Finite Group Databasebeta

← #13180070

Subgroups — #13180070

Order 1728.

824 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 30 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
1C22×C3×C22⋊A4C311yesmaximal, characteristic
2C22⋊(C22⋊(C22⋊C9))C311yesmaximal, characteristic, complemented
3C22×C22⋊(C22⋊C9)C312yesmaximal, complemented
4(C22⋊A4)⋊C9—41nomaximal
5A4×C22⋊C9—42nomaximal
6C2×C3×C22⋊A4—31no—
7C2×C22⋊(C22⋊C9)—32no—
8C26×C3C3211yescharacteristic
9C22×C22⋊A4C913yescomplemented
10C3×C22⋊A4A411yescharacteristic
11C22×C3×A4—42no—
12C22×C3×A4—121no—
13C22⋊(C22⋊C9)—42no—
14C22⋊(C22⋊C9)—41no—
15C22⋊(C22⋊C9)—122no—
16C22⋊(C22⋊C9)—121no—
17C22⋊(C22⋊C9)—123no—
18C22⋊(C22⋊C9)A412yescomplemented
19C22×C22⋊C9—42no—
20C22×C22⋊C9—42no—
21C22×C22⋊C9—122no—
22C3×C22⋊C9—161no—
23C9×A4—162no—
24C25×C3—31no—
25C25×C3—32no—
26C25×C3—91no—
27C25×C3—92no—
28C25×C3—93no—
29C2×C22⋊A4—33no—
30C2×C3×A4—122no—
31C2×C3×A4—123no—
32C2×C22⋊C9—122no—
33C2×C22⋊C9—122no—
34C2×C22⋊C9—126no—
35C26C3×C911yescharacteristic, complemented, Sylow
36C24×C3—31no—
37C24×C3—32no—
38C24×C3—33no—
39C24×C3—32no—
40C24×C3—31no—
41C24×C3—91no—
42C24×C3—92no—
43C24×C3—92no—
44C24×C3—92no—
45C24×C3—92no—
46C24×C3—96no—
47C24×C3—93no—
48C24×C3—96no—
49C24×C3—93no—
50C24×C3—93no—
51C24×C3—96no—
52C24×C3—91no—
53C24×C3—92no—
54C24×C3—93no—
55C24×C3—912no—
56C24×C3—96no—
57C24×C3—93no—
58C24×C3—96no—
59C24×C3C3×A412yes—
60C24×C3C3×A411yescharacteristic
61C22⋊A4C22⋊C913yescomplemented
62C22×A4—46no—
63C22×A4—123no—
64C22×C32—161no—
65C3×A4—42no—
66C3×A4—121no—
67C22×C9—162no—
68C22⋊C9—42no—
69C22⋊C9—42no—
70C22⋊C9—122no—
71C22⋊C9—162no—
72C22⋊C9—161no—
73C22⋊C9—482no—
74C22⋊C9—481no—
75C22⋊C9—483no—
76C25—32no—
77C25—31no—
78C25—91no—
79C25—92no—
80C25—93no—
81C3×C9—641noSylow
82C23×C3—36no—
83C23×C3—33no—
84C23×C3—32no—
85C23×C3—32no—
86C23×C3—32no—
87C23×C3—92no—
88C23×C3—91no—
89C23×C3—92no—
90C23×C3—91no—
91C23×C3—96no—
92C23×C3—93no—
93C23×C3—96no—
94C23×C3—96no—
95C23×C3—93no—
96C23×C3—96no—
97C23×C3—93no—
98C23×C3—93no—
99C23×C3—96no—
100C23×C3—93no—
101C23×C3—96no—
102C23×C3—96no—
103C23×C3—96no—
104C23×C3—93no—
105C23×C3—96no—
106C23×C3—96no—
107C23×C3—93no—
108C23×C3—912no—
109C23×C3—912no—
110C23×C3—912no—
111C23×C3—96no—
112C23×C3—912no—
113C23×C3—96no—
114C23×C3—92no—
115C23×C3—91no—
116C2×A4—126no—
117C2×A4—129no—
118C2×C32—481no—
119C2×C9—482no—
120C24—31no—
121C24—32no—
122C24—33no—
123C24—32no—
124C24—31no—
125C24—92no—
