Finite Group Databasebeta

← #10588565⋊C3 #13142246

Subgroups — #10588565⋊C3 #13142246

Order 1728.

336 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 86 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
2#63454⋊C3C211yesmaximal, 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#63367⋊C3C211yesmaximal, characteristic, complemented
8C3×#4885—31nomaximal
9#10588565C311yesmaximal, characteristic, complemented
10C2×C4⋊AGL(1,ℤ/9)—21no—
11C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
12C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
13C2×C8×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15C4⋊C4×3−1+2C2211yescharacteristic
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17(C2×AGL(1,ℤ/9))⋊C4—21no—
18C4⋊(3−1+2⋊C4)C2211yescharacteristic
19C4×3−1+2⋊C4—21no—
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×#895—31no—
22C3×#892—31no—
23C32×#107D311yescharacteristic
24C3×#821—31no—
25C3×(C2×D3)⋊C8—31no—
26C3×#809—31no—
27C3×#808—31no—
28#63457C2×C311yescharacteristic
29#63454C2×C311yescharacteristic, complemented
30C9×#107—21no—
31C9×#107C2×C311yescharacteristic, complemented
32#63380C2×C311yescharacteristic, complemented
33(C2×D9)⋊C8C2×C311yescharacteristic, complemented
34#63369C2×C311yescharacteristic, complemented
35#63367C2×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—
48C8×3−1+2—21no—
493−1+2⋊C8—21no—
50C3×#482—91no—
51#4885—31no—
52C2×C3×D12—31no—
53C2×C3×D12—61no—
54C2×C3×C4×D3—31no—
55C2×C32×C8C2×D311yescharacteristic
56C32×C4⋊C4C2×D311yescharacteristic
57C32×C4⋊C4C2×D311yescharacteristic
58C3×(C2×D3)⋊C4—61no—
59C3×(C2×D3)⋊C4—61no—
60C3×C4⋊Dic3—31no—
61C3×C4×Dic3—61no—
62C2×C3×C3⋊C8—31no—
63C2×D36—21no—
64C2×D36C22×C311yescharacteristic
65C2×C4×D9C22×C311yescharacteristic
66C2×C8×C9—21no—
67C2×C8×C9C22×C311yescharacteristic
68C9×C4⋊C4—21no—
69C9×C4⋊C4—21no—
70C9×C4⋊C4C22×C311yescharacteristic
71C9×C4⋊C4C22×C311yescharacteristic
72(C2×D9)⋊C4—21no—
73(C2×D9)⋊C4—21no—
74C4⋊Dic9C22×C311yescharacteristic
75C4×Dic9—21no—
76C2×C9⋊C8C22×C311yescharacteristic
77C22×3−1+2C2×D411yescharacteristic
78C2×AGL(1,ℤ/9)—21no—
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—
84C4×3−1+2—41no—
85C4×3−1+2—41no—
86C4×3−1+2#4111yescharacteristic
87C4×3−1+2#4111yescharacteristic
883−1+2⋊C4—21no—
893−1+2⋊C4—41no—
90C3×#121—91no—
91C3×#117—91no—
92#895—31no—
93#892—31no—
94C3×#107—31no—
95C3×#107C3×D311yescharacteristic
96C3×#106—91no—
97C3×#102—91no—
98C3×#102—91no—
99C3×C22⋊C8—91no—
100#821—31no—
101(C2×D3)⋊C8—31no—
102#809—31no—
103#808—31no—
104C22×C3×D3—31no—
105C22×C3×D3—61no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3×Dic3—31no—
110C2×C3×Dic3—61no—
111C3×D12—61no—
112C3×D12—61no—
113C3×D12—122no—
114C3×C4×D3—61no—
115C3×C4×D3—61no—
116C22×D9—21no—
117C22×D9C3×D411yescharacteristic
118C32×C8—21no—
119C3×C3⋊C8—61no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—41no—
124C2×C4×C9—41no—
125C2×C4×C9C23×C311yescharacteristic
126C2×Dic9—21no—
127C2×Dic9C3×D411yescharacteristic
128D36—21no—
129D36—21no—
130D36—42no—
131C4×D9—21no—
132C4×D9—21no—
133C8×C9—22no—
134C8×C9—21no—
135C9⋊C8—21no—
136#482—91noSylow
137C2×3−1+2#12311yescharacteristic
138C2×3−1+2#13511yescharacteristic
139C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
140AGL(1,ℤ/9)—41no—
141AGL(1,ℤ/9)—81no—
142C2×C3×D4—91no—
143C2×C3×D4—181no—
144C22×C3×C4—91no—
145C2×D12—31no—
146C2×D12—61no—
147C2×C4×D3—31no—
148C2×C3×C8—31no—
149C2×C3×C8—91no—
150C2×C3×C8C2×C3×D311yescharacteristic
151C3×C4⋊C4—31no—
152C3×C4⋊C4—31no—
153C3×C4⋊C4—91no—
154C3×C4⋊C4C2×C3×D311yescharacteristic
155C3×C4⋊C4C2×C3×D311yescharacteristic
156C3×C22⋊C4—181no—
157C3×C22⋊C4—181no—
158C3×C42—181no—
159(C2×D3)⋊C4—61no—
160(C2×D3)⋊C4—61no—
161C4⋊Dic3—31no—
162C4×Dic3—61no—
163C2×C3⋊C8—31no—
164C22×C32D3×D411yescharacteristic
165C2×C3×D3—61no—
166C2×C3×D3—61no—
167C2×C3×D3—61no—
168C2×C3×D3—121no—
169C2×C3×D3—121no—
170C2×C3×D3—121no—
