Finite Group Databasebeta

← #10588413⋊C3 #13141637

Subgroups — #10588413⋊C3 #13141637

Order 1728.

589 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C22×C4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic
2#63486⋊C3C211yesmaximal, characteristic, complemented
3C2×#3664⋊C3C212yesmaximal, complemented
43−1+2×C22⋊C8C211yesmaximal, characteristic, complemented
5#63380⋊C3C212yesmaximal, complemented
6C3×#4731—31nomaximal
7#10588413C311yesmaximal, characteristic, complemented
8C23×AGL(1,ℤ/9)—21no—
9C22×C4×3−1+2C2211yescharacteristic
10C2×C4⋊AGL(1,ℤ/9)—21no—
11C2×C4⋊AGL(1,ℤ/9)—24no—
12C2×C4⋊AGL(1,ℤ/9)C2212yes—
13C2×3−1+2⋊Q8C2211yescharacteristic, complemented
14C2×C8×3−1+2C2212yescomplemented
15#3676⋊C3—21no—
16C4⋊(3−1+2⋊C4)C2211yescharacteristic, complemented
17(3−1+2⋊C4)⋊C4—21no—
18#3664⋊C3—24no—
19C22×C3×D12—31no—
20C3×#924—31no—
21C2×C3×#204—32no—
22C22×D36C2×C311yescharacteristic
23C32×C22⋊C8D311yescharacteristic
24C3×#821—32no—
25#63486C2×C311yescharacteristic, complemented
26C2×#3664C2×C312yescomplemented
27C9×C22⋊C8—21no—
28C9×C22⋊C8C2×C311yescharacteristic, complemented
29#63380C2×C312yescomplemented
30C23×3−1+2D411yescharacteristic
31C22×AGL(1,ℤ/9)—24no—
32C22×AGL(1,ℤ/9)—22no—
33C22×AGL(1,ℤ/9)—42no—
34C22×AGL(1,ℤ/9)—42no—
35C2×C4×3−1+2—22no—
36C2×C4×3−1+2D411yescharacteristic
37C2×C4×3−1+2C2311yescharacteristic
38C2×3−1+2⋊C4—21no—
39C2×3−1+2⋊C4—21no—
40C4⋊AGL(1,ℤ/9)—24no—
41C4⋊AGL(1,ℤ/9)—22no—
42C4⋊AGL(1,ℤ/9)D414yes—
433−1+2⋊Q8—22no—
44C8×3−1+2—22no—
45C3×#450—91no—
46#4731—31no—
47C23×C3×D3—61no—
48C22×C32×C4C2×D311yescharacteristic
49C2×C3×D12—32no—
50C2×C3×D12—64no—
51C2×C3×D12—61no—
52C2×C3×Dic6—31no—
53C23×D9—21no—
54C2×C32×C8C2×D312yes—
55C3×#217—61no—
56C3×C4⋊Dic3—31no—
57C3×Dic3⋊C4—61no—
58C3×#204—64no—
59C22×C4×C9—21no—
60C22×C4×C9C22×C311yescharacteristic
61C2×D36—21no—
62C2×D36—24no—
63C2×D36C22×C312yes—
64C2×Dic18C22×C311yescharacteristic, complemented
65C2×C8×C9—22no—
66C2×C8×C9C22×C312yescomplemented
67#3676—21no—
68C4⋊Dic9C22×C311yescharacteristic, complemented
69Dic9⋊C4—21no—
70#3664—24no—
71C22×3−1+2—21no—
72C22×3−1+2—21no—
73C22×3−1+2SD1612yescomplemented
74C22×3−1+2C2×D411yescharacteristic
75C2×AGL(1,ℤ/9)—24no—
76C2×AGL(1,ℤ/9)—48no—
77C2×AGL(1,ℤ/9)—42no—
78C2×AGL(1,ℤ/9)—42no—
79C4×3−1+2—21no—
80C4×3−1+2C2×D412yes—
813−1+2⋊C4—41no—
823−1+2⋊C4—41no—
83C22×C3×D4—91no—
84C22×D12—31no—
85C2×C3×SD16—92no—
86C3×C22⋊Q8—91no—
87#924—31no—
88C2×#204—32no—
89C3×#102—92no—
90C3×C22⋊C8—31no—
91C3×C22⋊C8C3×D311yescharacteristic
92#821—32no—
93C23×C32D1211yescharacteristic
94C22×C3×D3—64no—
95C22×C3×D3—62no—
96C22×C3×D3—122no—
97C22×C3×D3—122no—
98C2×C32×C4—22no—
99C2×C32×C4D1211yescharacteristic
