Finite Group Databasebeta

← #10588562⋊C3 #13142243

Subgroups — #10588562⋊C3 #13142243

Order 1728.

360 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#63457⋊C3C211yesmaximal, characteristic
2C4⋊C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
33−1+2×#107C211yesmaximal, characteristic, complemented
4#63380⋊C3C211yesmaximal, characteristic, complemented
5(C2×AGL(1,ℤ/9))⋊C8C211yesmaximal, characteristic, complemented
6#63369⋊C3C211yesmaximal, characteristic, complemented
7#63366⋊C3C211yesmaximal, characteristic, complemented
8C3×#4882—31nomaximal
9#10588562C311yesmaximal, characteristic, complemented
10C2×C4⋊AGL(1,ℤ/9)—21no—
11C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
12C2×C4×AGL(1,ℤ/9)—21no—
13C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
14C2×C8×3−1+2C2211yescharacteristic
15C4⋊C4×3−1+2C2211yescharacteristic
16C4⋊C4×3−1+2C2211yescharacteristic
17(C2×AGL(1,ℤ/9))⋊C4—21no—
18C4⋊(3−1+2⋊C4)C2211yescharacteristic
19(3−1+2⋊C4)⋊C4—21no—
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×#895—31no—
22C3×D3×C4⋊C4—31no—
23C32×#107D311yescharacteristic
24C3×#821—31no—
25C3×(C2×D3)⋊C8—31no—
26C3×#809—31no—
27C3×#807—31no—
28#63457C2×C311yescharacteristic
29C4⋊C4×D9C2×C311yescharacteristic, complemented
30C9×#107—21no—
31C9×#107C2×C311yescharacteristic, complemented
32#63380C2×C311yescharacteristic, complemented
33(C2×D9)⋊C8C2×C311yescharacteristic, complemented
34#63369C2×C311yescharacteristic, complemented
35#63366C2×C311yescharacteristic, complemented
36C22×AGL(1,ℤ/9)—21no—
37C22×AGL(1,ℤ/9)D411yescharacteristic
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2C2311yescharacteristic
41C2×3−1+2⋊C4—21no—
42C2×3−1+2⋊C4D411yescharacteristic
43C4⋊AGL(1,ℤ/9)—21no—
44C4⋊AGL(1,ℤ/9)—21no—
45C4⋊AGL(1,ℤ/9)—42no—
46C4×AGL(1,ℤ/9)—21no—
47C4×AGL(1,ℤ/9)—21no—
48C4×AGL(1,ℤ/9)—42no—
49C8×3−1+2—21no—
503−1+2⋊C8—21no—
51C3×#480—91no—
52#4882—31no—
53C2×C3×D12—31no—
54C2×C3×D12—61no—
55C2×C3×C4×D3—31no—
56C2×C3×C4×D3—61no—
57C2×C32×C8C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59C32×C4⋊C4C2×D311yescharacteristic
60C3×(C2×D3)⋊C4—61no—
61C3×C4⋊Dic3—31no—
62C3×Dic3⋊C4—61no—
63C2×C3×C3⋊C8—31no—
64C2×D36—21no—
65C2×D36C22×C311yescharacteristic
66C2×C4×D9—21no—
67C2×C4×D9C22×C311yescharacteristic
68C2×C8×C9—21no—
69C2×C8×C9C22×C311yescharacteristic
70C9×C4⋊C4—21no—
71C9×C4⋊C4—21no—
72C9×C4⋊C4C22×C311yescharacteristic
73C9×C4⋊C4C22×C311yescharacteristic
74(C2×D9)⋊C4—21no—
75C4⋊Dic9C22×C311yescharacteristic
76Dic9⋊C4—21no—
77C2×C9⋊C8C22×C311yescharacteristic
78C22×3−1+2C2×D411yescharacteristic
79C2×AGL(1,ℤ/9)—21no—
80C2×AGL(1,ℤ/9)—21no—
81C2×AGL(1,ℤ/9)—41no—
82C2×AGL(1,ℤ/9)—41no—
83C2×AGL(1,ℤ/9)—41no—
84C2×AGL(1,ℤ/9)D812yes—
85C4×3−1+2—41no—
86C4×3−1+2—41no—
87C4×3−1+2#4111yescharacteristic
88C4×3−1+2#4111yescharacteristic
893−1+2⋊C4—21no—
903−1+2⋊C4—41no—
91C3×#121—91no—
