Finite Group Databasebeta

← #13180073

Subgroups — #13180073

Order 1728.

808 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 24 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#88881—41nomaximal
5#88880—42nomaximal
6C2×C3×C22⋊A4—31no—
7C2×C22⋊(C22⋊C9)—32no—
8C26×C3C3211yescharacteristic
9C22×C22⋊A4—31no—
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—
22#1094—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—
35C263−1+211yescharacteristic, 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⋊A4—31no—
62C22×A4—123no—
63C22×A4—122no—
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—
813−1+2—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—129no—
117C2×A4—126no—
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
144C24#109411yescharacteristic, 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—481no—
169C22×C3A4212yes—
170C22×C3A4211yescharacteristic
171A4—123no—
172A4—122no—
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—
239C22#8888012yescomplemented
240C22#8888111yescharacteristic, complemented
241C3—481no—
242C3#1059678211yescharacteristic
243C2—32no—
244C2—31no—
245C2—91no—
246C2—92no—
247C2—93no—
subgroupG/H
center, Frattini, hypercenterC3#10596782
commutator, Fitting, socle, cosocleC26×C3C32
prime radicalC22×C3×C22⋊A4C3
hypocenterC263−1+2
LCSG ▷ C26×C3 ▷ 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
18. C22⋊(C22⋊C9)A4semidirect
35. C263−1+2semidirect
143. C24C9⋊A4semidirect
144. C24#1094semidirect
239. C22#88880semidirect
240. C22#88881semidirect

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
#888814
#888804