Finite Group Databasebeta

← #13180940

Subgroups — #13180940

Order 1728.

306 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 91 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×#157C211yesmaximal, characteristic
2#11724243C212yesmaximal
3C4⋊(C3⋊AGL(1,𝔽9))C212yesmaximal, complemented
4C4×C3⋊AGL(1,𝔽9)C212yesmaximal, complemented
5#10596484—31nomaximal
6C22×C3×#157C411yescharacteristic
7C22×C3×#157C2211yescharacteristic
8C2×C3⋊AGL(1,𝔽9)C2214yes—
9C2×C3×C4×C3⋊D3C411yescharacteristic
10C3×C4×#157C411yescharacteristic
11C3×C4×#157C411yescharacteristic
12C3×C4×#157C2211yescharacteristic
13C3×C4×#157C2211yescharacteristic
14C2×C4×#157D311yescharacteristic
15C22⋊AGL(1,𝔽9)—32no—
16C4⋊AGL(1,𝔽9)—32no—
17C4×AGL(1,𝔽9)—32no—
18C22×C3×C3⋊D3C2×C411yescharacteristic
19C2×C3×#157—22no—
20C2×C3×#157—22no—
21C2×C3×#157C2×C411yescharacteristic
22C2×C3×#157C2×C411yescharacteristic
23C2×C3×#157C2×C411yescharacteristic
24C2×C3×#157C2311yescharacteristic
25C3⋊AGL(1,𝔽9)—24no—
26C2×C33×C4C811yescharacteristic, complemented
27C2×C3×C3⋊Dic3C811yescharacteristic, complemented
28C3×C4×C3⋊D3C811yescharacteristic, complemented
29C3×C4×C3⋊D3C811yescharacteristic, complemented
30C3×C4×C3⋊D3C2×C411yescharacteristic
31C3×C4×C3⋊D3C2×C411yescharacteristic
32#4925—91nomaximal
33C22×#157Dic311yescharacteristic
34C22×#157C2×D311yescharacteristic
35C2×AGL(1,𝔽9)—34no—
36C2×C4×C3⋊D3Dic311yescharacteristic
37C2×C3×C4×D3—121no—
38C4×#157Dic311yescharacteristic
39C4×#157Dic311yescharacteristic
40C4×#157C2×D311yescharacteristic
41C4×#157C2×D311yescharacteristic
42C22×C33C2×C811yescharacteristic
43C2×C3×C3⋊D3—21no—
44C2×C3×C3⋊D3—21no—
45C2×C3×C3⋊D3C2×C811yescharacteristic
46C2×C3×C3⋊D3C22×C411yescharacteristic
47C3×#157—21no—
48C3×#157—21no—
49C3×#157M1612yes—
50C3×#157#4112yes—
51C33×C4C2×C811yescharacteristic
52C33×C4C2×C811yescharacteristic
53C3×C3⋊Dic3C2×C811yescharacteristic
54C3×C3⋊Dic3C2×C811yescharacteristic
55C2×C3×C42—91no—
56#830—92no—
57C4⋊(C3⋊C8)—92no—
58C4×C3⋊C8—92no—
59C22×C3⋊D3C2×Dic311yescharacteristic
60C22×C3×D3—121no—
61C2×#157—22no—
62C2×#157—22no—
63C2×#157C2×Dic311yescharacteristic
64C2×#157C2×Dic311yescharacteristic
65C2×#157C2×Dic311yescharacteristic
66C2×#157C22×D311yescharacteristic
67AGL(1,𝔽9)—64no—
68C2×C32×C4—41no—
69C2×C32×C4—81no—
70C2×C32×C4C3⋊C811yescharacteristic, complemented
71C2×C3⋊Dic3C3⋊C811yescharacteristic, complemented
72C4×C3⋊D3C3⋊C811yescharacteristic, complemented
73C4×C3⋊D3C3⋊C811yescharacteristic, complemented
74C4×C3⋊D3C2×Dic311yescharacteristic
75C4×C3⋊D3C2×Dic311yescharacteristic
76C2×C3×Dic3—121no—
77C3×C4×D3—121no—
78C3×C4×D3—121no—
79C3×C4×D3—121no—
80C3×C4×D3—121no—
81#431—271noSylow
82C2×C33—21no—
83C2×C33C22×C811yescharacteristic
84C3×C3⋊D3—21no—
85C3×C3⋊D3#11711yescharacteristic
86C3×C3⋊D3C2×M1611yescharacteristic
87C22×C3×C4—91no—
88C22×C3×C4—91no—
89C22×C3×C4—91no—
90C2×C4×D3—121no—
91C3×C42—91no—
92C3×C42—91no—
93C3×C42—91no—
94C3×C42—91no—
95C2×C3⋊C8—94no—
96C22×C32—41no—
97C22×C32—81no—
98C22×C32C2×C3⋊C811yescharacteristic
99C2×C3⋊D3—21no—
100C2×C3⋊D3—21no—
101C2×C3⋊D3C2×C3⋊C811yescharacteristic
102C2×C3⋊D3C22×Dic311yescharacteristic
103C2×C3×D3—121no—
104C2×C3×D3—121no—
105C2×C3×D3—122no—
106C2×C3×D3—122no—
107#157—21no—
108#157—21no—
109#157#20812yes—
110#157#23512yes—
111C32×C4—41no—
112C32×C4—41no—
113C32×C4—81no—
114C32×C4—81no—
