Finite Group Databasebeta

← C2×C3⋊(C3⋊(#120⋊C3)) #13173192

Subgroups — C2×C3⋊(C3⋊(#120⋊C3)) #13173192

Order 1728.

964 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 964 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C2×C32×C22⋊A4C211yesmaximal, characteristic, complemented
2C3⋊(C3⋊(#120⋊C3))C212yesmaximal, direct product factor
3C2×#64162C311yesmaximal, characteristic, complemented
4C2×C3⋊(#120⋊C3)—34nomaximal
5C32×C22⋊A4C2211yescharacteristic, complemented
6C2×A4×C3⋊D3—41nomaximal
7C25×C32C2×C311yescharacteristic, complemented
8C2×C3×C22⋊A4—24no—
9C2×C3×C22⋊A4D314yescomplemented
10C22×#638—31no—
11#64162—32no—
12#64162C2×C312yescomplemented
13C2×#3757—31no—
14C3⋊(#120⋊C3)—38no—
15C2×C32×A4—41no—
16C2×C32×A4—82no—
17A4×C3⋊D3—42no—
18C2×#953—34no—
19C2×#120⋊C3—91no—
20C24×C32—31no—
21C24×C32—31no—
22C24×C32—62no—
23C24×C32—62no—
24C24×C32C22×C311yescharacteristic, complemented
25C23×C3⋊D3A411yescharacteristic, complemented
26C3×C22⋊A4—24no—
27C3×C22⋊A4C2×D314yescomplemented
28C2×D3×A4—124no—
29C2×#638—32no—
30C2×#638—32no—
31C2×#638—62no—
32C2×#638—62no—
33C22×C3⋊Dic3—31no—
34#3757—32no—
35#3757—32no—
36C2×C3×C3⋊D3—161no—
37C32×A4—41no—
38C32×A4—82no—
39C25×C3C3×D314yescomplemented
40C2×C22⋊A4—24no—
41C2×C22⋊A4C3⋊D311yescharacteristic, complemented
42C22×#67—94no—
43#953—38no—
44#953—98no—
45C2×#217—94no—
46#120⋊C3—92no—
47C23×C32—22no—
48C23×C32—32no—
49C23×C32—32no—
50C23×C32—31no—
51C23×C32—31no—
52C23×C32—62no—
53C23×C32—62no—
54C23×C32—62no—
55C23×C32—62no—
56C23×C32—62no—
57C23×C32—64no—
58C23×C32—62no—
59C23×C32—62no—
60C23×C32—64no—
61C23×C32C2×A411yescharacteristic
62C22×C3⋊D3—61no—
63C22×C3⋊D3—61no—
64C22×C3⋊D3C2×A412yescomplemented
65C2×C3×A4—44no—
66C2×C3×A4—88no—
67C2×C3×A4—816no—
68C2×C3×A4—84no—
69D3×A4—128no—
70#638—64no—
71#638—64no—
72C2×C3⋊Dic3—32no—
73C2×C3⋊Dic3—61no—
74C2×C3⋊Dic3—61no—
75C2×#120—91noSylow
76C2×C33—161no—
77C3×C3⋊D3—162no—
78C24×C3—34no—
79C24×C3—34no—
80C24×C3—68no—
81C24×C3—68no—
82C24×C3C2×C3×D314yescomplemented
83C23×D3—34no—
84C22⋊A4—24no—
85C22⋊A4C2×C3⋊D311yescharacteristic, complemented
86C22×A4—361no—
87C2×#67—98no—
88C2×#67—98no—
89C2×#67—188no—
90C2×#67—188no—
91C22×Dic3—94no—
92#217—98no—
93#217—98no—
94C22×C32—22no—
95C22×C32—31no—
96C22×C32—31no—
97C22×C32—32no—
98C22×C32—32no—
99C22×C32—62no—
100C22×C32—62no—
101C22×C32—64no—
102C22×C32—62no—
103C22×C32—62no—
104C22×C32—62no—
105C22×C32—62no—
106C22×C32—64no—
107C22×C32—62no—
108C22×C32C22×A411yescharacteristic
109C2×C3⋊D3—41no—
110C2×C3⋊D3—62no—
111C2×C3⋊D3—121no—
112C2×C3×D3—484no—
113C3×A4—44no—
114C3×A4—88no—
115C3×A4—816no—
116C3×A4—84no—
117C3⋊Dic3—62no—
118C25C3×C3⋊D311yescharacteristic, complemented
119C22×D4—271no—
120#120—92no—
121#120—272no—
122C2×C22⋊C4—271no—
123C33—161noSylow
124C23×C3—28no—
125C23×C3—38no—
126C23×C3—38no—
127C23×C3—34no—
128C23×C3—34no—
129C23×C3—68no—
130C23×C3—68no—
131C23×C3—68no—
132C23×C3—68no—
133C23×C3—68no—
134C23×C3—616no—
135C23×C3—68no—
136C23×C3—68no—
137C23×C3—616no—
138C23×C3D3×A414yes—
