Finite Group Databasebeta

← C3⋊#10589552 #13147221

Subgroups — C3⋊#10589552 #13147221

Order 1728.

283 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 39 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
1C3×#120⋊C9C211yesmaximal, characteristic, complemented
2C3⋊(#120⋊C9)C212yesmaximal, complemented
3C3×#5594C311yesmaximal, characteristic
4#10589552—31nomaximal
5C3×C22⋊(C22⋊C9)C2211yescharacteristic, complemented
6C2×D3×C22⋊C9—41nomaximal
7C32×#120C2×C311yescharacteristic
8C3×#953C2×C312yes—
9C3×#937—31no—
10C3×#936—31no—
11C3×#885—32no—
12#120⋊C9—21no—
13#120⋊C9—32no—
14#120⋊C9D311yescharacteristic, complemented
15C2×C3×C22⋊C9—41no—
16D3×C22⋊C9—42no—
17C3×H(3,𝔽4)—31no—
18#5594C911yescharacteristic, complemented
19C24×C32C22×C311yescharacteristic
20C23×C3×D3A411yescharacteristic
21C2×C32×D4—31no—
22C2×C3×#67—32no—
23C2×C3×Dic6—31no—
24C22⋊(C22⋊C9)—21no—
25C22⋊(C22⋊C9)C2×D311yescharacteristic, complemented
26C22×C22⋊C9—121no—
27C32×C22⋊C4—31no—
28C3×#217—31no—
29C3×#217—32no—
30C3×(C2×D3)⋊C4—32no—
31C3×C4×Dic3—31no—
32C2×D3×C9—161no—
33C3×C22⋊C9—41no—
34C3×C22⋊C9—161no—
35C3×#124—92no—
36C3×#124—91no—
37C3×#120—21no—
38C3×#120—32no—
39C3×#120—91no—
40C3×#120C2×C911yescharacteristic, complemented
41C3×#120C3×D311yescharacteristic
42#953C2×C912yescomplemented
43#937—31no—
44#936—31no—
45#885—32no—
46C23×C32—31no—
47C23×C32—121no—
48C23×C32C2×A411yescharacteristic
49C22×C3×D3—121no—
50C22×C3×D3C2×A412yes—
51C32×D4—121no—
52C2×C32×C4—31no—
53C3×#67—122no—
54C2×C3×Dic3—32no—
55C3×Dic6—121no—
56C2×C22⋊C9—41no—
57C2×C22⋊C9—81no—
58C2×C22⋊C9—122no—
59H(3,𝔽4)—31noSylow
60C2×C3×C9—161no—
61D3×C9—162no—
62C24×C3—21no—
63C24×C3—31no—
64C24×C3C22×C911yescharacteristic, complemented
65C24×C3C2×C3×D311yescharacteristic
66C23×D3C22⋊C911yescharacteristic, complemented
67C2×C3×Q8—91no—
68C2×C3×D4—31no—
69C2×C3×D4—31no—
70C2×C3×D4—61no—
71C2×C3×D4—92no—
72C2×#67—32no—
73C2×Dic6—31no—
74C3×C22⋊C4—31no—
75C3×C22⋊C4—31no—
76C3×C22⋊C4—61no—
77C3×C22⋊C4—92no—
78C3×C22⋊C4—91no—
79C3×C22⋊C4—92no—
80C3×C42—91no—
81#217—31no—
82#217—32no—
83(C2×D3)⋊C4—32no—
84C4×Dic3—31no—
85C22×C32—41no—
86C22×C32—61no—
87C22×C32—61no—
88C22×C32—62no—
89C22×C32—121no—
90C22×C32C22×A411yescharacteristic
91C2×C3×D3—41no—
92C2×C3×D3—62no—
93C2×C3×D3—121no—
94C32×C4—61no—
95C3×Dic3—62no—
96C22×C9—481no—
97C22⋊C9—41no—
98C22⋊C9—81no—
99C22⋊C9—161no—
100C22⋊C9—162no—
101#124—92no—
102#124—91no—
103#120—32no—
104#120—91no—
105#120D3×C911yescharacteristic, complemented
106C3×C9—161noSylow
107C23×C3—21no—
108C23×C3—31no—
109C23×C3—31no—
110C23×C3—32no—
111C23×C3—61no—
112C23×C3—121no—
