Finite Group Databasebeta

← #10588813⋊C3 #13143237

Subgroups — #10588813⋊C3 #13143237

Order 1728.

604 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 148 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×C4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic
2C2×M16×3−1+2C211yesmaximal, characteristic, complemented
3#63488⋊C3C211yesmaximal, characteristic, complemented
4#63380⋊C3C214yesmaximal, complemented
5C3×#5131—31nomaximal
6#10588813C311yesmaximal, characteristic, complemented
7C23×AGL(1,ℤ/9)—21no—
8C22×C4×3−1+2C2211yescharacteristic
9C2×C4⋊AGL(1,ℤ/9)—21no—
10C2×C4⋊AGL(1,ℤ/9)—22no—
11C2×C4⋊AGL(1,ℤ/9)C414yescomplemented
12C2×C4⋊AGL(1,ℤ/9)C2212yes—
13M16×3−1+2—22no—
14C2×C8×3−1+2C2212yescomplemented
15#3676⋊C3—21no—
16C4⋊(3−1+2⋊C4)C2212yescomplemented
17C4×3−1+2⋊C4—21no—
18C22×C3×D12—31no—
19C2×C32×M16D311yescharacteristic
20C3×#926—31no—
21C22×D36C2×C311yescharacteristic
22C3×#821—34no—
23C2×C9×M16—21no—
24C2×C9×M16C2×C311yescharacteristic, complemented
25#63488C2×C311yescharacteristic, complemented
26#63380C2×C314yescomplemented
27C23×3−1+2D411yescharacteristic
28C22×AGL(1,ℤ/9)—24no—
29C22×AGL(1,ℤ/9)—22no—
30C22×AGL(1,ℤ/9)—44no—
31C2×C4×3−1+2C2×C412yes—
32C2×C4×3−1+2D411yescharacteristic
33C2×C4×3−1+2D412yes—
34C2×C4×3−1+2C2311yescharacteristic
35C2×3−1+2⋊C4—22no—
36C4⋊AGL(1,ℤ/9)—24no—
37C4⋊AGL(1,ℤ/9)—22no—
38C4⋊AGL(1,ℤ/9)C2×C414yes—
39C8×3−1+2—22no—
40C3×#418—91no—
41#5131—31no—
42C23×C3×D3—61no—
43C22×C32×C4C2×D311yescharacteristic
44C2×C3×D12—34no—
45C2×C3×D12—32no—
46C2×C3×D12—61no—
47C2×C3×D12—62no—
48C23×D9—21no—
49C32×M16—22no—
50C2×C32×C8C2×D312yes—
51C3×#217—61no—
52C3×C4⋊Dic3—32no—
53C3×C4×Dic3—61no—
54C22×C4×C9—21no—
55C22×C4×C9C22×C311yescharacteristic
56C2×D36—21no—
57C2×D36—22no—
58C2×D36C3×C414yescomplemented
59C2×D36C22×C312yes—
60C9×M16—24no—
61C9×M16—22no—
62C2×C8×C9—22no—
63C2×C8×C9C22×C312yescomplemented
64#3676—21no—
65C4⋊Dic9C22×C312yescomplemented
66C4×Dic9—21no—
67C22×3−1+2—22no—
68C22×3−1+2C22⋊C412yes—
69C22×3−1+2C2×D411yescharacteristic
70C2×AGL(1,ℤ/9)—24no—
71C2×AGL(1,ℤ/9)—48no—
72C2×AGL(1,ℤ/9)—44no—
73C4×3−1+2C22⋊C412yes—
74C4×3−1+2C22×C411yescharacteristic
75C4×3−1+2C2×D411yescharacteristic
763−1+2⋊C4—42no—
77C22×C3×D4—91no—
78C22×D12—31no—
79C2×C3×M16—31no—
80C2×C3×M16C3×D311yescharacteristic
81C3×#117—91no—
82#926—31no—
83C3×#102—94no—
84#821—34no—
85C23×C32D1211yescharacteristic
86C22×C3×D3—64no—
87C22×C3×D3—62no—
88C22×C3×D3—124no—
89C2×C32×C4C4×D312yes—
90C2×C32×C4D1211yescharacteristic