92C2×C3×C4⋊C4—91no—
93#895—31no—
94D3×C4⋊C4—31no—
95C3×#107—31no—
96C3×#107—91no—
97C3×#107C3×D311yescharacteristic
98C3×#102—91no—
99C3×#102—91no—
100C3×C22⋊C8—91no—
101#821—31no—
102(C2×D3)⋊C8—31no—
103#809—31no—
104#807—31no—
105C22×C3×D3—31no—
106C22×C3×D3—61no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3×Dic3—31no—
111C2×C3×Dic3—61no—
112C3×D12—61no—
113C3×D12—61no—
114C3×D12—122no—
115C3×C4×D3—61no—
116C3×C4×D3—61no—
117C3×C4×D3—122no—
118C22×D9—21no—
119C22×D9C3×D411yescharacteristic
120C32×C8—21no—
121C3×C3⋊C8—61no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—21no—
125C2×C4×C9—41no—
126C2×C4×C9—41no—
127C2×C4×C9C23×C311yescharacteristic
128C2×Dic9—21no—
129C2×Dic9C3×D411yescharacteristic
130D36—21no—
131D36—21no—
132D36—42no—
133C4×D9—21no—
134C4×D9—21no—
135C4×D9—42no—
136C8×C9—22no—
137C8×C9—21no—
138C9⋊C8—21no—
139#480—91noSylow
140C2×3−1+2#12311yescharacteristic
141C2×3−1+2C2×D811yescharacteristic
142C2×3−1+2#13711yescharacteristic
143AGL(1,ℤ/9)—22no—
144AGL(1,ℤ/9)—81no—
145C2×C3×D4—91no—
146C2×C3×D4—181no—
147C22×C3×C4—91no—
148C22×C3×C4—181no—
149C2×D12—31no—
150C2×D12—61no—
151C2×C4×D3—31no—
152C2×C4×D3—61no—
153C2×C3×C8—31no—
154C2×C3×C8—91no—
155C2×C3×C8C2×C3×D311yescharacteristic
156C3×C4⋊C4—31no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—181no—
160C3×C4⋊C4C2×C3×D311yescharacteristic
161C3×C4⋊C4C2×C3×D311yescharacteristic
162C3×C22⋊C4—181no—
163(C2×D3)⋊C4—61no—
164C4⋊Dic3—31no—
165Dic3⋊C4—61no—
166C2×C3⋊C8—31no—
167C22×C32D3×D411yescharacteristic
168C2×C3×D3—32no—
169C2×C3×D3—61no—
170C2×C3×D3—61no—
171C2×C3×D3—121no—
172C2×C3×D3—121no—
173C2×C3×D3—121no—
174C32×C4—41no—
175C32×C4—41no—
176C32×C4#23511yescharacteristic
177C32×C4#23911yescharacteristic
178C3×Dic3—61no—
179C3×Dic3—121no—
180C22×C9—21no—
181C22×C9C2×C3×D411yescharacteristic
182C2×D9—21no—
183C2×D9—21no—
184C2×D9—41no—
185C2×D9—41no—
186C2×D9—41no—
187C2×D9C3×D812yes—
188C4×C9—21no—
189C4×C9—21no—
190C4×C9—41no—
191C4×C9—41no—
192C4×C9—81no—
193C4×C9—81no—
194C4×C9C3×#4111yescharacteristic
195C4×C9C3×#4111yescharacteristic
196Dic9—21no—
197Dic9—41no—
198#121—91no—
199C2×C4⋊C4—91no—
200#107—91no—
201#107AGL(1,ℤ/9)11yescharacteristic, complemented
202#102—91no—
203#102—91no—
204C22⋊C8—91no—
2053−1+2#48011yescharacteristic, complemented, Sylow
206C23×C3—91no—
207C23×C3—181no—
208C22×D3—31no—
209C22×D3—61no—
210C3×D4—181no—
211C3×D4—181no—
212C3×D4—362no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—31no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—91no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221C2×C3×C4—181no—
222C2×C3×C4—362no—
223C2×C3×C4C22×C3×D311yescharacteristic
224C2×Dic3—31no—
225C2×Dic3—61no—
226D12—61no—
227D12—61no—
228D12—122no—
229C4×D3—61no—
230C4×D3—61no—
231C4×D3—122no—
232C3×C8—21no—
233C3×C8—61no—
234C3×C8—181no—
235C3⋊C8—61no—
236C2×C32#89411yescharacteristic
