Finite Group Databasebeta

← #13135187

Subgroups — #13135187

Order 1728.

334 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C3×#63499C211yesmaximal, characteristic
2#11720577C211yesmaximal, characteristic, complemented
3C9×#833C211yesmaximal, characteristic, complemented
4#10589173—31nomaximal
5#10588153—31nomaximal
6C23×C3×D9C411yescharacteristic, complemented
7C2×C3×D4×C9C2211yescharacteristic
8C2×C3×#611—21no—
9#88551—21no—
10C3×#3676C411yescharacteristic, complemented
11C3×(C2×D9)⋊C4—21no—
12C9×#208—21no—
13C3×#937—31no—
14#63661—31no—
15C3×#833D311yescharacteristic
16#63499D311yescharacteristic
17C9×#100—31no—
18#63393—31no—
19#63393—61no—
20C23×C3×C9C2×C411yescharacteristic
21C23×C3×C9D411yescharacteristic
22C22×C3×D9—21no—
23C22×C3×D9—42no—
24C22×C3×D9—41no—
25C3×D4×C9—41no—
26C2×C3×C4×C9D411yescharacteristic
27C2×C3⋊Dic9—41no—
28C3×#611—42no—
29C2×C3×Dic9—21no—
30C9×C3⋊C8—41no—
31#4536—91no—
32#4475—91no—
33C23×C3×D3—31no—
34C2×C32×D4C2×D311yescharacteristic
35C2×C3×#67—61no—
36C23×D9Dic311yescharacteristic, complemented
37#3757—61no—
38C3×#217—31no—
39C3×(C2×D3)⋊C4—61no—
40C3×#208—21no—
41C2×D4×C9—21no—
42C2×D4×C9C2×D311yescharacteristic
43C2×#611—21no—
44C9×M16—61no—
45#3676—62no—
46#3676—61no—
47#3676Dic311yescharacteristic, complemented
48(C2×D9)⋊C4—21no—
49C22×C3×C9—21no—
50C22×C3×C9—21no—
51C22×C3×C9—41no—
52C22×C3×C9—41no—
53C22×C3×C9C22⋊C411yescharacteristic
54C2×C3×D9—22no—
55C2×C3×D9—41no—
56C2×C3×D9—42no—
57C2×C3×D9—41no—
58C2×C3×D9—42no—
59C3×C4×C9—21no—
60C3⋊Dic9—81no—
61C3×Dic9—41no—
62C3×#120—91no—
63#937—31no—
64C3×#100—31no—
65#834—91no—
66#834—91no—
67#834—181no—
68#833D911yescharacteristic, complemented
69C23×C32C4×D311yescharacteristic
70C23×C32D1211yescharacteristic
71C22×C3×D3—61no—
72C22×C3×D3—122no—
73C22×C3×D3—121no—
74C32×D4—41no—
75C2×C32×C4#6711yescharacteristic
76C2×C3⋊Dic3—121no—
77C3×#67—122no—
78C2×C3×Dic3—61no—
79C23×C9—21no—
80C23×C9—21no—
81C23×C9C2×Dic311yescharacteristic
82C23×C9#6711yescharacteristic
83C22×D9—21no—
84C22×D9—42no—
85C22×D9—41no—
86C3×C3⋊C8—41no—
87D4×C9—41no—
88D4×C9—42no—
89C2×C4×C9—21no—
90C2×C4×C9#6711yescharacteristic
91#611—42no—
92C2×Dic9—21no—
93C2×Dic9—122no—
94C2×Dic9—121no—
95C8×C9—121no—
96AΣL(1,𝔽16)—271noSylow
97C2×C3×C9—21no—
98C2×C3×C9—41no—
99C2×C3×C9—41no—
100C2×C3×C9#9911yescharacteristic
101C3×D9—42no—
102C24×C3—91no—
103C23×D3—31no—
104C2×C3×D4—21no—
105C2×C3×D4—181no—
106C2×C3×D4C2×D911yescharacteristic
107C2×C3×D4D3211yescharacteristic
108C2×#67—61no—
109C3×M16—61no—
110C3×C22⋊C4—91no—
111C3×C22⋊C4—181no—
112#217—31no—
113#217—182no—
114#217—181no—
115#217—181no—
116(C2×D3)⋊C4—61no—
117#208—21no—
118C22×C32—21no—
119C22×C32—21no—
120C22×C32—41no—
121C22×C32—41no—
122C22×C32(C2×D3)⋊C411yescharacteristic
123C2×C3×D3—62no—
124C2×C3×D3—121no—
125C2×C3×D3—122no—
126C2×C3×D3—121no—
127C2×C3×D3—122no—
128C32×C4—21no—
129C3⋊Dic3—241no—
130C3×Dic3—121no—
131C22×C9—22no—
132C22×C9—21no—
133C22×C9—21no—
134C22×C9—21no—
135C22×C9—41no—
136C22×C9—41no—
137C22×C9—41no—
138C22×C9—81no—
139C22×C9—81no—
140C22×C9#21711yescharacteristic
141C2×D9—22no—
142C2×D9—41no—
143C2×D9—42no—
