Finite Group Databasebeta

← #10588414⋊C3 #13141638

Subgroups — #10588414⋊C3 #13141638

Order 1728.

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

normal subgroups only

This group has 619 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, complemented
2#63492⋊C3C211yesmaximal, characteristic, complemented
3C2×D72⋊C3C212yesmaximal, complemented
43−1+2×C22⋊C8C211yesmaximal, characteristic, complemented
5#63380⋊C3C212yesmaximal, complemented
6C3×#4732—31nomaximal
7#10588414C311yesmaximal, characteristic, complemented
8C23×AGL(1,ℤ/9)—21no—
9C22×C4×3−1+2C2211yescharacteristic
10C2×#611⋊C3—21no—
11C2×C4⋊AGL(1,ℤ/9)—21no—
12C2×C4⋊AGL(1,ℤ/9)—24no—
13C2×C4⋊AGL(1,ℤ/9)C2212yes—
14C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic, complemented
15C2×C8×3−1+2C2212yescomplemented
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17C4⋊(3−1+2⋊C4)C2211yescharacteristic, complemented
18D72⋊C3—24no—
19C22×C3×D12—31no—
20C3×#930—31no—
21C2×C3×D24—32no—
22C22×D36C2×C311yescharacteristic, complemented
23C32×C22⋊C8D311yescharacteristic
24C3×#821—32no—
25#63492C2×C311yescharacteristic, complemented
26C2×D72C2×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)—21no—
34C22×AGL(1,ℤ/9)—42no—
35C22×AGL(1,ℤ/9)—42no—
36C2×C4×3−1+2—22no—
37C2×C4×3−1+2D411yescharacteristic
38C2×C4×3−1+2C2311yescharacteristic
39#611⋊C3—42no—
40C2×3−1+2⋊C4—21no—
41C4⋊AGL(1,ℤ/9)—24no—
42C4⋊AGL(1,ℤ/9)—22no—
43C4⋊AGL(1,ℤ/9)—22no—
44C4⋊AGL(1,ℤ/9)D414yescomplemented
45C8×3−1+2—22no—
46C3×#447—91no—
47#4732—31no—
48C23×C3×D3—61no—
49C22×C32×C4C2×D311yescharacteristic
50C2×C3×#67—61no—
51C2×C3×D12—32no—
52C2×C3×D12—31no—
53C2×C3×D12—64no—
54C2×C3×D12—61no—
55C23×D9—21no—
56C2×C32×C8C2×D312yes—
57C3×(C2×D3)⋊C4—61no—
58C3×C4⋊Dic3—31no—
59C3×D24—64no—
60C22×C4×C9—21no—
61C22×C4×C9C22×C311yescharacteristic
62C2×#611—21no—
63C2×D36—21no—
64C2×D36—24no—
65C2×D36C22×C312yes—
66C2×D36C22×C311yescharacteristic, complemented
67C2×C8×C9—22no—
68C2×C8×C9C22×C312yescomplemented
69(C2×D9)⋊C4—21no—
70C4⋊Dic9C22×C311yescharacteristic, complemented
71D72—24no—
72C22×3−1+2—21no—
73C22×3−1+2—21no—
74C22×3−1+2D812yescomplemented
75C22×3−1+2C2×D411yescharacteristic
76C2×AGL(1,ℤ/9)—24no—
77C2×AGL(1,ℤ/9)—48no—
78C2×AGL(1,ℤ/9)—42no—
79C2×AGL(1,ℤ/9)—42no—
80C2×AGL(1,ℤ/9)—41no—
81C2×AGL(1,ℤ/9)—41no—
82C2×AGL(1,ℤ/9)—41no—
83C4×3−1+2—21no—
84C4×3−1+2C2×D412yes—
853−1+2⋊C4—41no—
86C22×C3×D4—91no—
87C22×D12—31no—
88C2×C3×D8—92no—
89C3×#121—91no—
90#930—31no—
91C2×D24—32no—
92C3×#102—92no—
93C3×C22⋊C8—31no—
94C3×C22⋊C8C3×D311yescharacteristic
95#821—32no—
96C23×C32D1211yescharacteristic
97C22×C3×D3—64no—
98C22×C3×D3—62no—
99C22×C3×D3—61no—
100C22×C3×D3—122no—
101C22×C3×D3—122no—
102C2×C32×C4—22no—
103C2×C32×C4D1211yescharacteristic
104C2×C32×C4C22×D311yescharacteristic
105C3×#67—122no—
106C2×C3×Dic3—61no—
107C3×D12—34no—
108C3×D12—64no—
109C3×D12—62no—
110C3×D12—62no—
111C23×C9—21no—
112C23×C9C3×D411yescharacteristic
113C22×D9—24no—
114C22×D9—22no—
115C22×D9—21no—
116C22×D9—42no—
117C22×D9—42no—
118C32×C8—22no—
119C2×C4×C9—22no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—42no—
123C2×C4×C9C3×D411yescharacteristic
124C2×C4×C9C23×C311yescharacteristic
125#611—42no—
126C2×Dic9—21no—
127D36—24no—
128D36—22no—
129D36—22no—
130D36C3×D414yescomplemented
131C8×C9—22no—
132C8×C9—42no—
133#447—91noSylow
