Finite Group Databasebeta

← #13147036

Subgroups — #13147036

Order 1728.

446 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 99 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×C3×C4×C22⋊C9C211yesmaximal, characteristic
2C3⋊((C22⋊C9)⋊C8)C212yesmaximal, complemented
3#10592906—31nomaximal
4#10589511—33nomaximal
5C22×C3×C22⋊C9C411yescharacteristic
6C3×C4×C22⋊C9C411yescharacteristic
7C3×C4×C22⋊C9C2211yescharacteristic
8#88532—41nomaximal
9C23×C32×C4D311yescharacteristic
10C2×#3748—31no—
11C2×#3748—31no—
12C2×C4×C22⋊C9D313yes—
13#63657—32no—
14#63657—32no—
15(C22⋊C9)⋊C8—36no—
16C2×C3×C22⋊C9—21no—
17C2×C3×C22⋊C9C2×C411yescharacteristic
18C2×C3×C4×C9—41no—
19C3⋊(C9⋊C8)—42no—
20#5207—93no—
21#5207—91no—
22C24×C32Dic311yescharacteristic
23C22×C32×C4—31no—
24C22×C32×C4—31no—
25C22×C32×C4—31no—
26C22×C32×C4—31no—
27C22×C32×C4Dic311yescharacteristic
28C22×C32×C4C2×D311yescharacteristic
29C22×C22⋊C9Dic313yes—
30#3748—61no—
31#3748—61no—
32#3748—61no—
33#3748—61no—
34C2×C3⋊(C3⋊C8)—32no—
35C2×C3⋊(C3⋊C8)—32no—
36C4×C22⋊C9Dic313yes—
37C4×C22⋊C9C2×D313yes—
38#3667—123no—
39C22×C3×C9—41no—
40C3×C22⋊C9M1611yescharacteristic, complemented
41C3×C4×C9—41no—
42C3×C4×C9—41no—
43C23×C3×C4D913yes—
44C23×C3×C4C3⋊D311yescharacteristic
45C2×#208—93no—
46C2×#208—91no—
47C2×#208—93no—
48C2×#208—91no—
49#830—96no—
50#830—92no—
51#830—96no—
52#830—92no—
53C23×C32—21no—
54C23×C32—31no—
55C23×C32—31no—
56C23×C32—61no—
57C23×C32C2×Dic311yescharacteristic
58C2×C32×C4—31no—
59C2×C32×C4—31no—
60C2×C32×C4—31no—
61C2×C32×C4—31no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C32×C4—31no—
65C2×C32×C4—61no—
66C2×C32×C4S411yescharacteristic
67C2×C22⋊C9—23no—
68C2×C22⋊C9C2×Dic313yes—
69C3⋊(C3⋊C8)—62no—
70C3⋊(C3⋊C8)—62no—
71C2×C4×C9—43no—
72C9⋊C8—126no—
73#407—271noSylow
74C2×C3×C9—41no—
75C2×C3×C9—81no—
76C24×C3Dic913yes—
77C24×C3C3⋊Dic311yescharacteristic
78C22×C3×C4—33no—
79C22×C3×C4—31no—
80C22×C3×C4—33no—
81C22×C3×C4—31no—
82C22×C3×C4—33no—
83C22×C3×C4—33no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4Dic913yes—
87C22×C3×C4C2×D913yes—
88C22×C3×C4C3⋊Dic311yescharacteristic
89C22×C3×C4C2×C3⋊D311yescharacteristic
90#208—183no—
91#208—181no—
92#208—183no—
93#208—181no—
94#208—181no—
95#208—183no—
96#208—183no—
97#208—181no—
98C2×C3⋊C8—92no—
99C2×C3⋊C8—96no—
100C2×C3⋊C8—92no—
101C2×C3⋊C8—96no—
102C22×C32—31no—
103C22×C32—31no—
104C22×C32—32no—
105C22×C32—31no—
106C22×C32—61no—
107C22×C32—61no—
108C22×C32—61no—
109C22×C32#20811yescharacteristic
110C22×C32SL(2,ℤ/4)11yescharacteristic
111C32×C4—31no—
112C32×C4—31no—
113C32×C4SL(2,ℤ/4)11yescharacteristic
114C32×C4C2×S411yescharacteristic
115C22×C9—43no—
116C22⋊C9#20813yescomplemented
117C4×C9—43no—
118C4×C9—43no—
119C23×C4C3⋊D911yescharacteristic
120C2×M16—271no—
121C2×M16—271no—
122C22⋊C8—272no—
123C22⋊C8—272no—
124C3×C9—41noSylow
125C23×C3—23no—
126C23×C3—21no—
127C23×C3—33no—
128C23×C3—31no—
129C23×C3—33no—
130C23×C3—31no—
131C23×C3—63no—
132C23×C3—61no—