139C22×D3—38no—
140C22×D3—184no—
141C22×D3—184no—
142C2×A4—41no—
143C2×A4—82no—
144C2×A4—816no—
145C2×A4—84no—
146C2×A4—362no—
147#67—1816no—
148#67—1816no—
149C2×Dic3—98no—
150C2×Dic3—184no—
151C2×Dic3—184no—
152C2×C32—31no—
153C2×C32—31no—
154C2×C32—62no—
155C2×C32—62no—
156C2×C32—164no—
157C2×C32—324no—
158C2×C32#120⋊C311yescharacteristic, complemented
159C3⋊D3—42no—
160C3×D3—488no—
161C24—31no—
162C24—31no—
163C24—62no—
164C24—62no—
165C24—91no—
166C24C2×C3×C3⋊D311yescharacteristic, complemented
167C2×D4—272no—
168C2×D4—272no—
169C2×D4—542no—
170C2×D4—542no—
171C22×C4—271no—
172C22⋊C4—272no—
173C22⋊C4—272no—
174C22×C3—28no—
175C22×C3—34no—
176C22×C3—34no—
177C22×C3—38no—
178C22×C3—38no—
179C22×C3—68no—
180C22×C3—68no—
181C22×C3—616no—
182C22×C3—68no—
183C22×C3—68no—
184C22×C3—68no—
185C22×C3—68no—
186C22×C3—616no—
187C22×C3—68no—
188C22×C3—1441no—
189C22×C3C2×D3×A414yes—
190C2×D3—124no—
191C2×D3—188no—
192C2×D3—364no—
193A4—41no—
194A4—82no—
195A4—816no—
196A4—84no—
197Dic3—188no—
198C32—164no—
199C32—324no—
200C32C2×#120⋊C311yescharacteristic, complemented
201C23—22no—
202C23—32no—
203C23—32no—
204C23—31no—
205C23—31no—
206C23—62no—
207C23—62no—
208C23—62no—
209C23—62no—
210C23—62no—
211C23—64no—
212C23—62no—
213C23—62no—
214C23—64no—
215C23—92no—
216C23—541no—
217C23—541no—
218C23A4×C3⋊D311yescharacteristic
219D4—544no—
220D4—544no—
221C2×C4—272no—
222C2×C4—541no—
223C2×C4—541no—
224C2×C3—34no—
225C2×C3—34no—
226C2×C3—68no—
227C2×C3—68no—
228C2×C3—161no—
229C2×C3—324no—
230C2×C3—1442no—
231C2×C3C3⋊(#120⋊C3)14yescomplemented
232D3—128no—
233C22—22no—
234C22—31no—
235C22—31no—
236C22—32no—
237C22—32no—
238C22—62no—
239C22—62no—
240C22—64no—
241C22—62no—
242C22—62no—
243C22—62no—
244C22—62no—
245C22—64no—
246C22—62no—
247C22—361no—
248C22—542no—
249C22—1081no—
250C22C2×A4×C3⋊D311yescharacteristic
251C4—542no—
252C3—161no—
253C3—324no—
254C3C2×C3⋊(#120⋊C3)14yescomplemented
255C2—31no—
256C2—31no—
257C2—62no—
258C2—62no—
259C2—362no—
260C2C3⋊(C3⋊(#120⋊C3))11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2C3⋊(C3⋊(#120⋊C3))
commutator, cosocle, hypocenterC24×C32C22×C3
FrattiniC22C2×A4×C3⋊D3
FittingC25×C32C2×C3
socleC23×C32C2×A4
LCSG ▷ C24×C32
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C32×C22⋊A4C2semidirect
2. C3⋊(C3⋊(#120⋊C3))C2direct (N × H)
3. C2×#64162C3semidirect
5. C32×C22⋊A4C22semidirect
7. C25×C32C2×C3semidirect
9. C2×C3×C22⋊A4D3semidirect
12. #64162C2×C3semidirect
24. C24×C32C22×C3semidirect
25. C23×C3⋊D3A4semidirect
27. C3×C22⋊A4C2×D3semidirect
39. C25×C3C3×D3semidirect
41. C2×C22⋊A4C3⋊D3semidirect
64. C22×C3⋊D3C2×A4semidirect
82. C24×C3C2×C3×D3semidirect
85. C22⋊A4C2×C3⋊D3semidirect
118. C25C3×C3⋊D3semidirect
158. C2×C32#120⋊C3semidirect
166. C24C2×C3×C3⋊D3semidirect
200. C32C2×#120⋊C3semidirect
231. C2×C3C3⋊(#120⋊C3)semidirect
254. C3C2×C3⋊(#120⋊C3)semidirect
260. C2C3⋊(C3⋊(#120⋊C3))direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C22⋊A42
C3⋊(C3⋊(#120⋊C3))2
C2×#641623
C2×C3⋊(#120⋊C3)3
C2×A4×C3⋊D34