237C2×C32D3×D811yescharacteristic
238C2×C32#91811yescharacteristic
239C3×D3—62no—
240C3×D3—241no—
241C2×C9—21no—
242C2×C9—21no—
243C2×C9—21no—
244C2×C9C3×#12311yescharacteristic
245C2×C9C2×C3×D811yescharacteristic
246C2×C9C3×#13711yescharacteristic
247D9—22no—
248D9—81no—
249C2×D4—91no—
250C2×D4—181no—
251C22×C4—91no—
252C22×C4—181no—
253C2×C8—91no—
254C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
255C4⋊C4—91no—
256C4⋊C4—181no—
257C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
258C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
259C22⋊C4—181no—
260C22×C3—31no—
261C22×C3—92no—
262C22×C3—181no—
263C22×C3—181no—
264C22×C3—361no—
265C22×C3—361no—
266C22×C3—361no—
267C22×C3C3×D3×D411yescharacteristic
268C2×D3—32no—
269C2×D3—61no—
270C2×D3—61no—
271C2×D3—121no—
272C2×D3—121no—
273C2×D3—121no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—121no—
279C3×C4—121no—
280C3×C4—181no—
281C3×C4—361no—
282C3×C4C3×#23511yescharacteristic
283C3×C4C3×#23911yescharacteristic
284Dic3—61no—
285Dic3—121no—
286C32#488211yescharacteristic
287C9—21no—
288C9C3×#48011yescharacteristic, complemented
289C23—91no—
290C23—181no—
291D4—181no—
292D4—181no—
293D4—362no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—91no—
297C2×C4—181no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4—362no—
301C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
302C8—21no—
303C8—181no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—182no—
308C2×C3—721no—
309C2×C3C3×#89411yescharacteristic
310C2×C3C3×D3×D811yescharacteristic
311C2×C3C3×#91811yescharacteristic
312D3—62no—
313D3—241no—
314C22—92no—
315C22—181no—
316C22—181no—
317C22—361no—
318C22—361no—
319C22—361no—
320C22D4×AGL(1,ℤ/9)11yescharacteristic
321C4—41no—
322C4—41no—
323C4—181no—
324C4—361no—
325C4#3697⋊C311yescharacteristic
326C4#8872411yescharacteristic
327C3—31no—
328C3C3×#488211yescharacteristic
329C2—182no—
330C2—721no—
331C2#63456⋊C311yescharacteristic
332C2D8×AGL(1,ℤ/9)11yescharacteristic
333C2#63480⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#107C2
socleC22×C3C3×D3×D4
prime radical#63457⋊C3C2
hypercenter#107AGL(1,ℤ/9)
hypocenterC9C3×#480
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×AGL(1,ℤ/9)C2semidirect
3. 3−1+2×#107C2semidirect
4. #63380⋊C3C2semidirect
5. (C2×AGL(1,ℤ/9))⋊C8C2semidirect
6. #63369⋊C3C2semidirect
7. #63366⋊C3C2semidirect
9. #10588562C3semidirect
29. C4⋊C4×D9C2×C3semidirect
31. C9×#107C2×C3semidirect
32. #63380C2×C3semidirect
33. (C2×D9)⋊C8C2×C3semidirect
34. #63369C2×C3semidirect
35. #63366C2×C3semidirect
201. #107AGL(1,ℤ/9)semidirect
205. 3−1+2#480semidirect
288. C9C3×#480semidirect

Maximal subgroups

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

subgroupindex
#63457⋊C32
C4⋊C4×AGL(1,ℤ/9)2
3−1+2×#1072
#63380⋊C32
(C2×AGL(1,ℤ/9))⋊C82
#63369⋊C32
#63366⋊C32
C3×#48823
#105885623