Finite Group Databasebeta

← #13166226

Subgroups — #13166226

Order 1728.

362 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 43 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×C22⋊#157C211yesmaximal, characteristic
2#11724243C212yesmaximal, complemented
3#10593272—31nomaximal
4C23×C3×C3⋊D3C411yescharacteristic
5C22×C3×#157C411yescharacteristic
6C22×C3×#157C2211yescharacteristic
7C2×C3⋊AGL(1,𝔽9)—22no—
8C3×C22⋊#157—22no—
9C2×C22⋊#157D311yescharacteristic
10C22⋊AGL(1,𝔽9)—32no—
11C23×C33C811yescharacteristic, complemented
12C22×C3×C3⋊D3—21no—
13C22×C3×C3⋊D3—21no—
14C22×C3×C3⋊D3—41no—
15C22×C3×C3⋊D3—41no—
16C22×C3×C3⋊D3C811yescharacteristic, complemented
17C22×C3×C3⋊D3C2×C411yescharacteristic
18C2×C3×#157—21no—
19C2×C3×#157—41no—
20C2×C3×#157—41no—
21C2×C3×#157D412yes—
22C3⋊AGL(1,𝔽9)—42no—
23#4525—91nomaximal
24C23×C3⋊D3Dic311yescharacteristic
25C23×C3×D3—121no—
26C22×#157Dic311yescharacteristic
27C22×#157C2×D311yescharacteristic
28C2×AGL(1,𝔽9)—62no—
29C22⋊#157—22no—
30C22×C33—21no—
31C22×C33—41no—
32C22×C33C2×C811yescharacteristic
33C2×C3×C3⋊D3—21no—
34C2×C3×C3⋊D3—21no—
35C2×C3×C3⋊D3—21no—
36C2×C3×C3⋊D3—42no—
37C2×C3×C3⋊D3—41no—
38C2×C3×C3⋊D3—41no—
39C2×C3×C3⋊D3—41no—
40C2×C3×C3⋊D3C22⋊C411yescharacteristic
41C2×C3×C3⋊D3M1611yescharacteristic
42C3×#157—22no—
43C3×#157—41no—
44C2×C3×C22⋊C4—91no—
45#830—92no—
46C23×C32—41no—
47C23×C32—81no—
48C23×C32C3⋊C811yescharacteristic, complemented
49C22×C3⋊D3—21no—
50C22×C3⋊D3—21no—
51C22×C3⋊D3—41no—
52C22×C3⋊D3—41no—
53C22×C3⋊D3C3⋊C811yescharacteristic, complemented
54C22×C3⋊D3C2×Dic311yescharacteristic
55C22×C3×D3—121no—
56C22×C3×D3—121no—
57C22×C3×D3—122no—
58C22×C3×D3—122no—
59C22×C3×D3—124no—
60C22×C3×D3—124no—
61C2×#157—21no—
62C2×#157—41no—
63C2×#157—41no—
64C2×#157#6712yes—
65AGL(1,𝔽9)—122no—
66#323—271noSylow
67C2×C33—21no—
68C2×C33—41no—
69C2×C33C22⋊C811yescharacteristic
70C3×C3⋊D3—21no—
71C3×C3⋊D3—41no—
72C3×C3⋊D3#9911yescharacteristic
73C3×C3⋊D3#10011yescharacteristic
74C24×C3—91no—
75C23×D3—121no—
76C22×C3×C4—91no—
77C22×C3×C4—91no—
78C3×C22⋊C4—182no—
79C2×C3⋊C8—182no—
80C22×C32—21no—
81C22×C32—42no—
82C22×C32—41no—
83C22×C32—44no—
84C22×C32—41no—
85C22×C32—84no—
86C22×C32—82no—
87C22×C32—81no—
88C22×C32C2×C3⋊C811yescharacteristic
89C2×C3⋊D3—21no—
90C2×C3⋊D3—21no—
91C2×C3⋊D3—21no—
92C2×C3⋊D3—42no—
93C2×C3⋊D3—41no—
94C2×C3⋊D3—41no—
95C2×C3⋊D3—41no—
96C2×C3⋊D3#20811yescharacteristic
97C2×C3⋊D3#21711yescharacteristic
98C2×C3×D3—121no—
99C2×C3×D3—121no—
100C2×C3×D3—122no—
101C2×C3×D3—122no—
102C2×C3×D3—122no—
103C2×C3×D3—124no—
104C2×C3×D3—124no—
105C2×C3×D3—124no—
106C2×C3×D3—128no—
107#157—22no—
108#157—41no—
109C2×C22⋊C4—91no—
110C22⋊C8—272no—
111C33#32311yescharacteristic, complemented, Sylow
112C23×C3—41no—
113C23×C3—81no—
114C23×C3—91no—
115C23×C3—91no—