113C23×C3—121no—
114C23×C3—122no—
115C23×C3—361no—
116C23×C3C2×C22⋊C911yescharacteristic
117C23×C3D3×A411yescharacteristic
118C22×D3—121no—
119C22×D3C2×C22⋊C912yes—
120C3×Q8—361no—
121C3×D4—121no—
122C3×D4—121no—
123C3×D4—122no—
124C3×D4—362no—
125C2×C3×C4—31no—
126C2×C3×C4—31no—
127C2×C3×C4—61no—
128C2×C3×C4—92no—
129#67—122no—
130C2×Dic3—32no—
131Dic6—121no—
132C2×C32—31no—
133C2×C32—41no—
134C2×C32—121no—
135C3×D3—42no—
136C2×C9—161no—
137C2×C9—321no—
138C2×C9—482no—
139C24—31no—
140C24C2×D3×C911yescharacteristic, complemented
141C2×Q8—91no—
142C2×D4—31no—
143C2×D4—92no—
144C22⋊C4—31no—
145C22⋊C4—92no—
146C22⋊C4—91no—
147C22⋊C4—92no—
148C42—91no—
149C22×C3—21no—
150C22×C3—41no—
151C22×C3—42no—
152C22×C3—41no—
153C22×C3—61no—
154C22×C3—61no—
155C22×C3—62no—
156C22×C3—61no—
157C22×C3—61no—
158C22×C3—62no—
159C22×C3—62no—
160C22×C3—121no—
161C22×C3—122no—
162C22×C3—121no—
163C22×C3—121no—
164C22×C3—121no—
165C22×C3—122no—
166C22×C3—182no—
167C22×C3—361no—
168C22×C3C22×C22⋊C911yescharacteristic
169C22×C3C2×D3×A411yescharacteristic
170C2×D3—41no—
171C2×D3—62no—
172C2×D3—121no—
173C3×C4—61no—
174C3×C4—61no—
175C3×C4—121no—
176C3×C4—182no—
177Dic3—62no—
178C32#545011yescharacteristic
179C9—161no—
180C9—321no—
181C23—31no—
182C23—32no—
183C23—121no—
184C23—361no—
185C23D3×C22⋊C911yescharacteristic
186Q8—361no—
187D4—121no—
188D4—362no—
189C2×C4—31no—
190C2×C4—92no—
191C2×C3—31no—
192C2×C3—31no—
193C2×C3—41no—
194C2×C3—41no—
195C2×C3—61no—
196C2×C3—81no—
197C2×C3—122no—
198C2×C3—121no—
199C2×C3—121no—
200C2×C3—122no—
201D3—42no—
202C22—41no—
203C22—61no—
204C22—61no—
205C22—62no—
206C22—121no—
207C22—121no—
208C22—182no—
209C22—361no—
210C22C2×D3×C22⋊C911yescharacteristic
211C4—61no—
212C4—182no—
213C3—21no—
214C3#1058955211yescharacteristic, complemented
215C3C3⋊#545011yescharacteristic
216C2—31no—
217C2—41no—
218C2—121no—
219C2—122no—
subgroupG/H
center, hypercenterC3C3⋊#5450
commutator, hypocenterC24×C3C22×C9
FrattiniC22×C3C2×D3×A4
FittingC32×#120C2×C3
socleC22×C32C22×A4
prime radicalC3×#5594C3
cosocleC24×C32C22×C3
LCSG ▷ C24×C3
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×#120⋊C9C2semidirect
2. C3⋊(#120⋊C9)C2semidirect
5. C3×C22⋊(C22⋊C9)C22semidirect
14. #120⋊C9D3semidirect
18. #5594C9semidirect
25. C22⋊(C22⋊C9)C2×D3semidirect
40. C3×#120C2×C9semidirect
42. #953C2×C9semidirect
64. C24×C3C22×C9semidirect
66. C23×D3C22⋊C9semidirect
105. #120D3×C9semidirect
140. C24C2×D3×C9semidirect
214. C3#10589552semidirect

Maximal subgroups

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

subgroupindex
C3×#120⋊C92
C3⋊(#120⋊C9)2
C3×#55943
#105895523
C2×D3×C22⋊C94