100C2×C32×C4C22×D311yescharacteristic
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C3×D12—34no—
104C3×D12—64no—
105C3×D12—62no—
106C3×Dic6—62no—
107C23×C9—21no—
108C23×C9C3×D411yescharacteristic
109C22×D9—24no—
110C22×D9—22no—
111C22×D9—42no—
112C22×D9—42no—
113C32×C8—22no—
114C2×C4×C9—22no—
115C2×C4×C9—21no—
116C2×C4×C9—21no—
117C2×C4×C9—42no—
118C2×C4×C9C3×D411yescharacteristic
119C2×C4×C9C23×C311yescharacteristic
120C2×Dic9—21no—
121C2×Dic9—21no—
122D36—24no—
123D36—22no—
124D36C3×D414yes—
125Dic18—22no—
126C8×C9—22no—
127C8×C9—42no—
128#450—91noSylow
129C2×3−1+2—22no—
130C2×3−1+2#12011yescharacteristic
131C2×3−1+2C2×SD1611yescharacteristic
132C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
133AGL(1,ℤ/9)—44no—
134C24×C3—181no—
135C23×D3—61no—
136C2×C3×Q8—91no—
137C2×C3×D4—92no—
138C2×C3×D4—181no—
139C2×C3×D4—184no—
140C22×C3×C4—31no—
141C22×C3×C4C2×C3×D311yescharacteristic
142C2×D12—32no—
143C2×D12—64no—
144C2×D12—61no—
145C2×Dic6—31no—
146C3×SD16—184no—
147C2×C3×C8—32no—
148C2×C3×C8C2×C3×D312yes—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—181no—
151C3×C22⋊C4—181no—
152#217—61no—
153C4⋊Dic3—31no—
154Dic3⋊C4—61no—
155#204—64no—
156C22×C32—21no—
157C22×C32—21no—
158C22×C32#20412yes—
159C22×C32C2×D1211yescharacteristic
160C2×C3×D3—64no—
161C2×C3×D3—128no—
162C2×C3×D3—122no—
163C2×C3×D3—122no—
164C32×C4—21no—
165C32×C4D3×D412yes—
166C3×Dic3—121no—
167C3×Dic3—121no—
168C22×C9—21no—
169C22×C9—21no—
170C22×C9—22no—
171C22×C9—21no—
172C22×C9—41no—
173C22×C9—41no—
174C22×C9C3×SD1612yescomplemented
175C22×C9C2×C3×D411yescharacteristic
176C2×D9—24no—
177C2×D9—48no—
178C2×D9—42no—
179C2×D9—42no—
180C4×C9—21no—
181C4×C9—22no—
182C4×C9—41no—
183C4×C9C2×C3×D412yes—
184Dic9—41no—
185Dic9—41no—
186C22×D4—91no—
187C2×SD16—92no—
188C22⋊Q8—91no—
189#102—92no—
190C22⋊C8AGL(1,ℤ/9)11yescharacteristic, complemented
1913−1+2#45011yescharacteristic, complemented, Sylow
192C23×C3—31no—
193C23×C3—182no—
194C23×C3—184no—
195C23×C3—362no—
196C23×C3—362no—
197C23×C3C3×D1211yescharacteristic
198C22×D3—64no—
199C22×D3—62no—
200C22×D3—122no—
201C22×D3—122no—
202C3×Q8—182no—
203C3×D4—94no—
204C3×D4—182no—
205C3×D4—184no—
206C2×C3×C4—22no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—62no—
210C2×C3×C4—181no—
211C2×C3×C4—181no—
212C2×C3×C4C3×D1211yescharacteristic
213C2×C3×C4C22×C3×D311yescharacteristic
214C2×Dic3—61no—
215C2×Dic3—61no—
216D12—34no—
217D12—64no—
218D12—62no—
219Dic6—62no—
220C3×C8—22no—
221C3×C8—62no—
222C2×C32—22no—
223C2×C32#88211yescharacteristic
224C2×C32C2×#20411yescharacteristic
225C2×C32#90811yescharacteristic
226C3×D3—124no—
227C2×C9—22no—
228C2×C9—21no—
229C2×C9—21no—
230C2×C9—21no—
231C2×C9—42no—
232C2×C9C3×#12011yescharacteristic