171C32×C4—41no—
172C32×C4—41no—
173C32×C4#23511yescharacteristic
174C32×C4#23911yescharacteristic
175C3×Dic3—61no—
176C3×Dic3—121no—
177C22×C9—21no—
178C22×C9C2×C3×D411yescharacteristic
179C2×D9—21no—
180C2×D9—21no—
181C2×D9—21no—
182C2×D9—41no—
183C2×D9—41no—
184C2×D9—41no—
185C4×C9—21no—
186C4×C9—21no—
187C4×C9—41no—
188C4×C9—41no—
189C4×C9—81no—
190C4×C9—81no—
191C4×C9C3×#4111yescharacteristic
192C4×C9C3×#4111yescharacteristic
193Dic9—21no—
194Dic9—41no—
195#121—91no—
196#117—91no—
197#107AGL(1,ℤ/9)11yescharacteristic, complemented
198#106—91no—
199#102—91no—
200#102—91no—
201C22⋊C8—91no—
2023−1+2#48211yescharacteristic, complemented, Sylow
203C23×C3—91no—
204C23×C3—181no—
205C22×D3—31no—
206C22×D3—61no—
207C3×D4—181no—
208C3×D4—181no—
209C3×D4—362no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—31no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—91no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4C22×C3×D311yescharacteristic
220C2×Dic3—31no—
221C2×Dic3—61no—
222D12—61no—
223D12—61no—
224D12—122no—
225C4×D3—61no—
226C4×D3—61no—
227C3×C8—21no—
228C3×C8—61no—
229C3×C8—181no—
230C3⋊C8—61no—
231C2×C32#89411yescharacteristic
232C2×C32#91111yescharacteristic
233C2×C32#91911yescharacteristic
234C3×D3—121no—
235C3×D3—241no—
236C2×C9—21no—
237C2×C9—21no—
238C2×C9—21no—
239C2×C9C3×#12311yescharacteristic
240C2×C9C3×#13511yescharacteristic
241C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
242D9—41no—
243D9—81no—
244C2×D4—91no—
245C2×D4—181no—
246C22×C4—91no—
247C2×C8—91no—
248C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
249C4⋊C4—91no—
250C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
251C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
252C22⋊C4—181no—
253C22⋊C4—181no—
254C42—181no—
255C22×C3—31no—
256C22×C3—181no—
257C22×C3—181no—
258C22×C3—181no—
259C22×C3—361no—
260C22×C3—361no—
261C22×C3—361no—
262C22×C3C3×D3×D411yescharacteristic
263C2×D3—61no—
264C2×D3—61no—
265C2×D3—61no—
266C2×D3—121no—
267C2×D3—121no—
268C2×D3—121no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—121no—
274C3×C4—121no—
275C3×C4—181no—
276C3×C4—361no—
277C3×C4C3×#23511yescharacteristic
278C3×C4C3×#23911yescharacteristic
279Dic3—61no—
280Dic3—121no—
281C32#488511yescharacteristic
282C9—21no—
283C9C3×#48211yescharacteristic, complemented
284C23—91no—
285C23—181no—
286D4—181no—
287D4—181no—
288D4—362no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—91no—
292C2×C4—181no—
293C2×C4—181no—
294C2×C4—181no—
295C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
296C8—21no—
297C8—181no—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—361no—
302C2×C3—721no—
303C2×C3C3×#89411yescharacteristic
304C2×C3C3×#91111yescharacteristic
305C2×C3C3×#91911yescharacteristic
306D3—121no—
307D3—241no—
308C22—181no—
309C22—181no—
310C22—181no—
311C22—361no—
312C22—361no—
313C22—361no—
314C22D4×AGL(1,ℤ/9)11yescharacteristic
315C4—41no—
316C4—41no—
317C4—181no—
318C4—361no—
319C4#3697⋊C311yescharacteristic
320C4#8872411yescharacteristic
321C3—31no—
322C3C3×#488511yescharacteristic
323C2—361no—
324C2—721no—
325C2#63456⋊C311yescharacteristic
326C2#63473⋊C311yescharacteristic
327C2#63481⋊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×#482
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. #63454⋊C3C2semidirect
3. 3−1+2×#107C2semidirect
4. #63380⋊C3C2semidirect
5. (C2×AGL(1,ℤ/9))⋊C8C2semidirect
6. #63369⋊C3C2semidirect
7. #63367⋊C3C2semidirect
9. #10588565C3semidirect
29. #63454C2×C3semidirect
31. C9×#107C2×C3semidirect
32. #63380C2×C3semidirect
33. (C2×D9)⋊C8C2×C3semidirect
34. #63369C2×C3semidirect
35. #63367C2×C3semidirect
197. #107AGL(1,ℤ/9)semidirect
202. 3−1+2#482semidirect
283. C9C3×#482semidirect

Maximal subgroups

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

subgroupindex
#63457⋊C32
#63454⋊C32
3−1+2×#1072
#63380⋊C32
(C2×AGL(1,ℤ/9))⋊C82
#63369⋊C32
#63367⋊C32
C3×#48853
#105885653