116C23×C3—181no—
117C23×C3—181no—
118C23×C3—361no—
119C23×C3—361no—
120C23×C3AGL(1,𝔽9)11yescharacteristic, complemented
121C22×D3—121no—
122C22×D3—121no—
123C22×D3—122no—
124C22×D3—122no—
125C22×D3—124no—
126C22×D3—124no—
127C2×C3×C4—92no—
128C2×C3×C4—181no—
129C2×C3×C4—361no—
130C2×C3×C4—361no—
131C3⋊C8—362no—
132C2×C32—21no—
133C2×C32—42no—
134C2×C32—41no—
135C2×C32—44no—
136C2×C32—41no—
137C2×C32—84no—
138C2×C32—82no—
139C2×C32—81no—
140C2×C32#83011yescharacteristic
141C3⋊D3—21no—
142C3⋊D3—41no—
143C3⋊D3#83311yescharacteristic
144C3⋊D3#83411yescharacteristic
145C3×D3—121no—
146C3×D3—121no—
147C3×D3—122no—
148C3×D3—124no—
149C24—91no—
150C22×C4—91no—
151C22×C4—91no—
152C2×C8—542no—
153C22⋊C4—182no—
154C22×C3—21no—
155C22×C3—42no—
156C22×C3—41no—
157C22×C3—44no—
158C22×C3—41no—
159C22×C3—82no—
160C22×C3—84no—
161C22×C3—81no—
162C22×C3—91no—
163C22×C3—91no—
164C22×C3—181no—
165C22×C3—181no—
166C22×C3—181no—
167C22×C3—361no—
168C22×C3—361no—
169C22×C3—361no—
170C22×C3—362no—
171C22×C3C2×AGL(1,𝔽9)11yescharacteristic
172C2×D3—121no—
173C2×D3—121no—
174C2×D3—122no—
175C2×D3—122no—
176C2×D3—122no—
177C2×D3—124no—
178C2×D3—124no—
179C2×D3—124no—
180C2×D3—128no—
181C3×C4—182no—
182C3×C4—361no—
183C32—41no—
184C32—81no—
185C32#452511yescharacteristic, complemented
186C23—91no—
187C23—91no—
188C23—181no—
189C23—181no—
190C23—361no—
191C23—361no—
192C23C3⋊AGL(1,𝔽9)11yescharacteristic, complemented
193C2×C4—92no—
194C2×C4—181no—
195C2×C4—361no—
196C2×C4—361no—
197C8—1082no—
198C2×C3—21no—
199C2×C3—42no—
200C2×C3—44no—
201C2×C3—41no—
202C2×C3—41no—
203C2×C3—82no—
204C2×C3—84no—
205C2×C3—81no—
206C2×C3—91no—
207C2×C3—91no—
208C2×C3—181no—
209C2×C3—361no—
210C2×C3C22⋊AGL(1,𝔽9)11yescharacteristic
211D3—121no—
212D3—121no—
213D3—122no—
214D3—124no—
215C22—21no—
216C22—41no—
217C22—91no—
218C22—91no—
219C22—181no—
220C22—181no—
221C22—181no—
222C22—361no—
223C22—361no—
224C22—361no—
225C22—362no—
226C22C2×C3⋊AGL(1,𝔽9)11yescharacteristic
227C4—182no—
228C4—361no—
229C3—41no—
230C3—81no—
231C3#1059327211yescharacteristic, complemented
232C2—21no—
233C2—41no—
234C2—91no—
235C2—91no—
236C2—181no—
237C2—361no—
238C2#1172424311yescharacteristic
subgroupG/H
centerC2#11724243
commutatorC22×C33C2×C8
FrattiniC22C2×C3⋊AGL(1,𝔽9)
FittingC23×C33C8
socleC2×C33C22⋊C8
prime radicalC23×C3×C3⋊D3C4
cosocleC22×C3×#157C22
hypercenterC23C3⋊AGL(1,𝔽9)
hypocenterC33#323
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11724243C2semidirect
11. C23×C33C8semidirect
16. C22×C3×C3⋊D3C8semidirect
48. C23×C32C3⋊C8semidirect
53. C22×C3⋊D3C3⋊C8semidirect
111. C33#323semidirect
120. C23×C3AGL(1,𝔽9)semidirect
185. C32#4525semidirect
192. C23C3⋊AGL(1,𝔽9)semidirect
231. C3#10593272semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊#1572
#117242432
#105932723
#45259