Finite Group Databasebeta

← C23×C3⋊((C22⋊C9)⋊C2) #13180083

Subgroups — C23×C3⋊((C22⋊C9)⋊C2) #13180083

Order 1728.

3938 conjugacy classes of subgroups; only the 307 normal subgroups are shown.

normal subgroups only

This group has 3938 conjugacy classes of subgroups — too many to draw. Only its 307 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×C3×C22⋊C9C211yesmaximal, characteristic, complemented
2C22×C3⋊((C22⋊C9)⋊C2)C2114yesmaximal, direct product factor
3C23×#638—31nomaximal
4C23×(C22⋊C9)⋊C2—33nomaximal
5C23×C3⋊D9—41nomaximal
6C22×C3×C22⋊C9C2217yescomplemented
7C2×C3⋊((C22⋊C9)⋊C2)C22128yesdirect product factor
8C25×C32D311yescharacteristic
9C24×C3⋊D3—31no—
10C22×#638—314no—
11C22×#638—314no—
12C23×C3⋊Dic3—31no—
13C23×C22⋊C9D313yescomplemented
14C22×(C22⋊C9)⋊C2—342no—
15C23×C3×C9—41no—
16C22×C3⋊D9—414no—
17C2×C3×C22⋊C9C2317yescomplemented
18C3⋊((C22⋊C9)⋊C2)C2318yesdirect product factor
19C23×#67—93no—
20C23×#67—91no—
21C24×C32—31no—
22C24×C32—37no—
23C24×C32C2×D317yes—
24C23×C3⋊D3—314no—
25C23×C3⋊D3—67no—
26C23×C3⋊D3—61no—
27C2×#638—384no—
28C2×#638—328no—
29C22×C3⋊Dic3—314no—
30C23×D9—123no—
31C22×C22⋊C9C2×D3121yescomplemented
32C2×(C22⋊C9)⋊C2—384no—
33C22×C3×C9—47no—
34C2×C3⋊D9—428no—
35C3×C22⋊C9C2411yescharacteristic, complemented
36C25×C3D913yescomplemented
37C25×C3C3⋊D311yescharacteristic
38C24×D3—93no—
39C24×D3—91no—
40C22×#67—942no—
41C22×#67—914no—
42C22×#67—942no—
43C22×#67—914no—
44C23×Dic3—93no—
45C23×Dic3—91no—
46C23×C32—37no—
47C23×C32—37no—
48C23×C32—321no—
49C23×C32—67no—
50C23×C32S411yescharacteristic
51C23×C32C22×D317yes—
52C22×C3⋊D3—328no—
53C22×C3⋊D3—642no—
54C22×C3⋊D3—614no—
55#638—356no—
56#638—38no—
57C2×C3⋊Dic3—328no—
58C23×C9—43no—
59C22×D9—1242no—
60C2×C22⋊C9C22×D3121yescomplemented
61(C22⋊C9)⋊C2—324no—
62C23×D4—271noSylow
63C2×C3×C9—47no—
64C3⋊D9—48no—
65C24×C3—33no—
66C24×C3—31no—
67C24×C3—321no—
68C24×C3—37no—
69C24×C3C2×D9121yescomplemented
70C24×C3C2×C3⋊D317yes—
71C23×D3—914no—
72C23×D3—942no—
73C23×D3—187no—
74C23×D3—1821no—
75C23×D3—183no—
76C23×D3—181no—
77C2×#67—9252no—
78C2×#67—984no—
79C2×#67—984no—
80C2×#67—928no—
81C22×Dic3—914no—
82C22×Dic3—942no—
83C22×C32—37no—
84C22×C32—37no—
85C22×C32—321no—
86C22×C32—67no—
87C22×C32C2×S417yes—
88C22×C32C23×D311yescharacteristic
89C2×C3⋊D3—38no—
90C2×C3⋊D3—628no—
91C2×C3⋊D3—628no—
92C3⋊Dic3—38no—
93C22×C9—421no—
94C2×D9—1284no—
95C22⋊C9C23×D313yescomplemented
96C25—271no—
97C25C3⋊D911yescharacteristic, complemented
98C22×D4—2714no—
99C22×D4—2714no—
100C23×C4—271no—
101C3×C9—41noSylow
102C23×C3—37no—
103C23×C3—37no—
104C23×C3—321no—
105C23×C3—321no—
106C23×C3—363no—
107C23×C3—321no—
108C23×C3—67no—
109C23×C3—621no—
110C23×C3(C22⋊C9)⋊C213yescomplemented
111C23×C3C22×D9121yescomplemented
112C23×C3C3⋊S411yescharacteristic
113C23×C3C22×C3⋊D317yes—