233C2×C9C2×C3×SD1611yescharacteristic
234C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
235D9—44no—
236C24—181no—
237C2×Q8—91no—
238C2×D4—92no—
239C2×D4—184no—
240C2×D4—181no—
241C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
242SD16—184no—
243C2×C8C2×AGL(1,ℤ/9)12yescomplemented
244C4⋊C4—91no—
245C4⋊C4—181no—
246C22⋊C4—181no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—61no—
252C22×C3—61no—
253C22×C3—184no—
254C22×C3—362no—
255C22×C3—362no—
256C22×C3—368no—
257C22×C3C3×#20412yes—
258C22×C3C2×C3×D1211yescharacteristic
259C2×D3—64no—
260C2×D3—128no—
261C2×D3—122no—
262C2×D3—122no—
263C3×C4—21no—
264C3×C4—32no—
265C3×C4—61no—
266C3×C4—361no—
267C3×C4—361no—
268C3×C4C3×D3×D412yes—
269Dic3—121no—
270Dic3—121no—
271C32#473111yescharacteristic
272C9—21no—
273C9C3×#45011yescharacteristic, complemented
274C23—184no—
275C23—182no—
276C23—362no—
277C23—362no—
278C23C4⋊AGL(1,ℤ/9)11yescharacteristic
279Q8—182no—
280D4—94no—
281D4—184no—
282D4—182no—
283C2×C4—22no—
284C2×C4—181no—
285C2×C4—181no—
286C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
287C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
288C8—22no—
289C2×C3—22no—
290C2×C3—31no—
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—62no—
294C2×C3—364no—
295C2×C3C3×#88211yescharacteristic
296C2×C3C2×C3×#20411yescharacteristic
297C2×C3C3×#90811yescharacteristic
298D3—124no—
299C22—21no—
300C22—21no—
301C22—184no—
302C22—368no—
303C22—362no—
304C22—362no—
305C22#3664⋊C312yescomplemented
306C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
307C4—21no—
308C4—361no—
309C4—361no—
310C4D4×AGL(1,ℤ/9)12yes—
311C3—31no—
312C3C3×#473111yescharacteristic
313C2—22no—
314C2—364no—
315C2#63444⋊C311yescharacteristic
316C2C2×#3664⋊C311yescharacteristic
317C2#63470⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC22×C4⋊AGL(1,ℤ/9)C2
hypercenterC22⋊C8AGL(1,ℤ/9)
hypocenterC9C3×#450
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #63486⋊C3C2semidirect
3. C2×#3664⋊C3C2semidirect
4. 3−1+2×C22⋊C8C2semidirect
5. #63380⋊C3C2semidirect
7. #10588413C3semidirect
13. C2×3−1+2⋊Q8C22semidirect
14. C2×C8×3−1+2C22semidirect
16. C4⋊(3−1+2⋊C4)C22semidirect
25. #63486C2×C3semidirect
26. C2×#3664C2×C3semidirect
28. C9×C22⋊C8C2×C3semidirect
29. #63380C2×C3semidirect
64. C2×Dic18C22×C3semidirect
66. C2×C8×C9C22×C3semidirect
68. C4⋊Dic9C22×C3semidirect
73. C22×3−1+2SD16semidirect
174. C22×C9C3×SD16semidirect
190. C22⋊C8AGL(1,ℤ/9)semidirect
191. 3−1+2#450semidirect
243. C2×C8C2×AGL(1,ℤ/9)semidirect
273. C9C3×#450semidirect
305. C22#3664⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊AGL(1,ℤ/9)2
#63486⋊C32
C2×#3664⋊C32
3−1+2×C22⋊C82
#63380⋊C32
C3×#47313
#105884133