Finite Group Databasebeta

← #10588487⋊C3 #13141952

Subgroups — #10588487⋊C3 #13141952

Order 1728.

393 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 94 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#63508⋊C3C211yesmaximal, characteristic, complemented
2C2×#3675⋊C3C211yesmaximal, characteristic, complemented
3C2×C8×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
4#63457⋊C3C211yesmaximal, characteristic
53−1+2×#103C211yesmaximal, characteristic, complemented
6#63380⋊C3C211yesmaximal, characteristic, complemented
7#63367⋊C3C211yesmaximal, characteristic, complemented
8C3×#4807—31nomaximal
9#10588487C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×C4⋊AGL(1,ℤ/9)—21no—
12C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
13C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
14C2×C8×3−1+2C2211yescharacteristic
15C4⋊C4×3−1+2C2211yescharacteristic
16#3675⋊C3—22no—
17(C2×AGL(1,ℤ/9))⋊C4—21no—
18(C2×AGL(1,ℤ/9))⋊C4—21no—
19C4⋊(3−1+2⋊C4)C2211yescharacteristic
20(3−1+2⋊C4)⋊C4—21no—
21C2×3−1+2⋊C8C2211yescharacteristic
22C8×AGL(1,ℤ/9)—22no—
23C3×#946—31no—
24C2×C3×#215—31no—
25C2×C3×D3×C8—31no—
26C3×#895—31no—
27C32×#103D311yescharacteristic
28C3×#821—31no—
29C3×#808—31no—
30#63508C2×C311yescharacteristic, complemented
31C2×#3675C2×C311yescharacteristic, complemented
32C2×C8×D9C2×C311yescharacteristic, complemented
33#63457C2×C311yescharacteristic
34C9×#103—21no—
35C9×#103C2×C311yescharacteristic, complemented
36#63380C2×C311yescharacteristic, complemented
37#63367C2×C311yescharacteristic, complemented
38C22×AGL(1,ℤ/9)—21no—
39C22×AGL(1,ℤ/9)D411yescharacteristic
40Q8×3−1+2—21no—
41Q8×3−1+2—21no—
42C2×C4×3−1+2—21no—
43C2×C4×3−1+2—21no—
44C2×C4×3−1+2C2311yescharacteristic
45C2×3−1+2⋊C4—21no—
46C2×3−1+2⋊C4D411yescharacteristic
47C4⋊AGL(1,ℤ/9)—21no—
48C4⋊AGL(1,ℤ/9)—21no—
49C4⋊AGL(1,ℤ/9)—42no—
50C4×AGL(1,ℤ/9)—21no—
51C4×AGL(1,ℤ/9)—21no—
52C8×3−1+2—21no—
533−1+2⋊C8D412yescomplemented
54C3×#465—91no—
55#4807—31no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×C3×D12—31no—
58C2×C3×D12—61no—
59C2×C3×C4×D3—31no—
60C2×C32×C8C2×D311yescharacteristic
61C32×C4⋊C4C2×D311yescharacteristic
62C3×#215—62no—
63C3×(C2×D3)⋊C4—61no—
64C3×(C2×D3)⋊C4—61no—
65C3×C4⋊Dic3—31no—
66C3×Dic3⋊C4—61no—
67C2×C3×C3⋊C8—31no—
68C3×D3×C8—62no—
69C2×Q8×C9—21no—
70C2×Q8×C9C22×C311yescharacteristic
71C2×D36—21no—
72C2×D36C22×C311yescharacteristic
73C2×C4×D9C22×C311yescharacteristic
74C2×C8×C9—21no—
75C2×C8×C9C22×C311yescharacteristic
76C9×C4⋊C4—21no—
77C9×C4⋊C4C22×C311yescharacteristic
78#3675—22no—
79(C2×D9)⋊C4—21no—
80(C2×D9)⋊C4—21no—
81C4⋊Dic9C22×C311yescharacteristic
82Dic9⋊C4—21no—
83C2×C9⋊C8C22×C311yescharacteristic
84C8×D9—22no—
85C22×3−1+2C2×D411yescharacteristic
86C2×AGL(1,ℤ/9)—21no—
87C2×AGL(1,ℤ/9)—21no—
88C2×AGL(1,ℤ/9)—41no—
89C2×AGL(1,ℤ/9)—41no—
90C2×AGL(1,ℤ/9)—41no—
91C2×AGL(1,ℤ/9)SD1612yescomplemented
92C4×3−1+2—41no—
93C4×3−1+2—41no—
94C4×3−1+2C2×D411yescharacteristic
95C4×3−1+2#4111yescharacteristic
963−1+2⋊C4—21no—
973−1+2⋊C4—41no—
