Finite Group Databasebeta

← #10588480⋊C3 #13141943

Subgroups — #10588480⋊C3 #13141943

Order 1728.

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

normal subgroups only

This group has 309 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#63509⋊C3C211yesmaximal, characteristic
2#63452⋊C3C211yesmaximal, characteristic, complemented
33−1+2×#103C211yesmaximal, characteristic, complemented
4#63395⋊C3C211yesmaximal, characteristic, complemented
5#63380⋊C3C211yesmaximal, characteristic
6C8×3−1+2⋊C4C211yesmaximal, characteristic, complemented
7#63369⋊C3C211yesmaximal, characteristic
8C3×#4800—31nomaximal
9#10588480C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
12C2×C8×3−1+2C2211yescharacteristic
13C4⋊C4×3−1+2C2211yescharacteristic
14(C2×AGL(1,ℤ/9))⋊C4—21no—
15(C2×AGL(1,ℤ/9))⋊C4—21no—
16C4⋊(3−1+2⋊C4)—21no—
17C4⋊(3−1+2⋊C4)C2211yescharacteristic
18(3−1+2⋊C4)⋊C4—21no—
19C4×3−1+2⋊C4C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×#947—31no—
22C3×#890—31no—
23C32×#103D311yescharacteristic
24C3×#835—31no—
25C3×#821—31no—
26C3×C8×Dic3—31no—
27C3×#809—31no—
28#63509C2×C311yescharacteristic
29#63452C2×C311yescharacteristic, complemented
30C9×#103—21no—
31C9×#103C2×C311yescharacteristic, complemented
32#63395C2×C311yescharacteristic, complemented
33#63380C2×C311yescharacteristic
34C8×Dic9C2×C311yescharacteristic, complemented
35#63369C2×C311yescharacteristic
36C22×AGL(1,ℤ/9)—21no—
37Q8×3−1+2—21no—
38Q8×3−1+2—21no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2—21no—
41C2×C4×3−1+2C2311yescharacteristic
42C2×3−1+2⋊C4—21no—
43C2×3−1+2⋊C4D411yescharacteristic
44C2×3−1+2⋊C4D411yescharacteristic
45C4⋊AGL(1,ℤ/9)—21no—
46C4⋊AGL(1,ℤ/9)—21no—
47C8×3−1+2—21no—
483−1+2⋊C8—21no—
49C3×#488—91no—
50#4800—31no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C3×D12—31no—
53C2×C32×C8C2×D311yescharacteristic
54C32×C4⋊C4C2×D311yescharacteristic
55C3×(C2×D3)⋊C4—61no—
56C3×(C2×D3)⋊C4—61no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—61no—
59C3×Dic3⋊C4—61no—
60C3×C4×Dic3—31no—
61C2×C3×C3⋊C8—31no—
62C2×Q8×C9—21no—
63C2×Q8×C9C22×C311yescharacteristic
64C2×D36C22×C311yescharacteristic
65C2×C8×C9—21no—
66C2×C8×C9C22×C311yescharacteristic
67C9×C4⋊C4—21no—
68C9×C4⋊C4C22×C311yescharacteristic
69(C2×D9)⋊C4—21no—
70(C2×D9)⋊C4—21no—
71C4⋊Dic9—21no—
72C4⋊Dic9C22×C311yescharacteristic
73Dic9⋊C4—21no—
74C4×Dic9C22×C311yescharacteristic
75C2×C9⋊C8C22×C311yescharacteristic
76C22×3−1+2C2×D411yescharacteristic
77C2×AGL(1,ℤ/9)—41no—
78C2×AGL(1,ℤ/9)—41no—
79C2×AGL(1,ℤ/9)—41no—
80C4×3−1+2—41no—
81C4×3−1+2—41no—
82C4×3−1+2#4111yescharacteristic
83C4×3−1+2#4111yescharacteristic
843−1+2⋊C4—21no—
853−1+2⋊C4—21no—
863−1+2⋊C4—41no—
87C3×#125—91no—
88C3×#124—91no—
89#947—31no—
90#890—31no—
91C3×#103—31no—
92C3×#103—91no—
93C3×#103C3×D311yescharacteristic
94C3×#102—91no—
95C3×#102—91no—
96C3×C4×C8—91no—
97#835—31no—
98#821—31no—
99C8×Dic3—31no—
100#809—31no—
101C22×C3×D3—61no—
102C32×Q8—21no—
103C32×Q8—21no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3×Dic3—31no—
108C2×C3×Dic3—31no—
109C2×C3×Dic3—61no—
110C3×D12—61no—
111C3×D12—61no—
112C22×D9—21no—
113C32×C8—21no—
114C3×C3⋊C8—61no—
115Q8×C9—22no—
116Q8×C9—21no—
117Q8×C9—21no—
118Q8×C9—41no—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—41no—
123C2×C4×C9—41no—
124C2×C4×C9C23×C311yescharacteristic
125C2×Dic9—21no—
126C2×Dic9C3×D411yescharacteristic
127C2×Dic9C3×D411yescharacteristic
128D36—21no—
129D36—21no—
130C8×C9—21no—
131C8×C9—41no—
132C9⋊C8—21no—
133#488—91noSylow
134C2×3−1+2#12411yescharacteristic
135C2×3−1+2#13511yescharacteristic
136C2×3−1+2#13511yescharacteristic
137AGL(1,ℤ/9)—81no—
138C2×C3×Q8—31no—
139C2×C3×Q8C2×C3×D311yescharacteristic
140C2×C3×D4—91no—
141C2×D12—31no—
142C2×C3×C8—31no—
143C2×C3×C8—91no—
144C2×C3×C8C2×C3×D311yescharacteristic
145C3×C4⋊C4—31no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—181no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4C2×C3×D311yescharacteristic