114C22×D3—928no—
115C22×D3—984no—
116C22×D3—1842no—
117C22×D3—18126no—
118C22×D3—1814no—
119C22×D3—1842no—
120#67—9168no—
121#67—956no—
122#67—98no—
123#67—924no—
124C2×Dic3—928no—
125C2×Dic3—984no—
126C2×C32—31no—
127C2×C32—37no—
128C2×C32C22×S417yes—
129C3⋊D3—68no—
130C2×C9—421no—
131D9—1224no—
132C24—31no—
133C24—37no—
134C24—2714no—
135C24—547no—
136C24—541no—
137C24C2×C3⋊D917yescomplemented
138C2×D4—2784no—
139C2×D4—2728no—
140C22×C4—2714no—
141C22×C3—321no—
142C22×C3—37no—
143C22×C3—37no—
144C22×C3—321no—
145C22×C3—321no—
146C22×C3—363no—
147C22×C3—621no—
148C22×C3—67no—
149C22×C3C2×(C22⋊C9)⋊C2121yescomplemented
150C22×C3C23×D913yescomplemented
151C22×C3C2×C3⋊S417yes—
152C22×C3C23×C3⋊D311yescharacteristic
153C2×D3—924no—
154C2×D3—98no—
155C2×D3—1884no—
156C2×D3—1828no—
157C2×D3—1828no—
158C2×D3—1884no—
159Dic3—924no—
160Dic3—98no—
161C32C23×S411yescharacteristic
162C9—43no—
163C23—37no—
164C23—37no—
165C23—321no—
166C23—67no—
167C23—2728no—
168C23—5442no—
169C23—5414no—
170C23C3⋊((C22⋊C9)⋊C2)11yescharacteristic, direct product factor
171C23C22×C3⋊D917yescomplemented
172D4—2756no—
173D4—278no—
174C2×C4—2728no—
175C2×C3—33no—
176C2×C3—321no—
177C2×C3—31no—
178C2×C3—37no—
179C2×C3C22×(C22⋊C9)⋊C2121yescomplemented
180C2×C3C22×C3⋊S417yes—
181D3—1824no—
182D3—188no—
183C22—37no—
184C22—37no—
185C22—321no—
186C22—67no—
187C22—278no—
188C22—5428no—
189C22—5428no—
190C22C2×C3⋊((C22⋊C9)⋊C2)17yesdirect product factor
191C22C23×C3⋊D911yescharacteristic, complemented
192C4—278no—
193C3C23×(C22⋊C9)⋊C213yescomplemented
194C3C23×C3⋊S411yescharacteristic
195C2—31no—
196C2—37no—
197C2—548no—
198C2C22×C3⋊((C22⋊C9)⋊C2)17yesdirect product factor
subgroupG/H
center, hypercenterC23C3⋊((C22⋊C9)⋊C2)
commutator, cosocle, hypocenterC3×C22⋊C9C24
FrattiniC3C23×C3⋊S4
Fitting, socleC25×C32D3
LCSG ▷ C3×C22⋊C9
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3×C22⋊C9C2semidirect
2. C22×C3⋊((C22⋊C9)⋊C2)C2direct (N × H)
6. C22×C3×C22⋊C9C22semidirect
7. C2×C3⋊((C22⋊C9)⋊C2)C22direct (N × H)
13. C23×C22⋊C9D3semidirect
17. C2×C3×C22⋊C9C23semidirect
18. C3⋊((C22⋊C9)⋊C2)C23direct (N × H)
31. C22×C22⋊C9C2×D3semidirect
35. C3×C22⋊C9C24semidirect
36. C25×C3D9semidirect
60. C2×C22⋊C9C22×D3semidirect
69. C24×C3C2×D9semidirect
95. C22⋊C9C23×D3semidirect
97. C25C3⋊D9semidirect
110. C23×C3(C22⋊C9)⋊C2semidirect
111. C23×C3C22×D9semidirect
137. C24C2×C3⋊D9semidirect
149. C22×C3C2×(C22⋊C9)⋊C2semidirect
150. C22×C3C23×D9semidirect
170. C23C3⋊((C22⋊C9)⋊C2)direct (N × H)
171. C23C22×C3⋊D9semidirect
179. C2×C3C22×(C22⋊C9)⋊C2semidirect
190. C22C2×C3⋊((C22⋊C9)⋊C2)direct (N × H)
191. C22C23×C3⋊D9semidirect
193. C3C23×(C22⋊C9)⋊C2semidirect
198. C2C22×C3⋊((C22⋊C9)⋊C2)direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C22⋊C92
C22×C3⋊((C22⋊C9)⋊C2)2
C23×#6383
C23×(C22⋊C9)⋊C23
C23×C3⋊D94