋊S411yescharacteristic
221C2—31no—
222C2—37no—
223C2—188no—
224C2C22×#625817yesdirect product factor
subgroupG/H
center, hypercenterC23×C3C3⋊S4
commutator, cosocle, hypocenterC22⋊H(3,𝔽3)C24
FrattiniC3C23×C3⋊S4
FittingC25×C32D3
socleC25×C3C3⋊D3
LCSG ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊H(3,𝔽3)C2semidirect
2. C22×#6258C2direct (N × H)
6. C22×C22⋊H(3,𝔽3)C22semidirect
7. C2×#6258C22direct (N × H)
8. C25×C32D3semidirect
10. C23×C3×A4D3semidirect
17. C2×C22⋊H(3,𝔽3)C23semidirect
18. #6258C23direct (N × H)
24. C24×C32C2×D3semidirect
29. C22×C3×A4C2×D3semidirect
36. C22⋊H(3,𝔽3)C24semidirect
54. C23×C32S4semidirect
55. C23×C32C22×D3semidirect
60. C2×C3×A4C22×D3semidirect
96. C22×C32C2×S4semidirect
97. C22×C32C23×D3semidirect
102. C3×A4C23×D3semidirect
105. C25#272semidirect
144. C2×C32C22×S4semidirect
152. C24C2×#272semidirect
183. C32C23×S4semidirect
191. C23#6258direct (N × H)
192. C23C22×#272semidirect
215. C22C2×#6258direct (N × H)
216. C22C23×#272semidirect
224. C2C22×#6258direct (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×#62582
C23×C3×S43
C23×C3×#673
C23×#2724