Finite Group Databasebeta

← #10588747⋊C3 #13142961

Subgroups — #10588747⋊C3 #13142961

Order 1728.

334 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 96 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
13−1+2×#125C211yesmaximal, characteristic, complemented
2C4×C4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic
3#63369⋊C3C211yesmaximal, characteristic
4#63369⋊C3C211yesmaximal, characteristic
5#63367⋊C3C211yesmaximal, characteristic, complemented
6#63366⋊C3C211yesmaximal, characteristic, complemented
7C4⋊(3−1+2⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5066—31nomaximal
9#10588747C311yesmaximal, characteristic, complemented
10C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
11C2×C4×AGL(1,ℤ/9)—21no—
12C4⋊C4×3−1+2—21no—
13C4⋊C4×3−1+2—21no—
14C4⋊C4×3−1+2C2211yescharacteristic
15C4⋊C4×3−1+2C2211yescharacteristic
16C42×3−1+2C2211yescharacteristic
17(C2×AGL(1,ℤ/9))⋊C4—21no—
18C4⋊(3−1+2⋊C4)C2211yescharacteristic
19C2×3−1+2⋊C8C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C32×#125D311yescharacteristic
22C3×C4×D12—31no—
23C3×#809—31no—
24C3×#809—31no—
25C3×#808—31no—
26C3×#807—31no—
27C3×C4⋊(C3⋊C8)—31no—
28C9×#125—21no—
29C9×#125C2×C311yescharacteristic, complemented
30C4×D36C2×C311yescharacteristic
31#63369C2×C311yescharacteristic
32#63369C2×C311yescharacteristic
33#63367C2×C311yescharacteristic, complemented
34#63366C2×C311yescharacteristic, complemented
35C4⋊(C9⋊C8)C2×C311yescharacteristic, complemented
36C22×AGL(1,ℤ/9)—21no—
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2D411yescharacteristic
40C2×C4×3−1+2D411yescharacteristic
41C2×C4×3−1+2C2311yescharacteristic
42C2×3−1+2⋊C4—21no—
43C4⋊AGL(1,ℤ/9)—21no—
44C4⋊AGL(1,ℤ/9)Q812yes—
45C4×AGL(1,ℤ/9)—42no—
463−1+2⋊C8—21no—
473−1+2⋊C8—21no—
48C3×#478—91no—
49#5066—31no—
50C2×C3×D12—31no—
51C2×C3×C4×D3—61no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C32×C42C2×D311yescharacteristic
57C3×(C2×D3)⋊C4—61no—
58C3×C4⋊Dic3—31no—
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8—31no—
61C2×D36C22×C311yescharacteristic
62C2×C4×D9—21no—
63C9×C4⋊C4—22no—
64C9×C4⋊C4—22no—
65C9×C4⋊C4—21no—
66C9×C4⋊C4—21no—
67C9×C4⋊C4—21no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4C22×C311yescharacteristic
70C9×C4⋊C4C22×C311yescharacteristic
71C42×C9—21no—
72C42×C9C22×C311yescharacteristic
73(C2×D9)⋊C4—21no—
74C4⋊Dic9C22×C311yescharacteristic
75C2×C9⋊C8C22×C311yescharacteristic
76C2×C9⋊C8C22×C311yescharacteristic
77C22×3−1+2C2×D411yescharacteristic
78C2×AGL(1,ℤ/9)—22no—
79C2×AGL(1,ℤ/9)—41no—
80C2×AGL(1,ℤ/9)—41no—
81C4×3−1+2—21no—
82C4×3−1+2—21no—
83C4×3−1+2—41no—
84C4×3−1+2—41no—
85C4×3−1+2C2×Q811yescharacteristic
86C4×3−1+2#4111yescharacteristic
873−1+2⋊C4—41no—
88C3×#125—31no—
89C3×#125C3×D311yescharacteristic
90C3×C4×D4—91no—
91C4×D12—31no—
92C3×#107—91no—
93C3×#106—91no—
94C3×C4⋊C8—91no—
95C3×#102—91no—
96C3×#102—91no—
97#809—31no—
98#809—31no—
99#808—31no—
100#807—31no—
101C4⋊(C3⋊C8)—31no—
102C22×C3×D3—61no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4#6711yescharacteristic
106C2×C32×C4#6711yescharacteristic
107C2×C32×C4C22×D311yescharacteristic
108C2×C3×Dic3—61no—
109C3×D12—32no—
110C3×D12—61no—
111C3×C4×D3—122no—
112C22×D9—21no—
113C3×C3⋊C8—61no—
114C3×C3⋊C8—61no—
115C2×C4×C9—22no—
116C2×C4×C9—22no—
117C2×C4×C9—21no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9C3×D411yescharacteristic
123C2×C4×C9C3×D411yescharacteristic
124C2×C4×C9C23×C311yescharacteristic
125C2×Dic9—21no—
126D36—21no—
127D36C3×Q812yes—
128C4×D9—42no—
129C9⋊C8—21no—
130C9⋊C8—21no—
131#478—91noSylow
132C2×3−1+2C22⋊Q811yescharacteristic
133C2×3−1+2#13511yescharacteristic
134C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
135AGL(1,ℤ/9)—42no—
136C2×C3×D4—91no—
137C22×C3×C4—181no—
138C2×D12—31no—
139C2×C4×D3—61no—
140C2×C3×C8—91no—
141C2×C3×C8—91no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—61no—
147C3×C4⋊C4—61no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4C2×C3×D311yescharacteristic