91C2×C32×C4#6712yes—
92C2×C32×C4C22×D311yescharacteristic
93C2×C3×Dic3—62no—
94C3×D12—34no—
95C3×D12—64no—
96C3×D12—62no—
97C23×C9—21no—
98C23×C9C3×D411yescharacteristic
99C22×D9—24no—
100C22×D9—22no—
101C22×D9—44no—
102C32×C8—22no—
103C2×C4×C9—22no—
104C2×C4×C9—21no—
105C2×C4×C9—22no—
106C2×C4×C9—21no—
107C2×C4×C9C2×C3×C412yes—
108C2×C4×C9C3×D411yescharacteristic
109C2×C4×C9C3×D412yes—
110C2×C4×C9C23×C311yescharacteristic
111C2×Dic9—22no—
112D36—24no—
113D36—22no—
114D36C2×C3×C414yes—
115C8×C9—24no—
116C8×C9—22no—
117#418—91noSylow
118C2×3−1+2—22no—
119C2×3−1+2C2×C22⋊C411yescharacteristic
120C2×3−1+2AGL(1,ℤ/8)12yes—
121AGL(1,ℤ/9)—44no—
122C24×C3—181no—
123C23×D3—61no—
124C2×C3×D4—92no—
125C2×C3×D4—94no—
126C2×C3×D4—181no—
127C2×C3×D4—182no—
128C22×C3×C4—31no—
129C22×C3×C4C2×C3×D311yescharacteristic
130C2×D12—34no—
131C2×D12—32no—
132C2×D12—61no—
133C2×D12—62no—
134C3×M16—22no—
135C3×M16—62no—
136C2×C3×C8—32no—
137C2×C3×C8C2×C3×D312yes—
138C3×C4⋊C4—92no—
139C3×C22⋊C4—181no—
140C3×C42—181no—
141#217—61no—
142C4⋊Dic3—32no—
143C4×Dic3—61no—
144C22×C32—22no—
145C22×C32(C2×D3)⋊C412yes—
146C22×C32C2×D1211yescharacteristic
147C2×C3×D3—64no—
148C2×C3×D3—128no—
149C2×C3×D3—124no—
150C32×C4(C2×D3)⋊C412yes—
151C32×C4C2×C4×D311yescharacteristic
152C32×C4C2×#6711yescharacteristic
153C3×Dic3—122no—
154C22×C9—22no—
155C22×C9—22no—
156C22×C9—21no—
157C22×C9—42no—
158C22×C9C3×C22⋊C412yes—
159C22×C9C2×C3×D411yescharacteristic
160C2×D9—24no—
161C2×D9—48no—
162C2×D9—44no—
163C4×C9—22no—
164C4×C9—21no—
165C4×C9—21no—
166C4×C9C3×C22⋊C412yes—
167C4×C9C22×C3×C411yescharacteristic
168C4×C9C2×C3×D411yescharacteristic
169Dic9—42no—
170C22×D4—91no—
171C2×M16AGL(1,ℤ/9)11yescharacteristic, complemented
172#117—91no—
173#102—94no—
1743−1+2#41811yescharacteristic, complemented, Sylow
175C23×C3—31no—
176C23×C3—182no—
177C23×C3—184no—
178C23×C3—364no—
179C23×C3C3×D1211yescharacteristic
180C22×D3—64no—
181C22×D3—62no—
182C22×D3—124no—
183C3×D4—94no—
184C3×D4—182no—
185C3×D4—184no—
186C2×C3×C4—32no—
187C2×C3×C4—31no—
188C2×C3×C4—32no—
189C2×C3×C4—31no—
190C2×C3×C4—182no—
191C2×C3×C4C3×C4×D312yes—
192C2×C3×C4C3×D1211yescharacteristic
193C2×C3×C4C3×#6712yes—
194C2×C3×C4C22×C3×D311yescharacteristic
195C2×Dic3—62no—
196D12—34no—
197D12—64no—
198D12—62no—
199C3×C8—22no—
200C3×C8—62no—
201C2×C32—22no—
202C2×C32#90812yes—
203C2×C32C2×(C2×D3)⋊C411yescharacteristic
204C3×D3—124no—
205C2×C9—22no—
206C2×C9—22no—
207C2×C9—21no—
208C2×C9—42no—
209C2×C9C2×C3×C22⋊C411yescharacteristic