133C23×C3C2×Dic913yes—
134C23×C3C2×C3⋊Dic311yescharacteristic
135C2×C3×C4—33no—
136C2×C3×C4—31no—
137C2×C3×C4—31no—
138C2×C3×C4—31no—
139C2×C3×C4—33no—
140C2×C3×C4—33no—
141C2×C3×C4—33no—
142C2×C3×C4—33no—
143C2×C3×C4—31no—
144C2×C3×C4—31no—
145C2×C3×C4—33no—
146C2×C3×C4—33no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—61no—
150C2×C3×C4—63no—
151C2×C3×C4(C22⋊C9)⋊C213yes—
152C2×C3×C4C3⋊S411yescharacteristic
153C3⋊C8—182no—
154C3⋊C8—186no—
155C3⋊C8—182no—
156C3⋊C8—186no—
157C2×C32—21no—
158C2×C32—31no—
159C2×C32—31no—
160C2×C32—61no—
161C2×C32C2×SL(2,ℤ/4)11yescharacteristic
162C2×C9—43no—
163C2×C9—83no—
164C24C3⋊Dic911yescharacteristic
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4C3⋊Dic911yescharacteristic
170C22×C4C2×C3⋊D911yescharacteristic
171M16—541no—
172M16—541no—
173M16—541no—
174M16—541no—
175C2×C8—272no—
176C2×C8—272no—
177C22×C3—33no—
178C22×C3—31no—
179C22×C3—31no—
180C22×C3—33no—
181C22×C3—36no—
182C22×C3—32no—
183C22×C3—31no—
184C22×C3—33no—
185C22×C3—63no—
186C22×C3—61no—
187C22×C3—63no—
188C22×C3—61no—
189C22×C3—63no—
190C22×C3—61no—
191C22×C3#366713yescomplemented
192C22×C3(C22⋊C9)⋊C413yes—
193C22×C3#374811yescharacteristic
194C22×C3C3⋊SL(2,ℤ/4)11yescharacteristic
195C3×C4—31no—
196C3×C4—31no—
197C3×C4—33no—
198C3×C4—33no—
199C3×C4(C22⋊C9)⋊C413yes—
200C3×C4C2×(C22⋊C9)⋊C213yes—
201C3×C4C3⋊SL(2,ℤ/4)11yescharacteristic
202C3×C4C2×C3⋊S411yescharacteristic
203C32#540911yescharacteristic
204C9—43no—
205C23—21no—
206C23—31no—
207C23—31no—
208C23—61no—
209C23C2×C3⋊Dic911yescharacteristic
210C2×C4—31no—
211C2×C4—31no—
212C2×C4—31no—
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—61no—
218C2×C4C3⋊((C22⋊C9)⋊C2)11yescharacteristic
219C8—542no—
220C8—542no—
221C2×C3—23no—
222C2×C3—21no—
223C2×C3—33no—
224C2×C3—33no—
225C2×C3—31no—
226C2×C3—31no—
227C2×C3—63no—
228C2×C3—61no—
229C2×C3C2×(C22⋊C9)⋊C413yes—
230C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
231C22—31no—
232C22—31no—
233C22—32no—
234C22—31no—
235C22—61no—
236C22—61no—
237C22—61no—
238C22#8853211yescharacteristic, complemented
239C22C3⋊((C22⋊C9)⋊C4)11yescharacteristic
240C4—31no—
241C4—31no—
242C4C3⋊((C22⋊C9)⋊C4)11yescharacteristic
243C4C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
244C3#1058951113yescomplemented
245C3#1059364311yescharacteristic
246C2—21no—
247C2—31no—
248C2—31no—
249C2—61no—
250C2C2×C3⋊((C22⋊C9)⋊C4)11yescharacteristic
subgroupG/H
centerC4C2×C3⋊((C22⋊C9)⋊C2)
commutatorC2×C3×C22⋊C9C2×C4
FrattiniC3×C4C2×C3⋊S4
FittingC23×C32×C4D3
socleC23×C32C2×Dic3
prime radicalC24×C32Dic3
cosocleC3×C4×C22⋊C9C22
hypercenterC2×C4C3⋊((C22⋊C9)⋊C2)
hypocenterC3×C22⋊C9M16
LCSG ▷ C2×C3×C22⋊C9 ▷ C3×C22⋊C9
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊((C22⋊C9)⋊C8)C2semidirect
40. C3×C22⋊C9M16semidirect
116. C22⋊C9#208semidirect
191. C22×C3#3667semidirect
238. C22#88532semidirect
244. C3#10589511semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C92
C3⋊((C22⋊C9)⋊C8)2
#105929063
#105895113
#885324