144C2×D9—41no—
145C2×D9—42no—
146C4×C9—21no—
147C4×C9—41no—
148Dic9—41no—
149Dic9—242no—
150Dic9—241no—
151#120—91no—
152#100—31no—
153#99—271no—
154C3×C9AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—181no—
158C23×C3—361no—
159C23×C3—362no—
160C23×C3C4×D911yescharacteristic
161C23×C3D3611yescharacteristic
162C23×C3D3×Dic311yescharacteristic
163C23×C3#62611yescharacteristic
164C22×D3—61no—
165C22×D3—122no—
166C22×D3—121no—
167C3×D4—41no—
168C3×D4—41no—
169C3×D4—42no—
170C3×D4—362no—
171C2×C3×C4—21no—
172C2×C3×C4—181no—
173C2×C3×C4#61111yescharacteristic
174C2×C3×C4#62511yescharacteristic
175#67—122no—
176C2×Dic3—61no—
177C2×Dic3—361no—
178C2×Dic3—362no—
179C2×Dic3—361no—
180C3×C8—121no—
181C3⋊C8—41no—
182C2×C32—21no—
183C2×C32—41no—
184C2×C32—41no—
185C2×C32#80611yescharacteristic
186C3×D3—122no—
187C2×C9—21no—
188C2×C9—21no—
189C2×C9—41no—
190C2×C9—42no—
191C2×C9—41no—
192C2×C9—41no—
193C2×C9—81no—
194C2×C9#83411yescharacteristic
195D9—42no—
196C24—91no—
197C2×D4—181no—
198C2×D4D3×D911yescharacteristic
199M16—61no—
200C22⋊C4—91no—
201C22⋊C4—181no—
202C22⋊C4—541no—
203C22×C3—22no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—81no—
215C22×C3—81no—
216C22×C3—182no—
217C22×C3—361no—
218C22×C3—362no—
219C22×C3—362no—
220C22×C3—361no—
221C22×C3(C2×D9)⋊C411yescharacteristic
222C22×C3(C2×D3)⋊Dic311yescharacteristic
223C2×D3—62no—
224C2×D3—121no—
225C2×D3—122no—
226C2×D3—121no—
227C2×D3—122no—
228C3×C4—21no—
229C3×C4—21no—
230C3×C4—41no—
231C3×C4—361no—
232Dic3—121no—
233Dic3—721no—
234Dic3—722no—
235Dic3—721no—
236C32#447511yescharacteristic
237C9—21no—
238C9#453611yescharacteristic, complemented
239C23—181no—
240C23—361no—
241C23—362no—
242C23Dic3×D911yescharacteristic
243C23#619211yescharacteristic
244D4—41no—
245D4—362no—
246C2×C4—181no—
247C2×C4—1081no—
248C2×C4#619411yescharacteristic
249C8—121no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—41no—
254C2×C3—42no—
255C2×C3—41no—
256C2×C3—41no—
257C2×C3—41no—
258C2×C3—41no—
259C2×C3—81no—
260C2×C3—362no—
261C2×C3#6336511yescharacteristic
262C2×C3#6358011yescharacteristic
263D3—122no—
264C22—21no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—182no—
269C22—361no—
270C22—362no—
271C22—362no—
272C22—361no—
273C22(C2×D9)⋊Dic311yescharacteristic
274C4—21no—
275C4—361no—
276C4—2161no—
277C3—21no—
278C3#1058815311yescharacteristic, complemented
279C3#1058917311yescharacteristic
280C2—21no—
281C2—41no—
282C2—41no—
283C2—362no—
284C2#1172032911yescharacteristic
subgroupG/H
centerC2#11720329
commutatorC23×C3×C9C2×C4
FrattiniC2×C3×D4D32
Fitting, cosocleC2×C3×D4×C9C22
socleC2×C32#806
prime radicalC3×#63499C2
hypercenterC2×D4D3×D9
hypocenterC3×C9AΣL(1,𝔽16)
LCSG ▷ C23×C3×C9 ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11720577C2semidirect
3. C9×#833C2semidirect
6. C23×C3×D9C4semidirect
10. C3×#3676C4semidirect
36. C23×D9Dic3semidirect
47. #3676Dic3semidirect
68. #833D9semidirect
154. C3×C9AΣL(1,𝔽16)semidirect
238. C9#4536semidirect
278. C3#10588153semidirect

Maximal subgroups

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

subgroupindex
C3×#634992
#117205772
C9×#8332
#105891733
#105881533