98C2×C3×SD16—91no—
99C22×C3×C8—91no—
100C3×C22⋊Q8—91no—
101C3×#121—91no—
102#946—31no—
103C2×#215—31no—
104C2×D3×C8—31no—
105#895—31no—
106C3×#106—91no—
107C3×#103—31no—
108C3×#103C3×D311yescharacteristic
109C3×#102—91no—
110#821—31no—
111#808—31no—
112C22×C3×D3—31no—
113C22×C3×D3—61no—
114C32×Q8—21no—
115C32×Q8—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3×Dic3—31no—
120C2×C3×Dic3—61no—
121C3×D12—61no—
122C3×D12—61no—
123C3×D12—122no—
124C3×C4×D3—61no—
125C3×C4×D3—61no—
126C22×D9—21no—
127C22×D9C3×D411yescharacteristic
128C32×C8—21no—
129C3×C3⋊C8—32no—
130Q8×C9—22no—
131Q8×C9—21no—
132Q8×C9—21no—
133Q8×C9—41no—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—41no—
138C2×C4×C9—41no—
139C2×C4×C9C23×C311yescharacteristic
140C2×Dic9—21no—
141C2×Dic9C3×D411yescharacteristic
142D36—21no—
143D36—21no—
144D36—42no—
145C4×D9—21no—
146C4×D9—21no—
147C8×C9—21no—
148C8×C9—41no—
149C9⋊C8C3×D412yescomplemented
150#465—91noSylow
151C2×3−1+2#12111yescharacteristic
152C2×3−1+2C2×SD1611yescharacteristic
153C2×3−1+2#13511yescharacteristic
154AGL(1,ℤ/9)—22no—
155AGL(1,ℤ/9)—81no—
156C2×C3×Q8—31no—
157C2×C3×Q8C2×C3×D311yescharacteristic
158C2×C3×D4—91no—
159C2×C3×D4—181no—
160C22×C3×C4—91no—
161C2×D12—31no—
162C2×D12—61no—
163C2×C4×D3—31no—
164C3×SD16—182no—
165C2×C3×C8—31no—
166C2×C3×C8—91no—
167C2×C3×C8—182no—
168C2×C3×C8C2×C3×D311yescharacteristic
169C3×C4⋊C4—31no—
170C3×C4⋊C4—91no—
171C3×C4⋊C4—181no—
172C3×C4⋊C4C2×C3×D311yescharacteristic
173C3×C22⋊C4—181no—
174C3×C22⋊C4—181no—
175#215—62no—
176(C2×D3)⋊C4—61no—
177(C2×D3)⋊C4—61no—
178C4⋊Dic3—31no—
179Dic3⋊C4—61no—
180C2×C3⋊C8—31no—
181D3×C8—62no—
182C22×C32D3×D411yescharacteristic
183C2×C3×D3—32no—
184C2×C3×D3—61no—
185C2×C3×D3—61no—
186C2×C3×D3—121no—
187C2×C3×D3—121no—
188C2×C3×D3—121no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4#23511yescharacteristic
192C32×C4D3×D411yescharacteristic
193C3×Dic3—61no—
194C3×Dic3—121no—
195C22×C9—21no—
196C22×C9C2×C3×D411yescharacteristic
197C2×D9—21no—
198C2×D9—21no—
199C2×D9—41no—
200C2×D9—41no—
201C2×D9—41no—
202C2×D9C3×SD1612yescomplemented
203C4×C9—21no—
204C4×C9—21no—
205C4×C9—42no—
206C4×C9—41no—
207C4×C9—41no—
208C4×C9—81no—
209C4×C9C2×C3×D411yescharacteristic
210C4×C9C3×#4111yescharacteristic
211Dic9—21no—
212Dic9—41no—
213C2×SD16—91no—
214C22×C8—91no—
215C22⋊Q8—91no—
216#121—91no—
217#106—91no—
218#103AGL(1,ℤ/9)11yescharacteristic, complemented
219#102—91no—
2203−1+2#46511yescharacteristic, complemented, Sylow
221C23×C3—91no—
222C23×C3—181no—
223C22×D3—31no—
224C22×D3—61no—
225C3×Q8—21no—
226C3×Q8—21no—
227C3×Q8—61no—
228C3×Q8—61no—
229C3×D4—181no—
230C3×D4—181no—
231C3×D4—362no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—31no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—91no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4C22×C3×D311yescharacteristic
242C2×Dic3—31no—
243C2×Dic3—61no—
244D12—61no—
245D12—61no—
246D12—122no—
247C4×D3—61no—
248C4×D3—61no—
249C3×C8—21no—
250C3×C8—61no—
251C3×C8—92no—
252C3⋊C8—32no—