150C3×C4⋊C4C2×C3×D311yescharacteristic
151C3×C22⋊C4—181no—
152C3×C42—31no—
153C3×C42C2×C3×D311yescharacteristic
154(C2×D3)⋊C4—61no—
155C4⋊Dic3—31no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8—31no—
158C22×C32C2×#6711yescharacteristic
159C2×C3×D3—62no—
160C2×C3×D3—121no—
161C2×C3×D3—121no—
162C32×C4—21no—
163C32×C4—21no—
164C32×C4—41no—
165C32×C4—41no—
166C32×C4D3×Q811yescharacteristic
167C32×C4#23911yescharacteristic
168C3×Dic3—121no—
169C22×C9—21no—
170C22×C9C2×C3×D411yescharacteristic
171C2×D9—22no—
172C2×D9—41no—
173C2×D9—41no—
174C4×C9—21no—
175C4×C9—21no—
176C4×C9—21no—
177C4×C9—21no—
178C4×C9—41no—
179C4×C9—41no—
180C4×C9—42no—
181C4×C9—42no—
182C4×C9—41no—
183C4×C9—41no—
184C4×C9C2×C3×Q811yescharacteristic
185C4×C9C3×#4111yescharacteristic
186Dic9—41no—
187#125AGL(1,ℤ/9)11yescharacteristic, complemented
188C4×D4—91no—
189#107—91no—
190#106—91no—
191C4⋊C8—91no—
192#102—91no—
193#102—91no—
1943−1+2#47811yescharacteristic, complemented, Sylow
195C23×C3—181no—
196C22×D3—61no—
197C3×D4—92no—
198C3×D4—181no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—61no—
205C2×C3×C4—61no—
206C2×C3×C4—181no—
207C2×C3×C4—362no—
208C2×C3×C4C3×#6711yescharacteristic
209C2×C3×C4C3×#6711yescharacteristic
210C2×C3×C4C22×C3×D311yescharacteristic
211C2×Dic3—61no—
212D12—32no—
213D12—61no—
214C4×D3—122no—
215C3×C8—181no—
216C3×C8—181no—
217C3⋊C8—61no—
218C3⋊C8—61no—
219C2×C32#94611yescharacteristic
220C2×C32#94911yescharacteristic
221C2×C32#95011yescharacteristic
222C3×D3—122no—
223C2×C9—21no—
224C2×C9—21no—
225C2×C9—21no—
226C2×C9C3×C22⋊Q811yescharacteristic
227C2×C9C3×#13511yescharacteristic
228C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
229D9—42no—
230C2×D4—91no—
231C22×C4—181no—
232C2×C8—91no—
233C2×C8—91no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—91no—
237C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
238C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
239C22⋊C4—181no—
240C42C2×AGL(1,ℤ/9)11yescharacteristic
241C22×C3—31no—
242C22×C3—182no—
243C22×C3—361no—
244C22×C3—361no—
245C22×C3C2×C3×#6711yescharacteristic
246C2×D3—62no—
247C2×D3—121no—
248C2×D3—121no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—31no—
252C3×C4—31no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—61no—
256C3×C4—61no—
257C3×C4—121no—
258C3×C4—121no—
259C3×C4—361no—
260C3×C4C3×D3×Q811yescharacteristic
261C3×C4C3×#23911yescharacteristic
262Dic3—121no—
263C32#506611yescharacteristic
264C9—21no—
265C9C3×#47811yescharacteristic, complemented
266C23—181no—
267D4—92no—
268D4—181no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—181no—
272C2×C4—362no—
273C2×C4#611⋊C311yescharacteristic
274C2×C4#611⋊C311yescharacteristic
275C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
276C8—181no—
277C8—181no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—362no—
282C2×C3C3×#94611yescharacteristic
283C2×C3C3×#94911yescharacteristic
284C2×C3C3×#95011yescharacteristic
285D3—122no—
286C22—182no—
287C22—361no—
288C22—361no—
289C22C2×#611⋊C311yescharacteristic
290C4—21no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—361no—
295C4Q8×AGL(1,ℤ/9)11yescharacteristic
296C4#8872411yescharacteristic
297C3—31no—
298C3C3×#506611yescharacteristic
299C2—362no—
300C2#63508⋊C311yescharacteristic
301C2#63512⋊C311yescharacteristic
302C2#63513⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#125C2
socleC22×C3C2×C3×#67
prime radicalC2×C4⋊AGL(1,ℤ/9)C22
hypercenter#125AGL(1,ℤ/9)
hypocenterC9C3×#478
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#125C2semidirect
5. #63367⋊C3C2semidirect
6. #63366⋊C3C2semidirect
7. C4⋊(3−1+2⋊C8)C2semidirect
9. #10588747C3semidirect
29. C9×#125C2×C3semidirect
33. #63367C2×C3semidirect
34. #63366C2×C3semidirect
35. C4⋊(C9⋊C8)C2×C3semidirect
187. #125AGL(1,ℤ/9)semidirect
194. 3−1+2#478semidirect
265. C9C3×#478semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1252
C4×C4⋊AGL(1,ℤ/9)2
#63369⋊C32
#63367⋊C32
#63366⋊C32
C4⋊(3−1+2⋊C8)2
C3×#50663
#105887473