115C32×C4C2×C3⋊C811yescharacteristic
116C32×C4C2×C3⋊C811yescharacteristic
117C3⋊Dic3C2×C3⋊C811yescharacteristic
118C3⋊Dic3C2×C3⋊C811yescharacteristic
119C3×Dic3—121no—
120C3×Dic3—121no—
121C2×C42—91no—
122C4⋊C8—272no—
123C22⋊C8—272no—
124C4×C8—272no—
125C33#43111yescharacteristic, complemented, Sylow
126C23×C3—91no—
127C22×D3—121no—
128C2×C3×C4—41no—
129C2×C3×C4—81no—
130C2×C3×C4—91no—
131C2×C3×C4—91no—
132C2×C3×C4—91no—
133C2×C3×C4—91no—
134C2×C3×C4—91no—
135C2×C3×C4—91no—
136C2×C3×C4—91no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4—182no—
140C2×C3×C4—182no—
141C2×C3×C4AGL(1,𝔽9)11yescharacteristic, complemented
142C2×Dic3—121no—
143C4×D3—121no—
144C4×D3—121no—
145C4×D3—121no—
146C4×D3—121no—
147C3⋊C8—184no—
148C2×C32—21no—
149C2×C32—42no—
150C2×C32—41no—
151C2×C32—82no—
152C2×C32—81no—
153C2×C32C22×C3⋊C811yescharacteristic
154C3⋊D3—21no—
155C3⋊D3C2×#20811yescharacteristic
156C3⋊D3#92611yescharacteristic
157C3×D3—121no—
158C3×D3—121no—
159C3×D3—122no—
160C22×C4—91no—
161C22×C4—91no—
162C22×C4—91no—
163C2×C8—274no—
164C42—91no—
165C42—91no—
166C42—91no—
167C42—91no—
168C22×C3—41no—
169C22×C3—81no—
170C22×C3—91no—
171C22×C3—91no—
172C22×C3—181no—
173C22×C3—181no—
174C22×C3C2×AGL(1,𝔽9)11yescharacteristic
175C2×D3—121no—
176C2×D3—121no—
177C2×D3—122no—
178C2×D3—122no—
179C3×C4—41no—
180C3×C4—41no—
181C3×C4—81no—
182C3×C4—81no—
183C3×C4—92no—
184C3×C4—92no—
185C3×C4—91no—
186C3×C4—91no—
187C3×C4—181no—
188C3×C4—181no—
189C3×C4C2×AGL(1,𝔽9)11yescharacteristic
190C3×C4C2×AGL(1,𝔽9)11yescharacteristic
191Dic3—121no—
192Dic3—121no—
193C32—41no—
194C32—81no—
195C32#492511yescharacteristic, complemented
196C23—91no—
197C2×C4—91no—
198C2×C4—91no—
199C2×C4—91no—
200C2×C4—91no—
201C2×C4—91no—
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C4—91no—
206C2×C4—182no—
207C2×C4—182no—
208C2×C4C3⋊AGL(1,𝔽9)11yescharacteristic, complemented
209C8—544no—
210C2×C3—21no—
211C2×C3—42no—
212C2×C3—41no—
213C2×C3—82no—
214C2×C3—81no—
215C2×C3—91no—
216C2×C3—91no—
217C2×C3—181no—
218C2×C3C22×AGL(1,𝔽9)11yescharacteristic
219D3—121no—
220D3—121no—
221D3—122no—
222C22—91no—
223C22—91no—
224C22—181no—
225C22—181no—
226C22C2×C3⋊AGL(1,𝔽9)11yescharacteristic
227C4—92no—
228C4—92no—
229C4—91no—
230C4—91no—
231C4—181no—
232C4—181no—
233C4C2×C3⋊AGL(1,𝔽9)11yescharacteristic
234C4C2×C3⋊AGL(1,𝔽9)11yescharacteristic
235C3—41no—
236C3—81no—
237C3#1059648411yescharacteristic, complemented
238C2—21no—
239C2—91no—
240C2—91no—
241C2—181no—
242C2C22×C3⋊AGL(1,𝔽9)11yescharacteristic
subgroupG/H
centerC4C2×C3⋊AGL(1,𝔽9)
commutator, socleC2×C33C22×C8
FrattiniC2C22×C3⋊AGL(1,𝔽9)
FittingC2×C33×C4C8
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×#157C23
hypercenterC2×C4C3⋊AGL(1,𝔽9)
hypocenterC33#431
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. C4⋊(C3⋊AGL(1,𝔽9))C2semidirect
4. C4×C3⋊AGL(1,𝔽9)C2semidirect
26. C2×C33×C4C8semidirect
27. C2×C3×C3⋊Dic3C8semidirect
28. C3×C4×C3⋊D3C8semidirect
29. C3×C4×C3⋊D3C8semidirect
70. C2×C32×C4C3⋊C8semidirect
71. C2×C3⋊Dic3C3⋊C8semidirect
72. C4×C3⋊D3C3⋊C8semidirect
73. C4×C3⋊D3C3⋊C8semidirect
125. C33#431semidirect
141. C2×C3×C4AGL(1,𝔽9)semidirect
195. C32#4925semidirect
208. C2×C4C3⋊AGL(1,𝔽9)semidirect
237. C3#10596484semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×#1572
#117242432
C4⋊(C3⋊AGL(1,𝔽9))2
C4×C3⋊AGL(1,𝔽9)2
#105964843
#49259