150C3×C22⋊C4—181no—
151C3×C22⋊C4—181no—
152C3×C42—91no—
153(C2×D3)⋊C4—61no—
154(C2×D3)⋊C4—61no—
155C4⋊Dic3—31no—
156C4⋊Dic3—61no—
157Dic3⋊C4—61no—
158C4×Dic3—31no—
159C2×C3⋊C8—31no—
160C22×C32D3×D411yescharacteristic
161C2×C3×D3—121no—
162C2×C3×D3—121no—
163C2×C3×D3—121no—
164C32×C4—41no—
165C32×C4—41no—
166C32×C4#23511yescharacteristic
167C32×C4#23711yescharacteristic
168C3×Dic3—61no—
169C3×Dic3—61no—
170C3×Dic3—121no—
171C22×C9—21no—
172C22×C9C2×C3×D411yescharacteristic
173C2×D9—41no—
174C2×D9—41no—
175C2×D9—41no—
176C4×C9—21no—
177C4×C9—21no—
178C4×C9—42no—
179C4×C9—41no—
180C4×C9—41no—
181C4×C9—81no—
182C4×C9C3×#4111yescharacteristic
183C4×C9C3×#4111yescharacteristic
184Dic9—21no—
185Dic9—21no—
186Dic9—41no—
187#125—91no—
188#124—91no—
189#103—91no—
190#103AGL(1,ℤ/9)11yescharacteristic, complemented
191#102—91no—
192#102—91no—
193C4×C8—91no—
1943−1+2#48811yescharacteristic, complemented, Sylow
195C23×C3—181no—
196C22×D3—61no—
197C3×Q8—21no—
198C3×Q8—21no—
199C3×Q8—61no—
200C3×Q8—61no—
201C3×D4—181no—
202C3×D4—181no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—31no—
206C2×C3×C4—61no—
207C2×C3×C4—61no—
208C2×C3×C4—91no—
209C2×C3×C4—91no—
210C2×C3×C4—181no—
211C2×C3×C4C22×C3×D311yescharacteristic
212C2×Dic3—31no—
213C2×Dic3—31no—
214C2×Dic3—61no—
215D12—61no—
216D12—61no—
217C3×C8—21no—
218C3×C8—61no—
219C3×C8—181no—
220C3⋊C8—61no—
221C2×C32#88511yescharacteristic
222C2×C32#91611yescharacteristic
223C2×C32#91911yescharacteristic
224C3×D3—241no—
225C2×C9—21no—
226C2×C9—21no—
227C2×C9—21no—
228C2×C9C3×#12411yescharacteristic
229C2×C9C3×#13511yescharacteristic
230C2×C9C3×#13511yescharacteristic
231D9—81no—
232C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
233C2×D4—91no—
234C2×C8—91no—
235C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
236C4⋊C4—91no—
237C4⋊C4—181no—
238C4⋊C4—181no—
239C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
240C22⋊C4—181no—
241C22⋊C4—181no—
242C42—91no—
243C22×C3—31no—
244C22×C3—361no—
245C22×C3—361no—
246C22×C3—361no—
247C22×C3C3×D3×D411yescharacteristic
248C2×D3—121no—
249C2×D3—121no—
250C2×D3—121no—
251C3×C4—31no—
252C3×C4—31no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—121no—
256C3×C4—121no—
257C3×C4—181no—
258C3×C4—181no—
259C3×C4—361no—
260C3×C4C3×#23511yescharacteristic
261C3×C4C3×#23711yescharacteristic
262Dic3—61no—
263Dic3—61no—
264Dic3—121no—
265C32#480011yescharacteristic
266C9—21no—
267C9C3×#48811yescharacteristic, complemented
268C23—181no—
269Q8—21no—
270Q8—21no—
271D4—181no—
272D4—181no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—91no—
276C2×C4—91no—
277C2×C4—181no—
278C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
279C8—21no—
280C8—181no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—31no—
284C2×C3—721no—
285C2×C3C3×#88511yescharacteristic
286C2×C3C3×#91611yescharacteristic
287C2×C3C3×#91911yescharacteristic
288D3—241no—
289C22—361no—
290C22—361no—
291C22—361no—
292C22D4×AGL(1,ℤ/9)11yescharacteristic
293C4—41no—
294C4—41no—
295C4—181no—
296C4—181no—
297C4—361no—
298C4#3697⋊C311yescharacteristic
299C4#3699⋊C311yescharacteristic
300C3—31no—
301C3C3×#480011yescharacteristic
302C2—721no—
303C2#63447⋊C311yescharacteristic
304C2#63478⋊C311yescharacteristic
305C2#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 radicalC2×C4⋊AGL(1,ℤ/9)C22
hypercenter#103AGL(1,ℤ/9)
hypocenterC9C3×#488
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
2. #63452⋊C3C2semidirect
3. 3−1+2×#103C2semidirect
4. #63395⋊C3C2semidirect
6. C8×3−1+2⋊C4C2semidirect
9. #10588480C3semidirect
29. #63452C2×C3semidirect
31. C9×#103C2×C3semidirect
32. #63395C2×C3semidirect
34. C8×Dic9C2×C3semidirect
190. #103AGL(1,ℤ/9)semidirect
194. 3−1+2#488semidirect
267. C9C3×#488semidirect

Maximal subgroups

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

subgroupindex
#63509⋊C32
#63452⋊C32
3−1+2×#1032
#63395⋊C32
#63380⋊C32
C8×3−1+2⋊C42
#63369⋊C32
C3×#48003
#105884803