210C2×C9C3×AGL(1,ℤ/8)12yes—
211D9—44no—
212C24—181no—
213C2×D4—94no—
214C2×D4—92no—
215C2×D4—181no—
216C2×D4—182no—
217C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
218M16—22no—
219C2×C8C2×AGL(1,ℤ/9)12yescomplemented
220C4⋊C4—92no—
221C22⋊C4—181no—
222C42—181no—
223C22×C3—22no—
224C22×C3—31no—
225C22×C3—32no—
226C22×C3—62no—
227C22×C3—184no—
228C22×C3—364no—
229C22×C3—368no—
230C22×C3C3×(C2×D3)⋊C412yes—
231C22×C3C2×C3×D1211yescharacteristic
232C2×D3—64no—
233C2×D3—128no—
234C2×D3—124no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—32no—
238C3×C4—362no—
239C3×C4C3×(C2×D3)⋊C412yes—
240C3×C4C2×C3×C4×D311yescharacteristic
241C3×C4C2×C3×#6711yescharacteristic
242Dic3—122no—
243C32#513111yescharacteristic
244C9—21no—
245C9C3×#41811yescharacteristic, complemented
246C23—182no—
247C23—184no—
248C23—364no—
249C23C4⋊AGL(1,ℤ/9)11yescharacteristic
250D4—94no—
251D4—182no—
252D4—184no—
253C2×C4—182no—
254C2×C4C4×AGL(1,ℤ/9)12yes—
255C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
256C2×C4#611⋊C312yes—
257C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
258C8—22no—
259C2×C3—22no—
260C2×C3—32no—
261C2×C3—31no—
262C2×C3—62no—
263C2×C3—364no—
264C2×C3C3×#90812yes—
265C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
266D3—124no—
267C22—22no—
268C22—184no—
269C22—368no—
270C22—364no—
271C22(C2×AGL(1,ℤ/9))⋊C412yes—
272C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
273C4—362no—
274C4(C2×AGL(1,ℤ/9))⋊C412yes—
275C4C2×C4×AGL(1,ℤ/9)11yescharacteristic
276C4C2×#611⋊C311yescharacteristic
277C3—31no—
278C3C3×#513111yescharacteristic
279C2—22no—
280C2—364no—
281C2#63470⋊C312yes—
282C2C2×(C2×AGL(1,ℤ/9))⋊C411yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×3−1+2C2
socleC22×C3C2×C3×D12
prime radicalC22×C4⋊AGL(1,ℤ/9)C2
cosocleC2×C4×C9C23×C3
hypercenterC2×M16AGL(1,ℤ/9)
hypocenterC9C3×#418
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×M16×3−1+2C2semidirect
3. #63488⋊C3C2semidirect
4. #63380⋊C3C2semidirect
6. #10588813C3semidirect
11. C2×C4⋊AGL(1,ℤ/9)C4semidirect
14. C2×C8×3−1+2C22semidirect
16. C4⋊(3−1+2⋊C4)C22semidirect
24. C2×C9×M16C2×C3semidirect
25. #63488C2×C3semidirect
26. #63380C2×C3semidirect
58. C2×D36C3×C4semidirect
63. C2×C8×C9C22×C3semidirect
65. C4⋊Dic9C22×C3semidirect
171. C2×M16AGL(1,ℤ/9)semidirect
174. 3−1+2#418semidirect
219. C2×C8C2×AGL(1,ℤ/9)semidirect
245. C9C3×#418semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊AGL(1,ℤ/9)2
C2×M16×3−1+22
#63488⋊C32
#63380⋊C32
C3×#51313
#105888133