253C2×C32#88411yescharacteristic
254C2×C32D3×SD1611yescharacteristic
255C2×C32#91911yescharacteristic
256C3×D3—62no—
257C3×D3—241no—
258C2×C9—21no—
259C2×C9—21no—
260C2×C9—21no—
261C2×C9C3×#12111yescharacteristic
262C2×C9C2×C3×SD1611yescharacteristic
263C2×C9C3×#13511yescharacteristic
264D9—22no—
265D9—81no—
266C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
267C2×D4—91no—
268C2×D4—181no—
269C22×C4—91no—
270SD16—182no—
271C2×C8—91no—
272C2×C8—182no—
273C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
274C4⋊C4—91no—
275C4⋊C4—181no—
276C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
277C22⋊C4—181no—
278C22⋊C4—181no—
279C22×C3—31no—
280C22×C3—92no—
281C22×C3—181no—
282C22×C3—181no—
283C22×C3—361no—
284C22×C3—361no—
285C22×C3—361no—
286C22×C3C3×D3×D411yescharacteristic
287C2×D3—32no—
288C2×D3—61no—
289C2×D3—61no—
290C2×D3—121no—
291C2×D3—121no—
292C2×D3—121no—
293C3×C4—31no—
294C3×C4—31no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—121no—
298C3×C4—121no—
299C3×C4—181no—
300C3×C4—361no—
301C3×C4C3×#23511yescharacteristic
302C3×C4C3×D3×D411yescharacteristic
303Dic3—61no—
304Dic3—121no—
305C32#480711yescharacteristic
306C9—21no—
307C9C3×#46511yescharacteristic, complemented
308C23—91no—
309C23—181no—
310Q8—21no—
311Q8—21no—
312D4—181no—
313D4—181no—
314D4—362no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—91no—
318C2×C4—181no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
322C8—21no—
323C8—92no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—182no—
328C2×C3—721no—
329C2×C3C3×#88411yescharacteristic
330C2×C3C3×D3×SD1611yescharacteristic
331C2×C3C3×#91911yescharacteristic
332D3—62no—
333D3—241no—
334C22—92no—
335C22—181no—
336C22—181no—
337C22—361no—
338C22—361no—
339C22—361no—
340C22D4×AGL(1,ℤ/9)11yescharacteristic
341C4—41no—
342C4—41no—
343C4—181no—
344C4—361no—
345C4#3697⋊C311yescharacteristic
346C4D4×AGL(1,ℤ/9)11yescharacteristic
347C3—31no—
348C3C3×#480711yescharacteristic
349C2—182no—
350C2—721no—
351C2#63446⋊C311yescharacteristic
352C2SD16×AGL(1,ℤ/9)11yescharacteristic
353C2#63481⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#103C2
socleC22×C3C3×D3×D4
prime radical#63457⋊C3C2
hypercenter#103AGL(1,ℤ/9)
hypocenterC9C3×#465
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. #63508⋊C3C2semidirect
2. C2×#3675⋊C3C2semidirect
3. C2×C8×AGL(1,ℤ/9)C2semidirect
5. 3−1+2×#103C2semidirect
6. #63380⋊C3C2semidirect
7. #63367⋊C3C2semidirect
9. #10588487C3semidirect
30. #63508C2×C3semidirect
31. C2×#3675C2×C3semidirect
32. C2×C8×D9C2×C3semidirect
35. C9×#103C2×C3semidirect
36. #63380C2×C3semidirect
37. #63367C2×C3semidirect
53. 3−1+2⋊C8D4semidirect
91. C2×AGL(1,ℤ/9)SD16semidirect
149. C9⋊C8C3×D4semidirect
202. C2×D9C3×SD16semidirect
218. #103AGL(1,ℤ/9)semidirect
220. 3−1+2#465semidirect
307. C9C3×#465semidirect

Maximal subgroups

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

subgroupindex
#63508⋊C32
C2×#3675⋊C32
C2×C8×AGL(1,ℤ/9)2
#63457⋊C32
3−1+2×#1032
#63380⋊C32
#63367⋊C32
C3×#48073
#105884873