126C24—91no—
127C24—92no—
128C24—92no—
129C24—92no—
130C24—93no—
131C24—96no—
132C24—93no—
133C24—96no—
134C24—96no—
135C24—93no—
136C24—92no—
137C24—91no—
138C24—93no—
139C24—912no—
140C24—96no—
141C24—93no—
142C24—96no—
143C24C9×A412yescomplemented
144C24C3×C22⋊C911yescharacteristic, complemented
145C22×C3—32no—
146C22×C3—31no—
147C22×C3—32no—
148C22×C3—31no—
149C22×C3—33no—
150C22×C3—91no—
151C22×C3—92no—
152C22×C3—92no—
153C22×C3—92no—
154C22×C3—92no—
155C22×C3—96no—
156C22×C3—93no—
157C22×C3—96no—
158C22×C3—93no—
159C22×C3—96no—
160C22×C3—93no—
161C22×C3—93no—
162C22×C3—96no—
163C22×C3—912no—
164C22×C3—96no—
165C22×C3—93no—
166C22×C3—92no—
167C22×C3—91no—
168C22×C3—163no—
169C22×C3A4212yes—
170C22×C3A4211yescharacteristic
171A4—46no—
172A4—123no—
173C32—161no—
174C9—162no—
175C9—641no—
176C23—33no—
177C23—36no—
178C23—32no—
179C23—32no—
180C23—32no—
181C23—92no—
182C23—91no—
183C23—92no—
184C23—91no—
185C23—96no—
186C23—93no—
187C23—96no—
188C23—96no—
189C23—93no—
190C23—93no—
191C23—96no—
192C23—93no—
193C23—96no—
194C23—93no—
195C23—96no—
196C23—96no—
197C23—96no—
198C23—93no—
199C23—96no—
200C23—96no—
201C23—93no—
202C23—912no—
203C23—912no—
204C23—912no—
205C23—96no—
206C23—96no—
207C23—912no—
208C23—92no—
209C23—91no—
210C2×C3—32no—
211C2×C3—31no—
212C2×C3—92no—
213C2×C3—91no—
214C2×C3—93no—
215C2×C3—483no—
216C22—31no—
217C22—32no—
218C22—32no—
219C22—31no—
220C22—33no—
221C22—91no—
222C22—92no—
223C22—92no—
224C22—92no—
225C22—92no—
226C22—96no—
227C22—93no—
228C22—96no—
229C22—93no—
230C22—93no—
231C22—96no—
232C22—93no—
233C22—912no—
234C22—96no—
235C22—93no—
236C22—96no—
237C22—92no—
238C22—91no—
239C22A4×C22⋊C912yescomplemented
240C22(C22⋊A4)⋊C911yescharacteristic, complemented
241C3—163no—
242C3#1059678211yescharacteristic
243C2—32no—
244C2—31no—
245C2—91no—
246C2—92no—
247C2—93no—
subgroupG/H
center, Frattini, hypercenterC3#10596782
commutator, hypocenterC26C3×C9
Fitting, socle, cosocleC26×C3C32
prime radicalC22×C3×C22⋊A4C3
LCSG ▷ C26
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
2. C22⋊(C22⋊(C22⋊C9))C3semidirect
3. C22×C22⋊(C22⋊C9)C3semidirect
9. C22×C22⋊A4C9semidirect
18. C22⋊(C22⋊C9)A4semidirect
35. C26C3×C9semidirect
61. C22⋊A4C22⋊C9semidirect
143. C24C9×A4semidirect
144. C24C3×C22⋊C9semidirect
239. C22A4×C22⋊C9semidirect
240. C22(C22⋊A4)⋊C9semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊A43
C22⋊(C22⋊(C22⋊C9))3
C22×C22⋊(C22⋊C9)3
(C22⋊A4)⋊C94
A4×C22⋊C94