134C2×3−1+2—22no—
135C2×3−1+2#12011yescharacteristic
136C2×3−1+2C2×D811yescharacteristic
137C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
138AGL(1,ℤ/9)—44no—
139AGL(1,ℤ/9)—81no—
140C24×C3—181no—
141C23×D3—61no—
142C2×C3×D4—92no—
143C2×C3×D4—91no—
144C2×C3×D4—181no—
145C2×C3×D4—184no—
146C2×C3×D4—181no—
147C22×C3×C4—31no—
148C22×C3×C4C2×C3×D311yescharacteristic
149C2×#67—61no—
150C2×D12—32no—
151C2×D12—31no—
152C2×D12—64no—
153C2×D12—61no—
154C3×D8—184no—
155C2×C3×C8—32no—
156C2×C3×C8C2×C3×D312yes—
157C3×C4⋊C4—91no—
158C3×C22⋊C4—181no—
159(C2×D3)⋊C4—61no—
160C4⋊Dic3—31no—
161D24—64no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32D2412yes—
165C22×C32C2×D1211yescharacteristic
166C2×C3×D3—64no—
167C2×C3×D3—128no—
168C2×C3×D3—122no—
169C2×C3×D3—122no—
170C2×C3×D3—121no—
171C2×C3×D3—121no—
172C2×C3×D3—121no—
173C32×C4—21no—
174C32×C4D3×D412yes—
175C3×Dic3—121no—
176C22×C9—21no—
177C22×C9—21no—
178C22×C9—22no—
179C22×C9—21no—
180C22×C9—41no—
181C22×C9—41no—
182C22×C9C3×D812yescomplemented
183C22×C9C2×C3×D411yescharacteristic
184C2×D9—24no—
185C2×D9—48no—
186C2×D9—42no—
187C2×D9—42no—
188C2×D9—41no—
189C2×D9—41no—
190C2×D9—41no—
191C4×C9—21no—
192C4×C9—22no—
193C4×C9—41no—
194C4×C9C2×C3×D412yes—
195Dic9—41no—
196C22×D4—91no—
197C2×D8—92no—
198#121—91no—
199#102—92no—
200C22⋊C8AGL(1,ℤ/9)11yescharacteristic, complemented
2013−1+2#44711yescharacteristic, complemented, Sylow
202C23×C3—31no—
203C23×C3—182no—
204C23×C3—181no—
205C23×C3—184no—
206C23×C3—362no—
207C23×C3—362no—
208C23×C3C3×D1211yescharacteristic
209C22×D3—64no—
210C22×D3—62no—
211C22×D3—61no—
212C22×D3—122no—
213C22×D3—122no—
214C3×D4—94no—
215C3×D4—182no—
216C3×D4—182no—
217C3×D4—184no—
218C3×D4—362no—
219C2×C3×C4—22no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—62no—
223C2×C3×C4—181no—
224C2×C3×C4C3×D1211yescharacteristic
225C2×C3×C4C22×C3×D311yescharacteristic
226#67—122no—
227C2×Dic3—61no—
228D12—34no—
229D12—64no—
230D12—62no—
231D12—62no—
232C3×C8—22no—
233C3×C8—62no—
234C2×C32—22no—
235C2×C32#88211yescharacteristic
236C2×C32C2×D2411yescharacteristic
237C2×C32#90811yescharacteristic
238C3×D3—124no—
239C3×D3—241no—
240C2×C9—22no—
241C2×C9—21no—
242C2×C9—21no—
243C2×C9—21no—
244C2×C9—42no—
245C2×C9C3×#12011yescharacteristic
246C2×C9C2×C3×D811yescharacteristic
247C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
248D9—44no—
249D9—81no—
250C24—181no—
251C2×D4—92no—
252C2×D4—91no—
253C2×D4—184no—
254C2×D4—181no—
255C2×D4—181no—
256C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
257D8—184no—
258C2×C8C2×AGL(1,ℤ/9)12yescomplemented
259C4⋊C4—91no—
260C22⋊C4—181no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—31no—
264C22×C3—32no—
265C22×C3—61no—
266C22×C3—61no—
267C22×C3—184no—
268C22×C3—362no—
269C22×C3—362no—
270C22×C3—361no—
271C22×C3—361no—
272C22×C3—361no—
273C22×C3—368no—
274C22×C3C3×D2412yes—
275C22×C3C2×C3×D1211yescharacteristic
276C2×D3—64no—
277C2×D3—128no—
278C2×D3—122no—
279C2×D3—122no—
280C2×D3—121no—
281C2×D3—121no—
282C2×D3—121no—
283C3×C4—21no—
284C3×C4—32no—
285C3×C4—61no—
286C3×C4—361no—
287C3×C4C3×D3×D412yes—
288Dic3—121no—
289C32#473211yescharacteristic
290C9—21no—
291C9C3×#44711yescharacteristic, complemented
292C23—184no—
293C23—182no—
294C23—181no—
295C23—362no—
296C23—362no—
297C23C4⋊AGL(1,ℤ/9)11yescharacteristic
298D4—94no—
299D4—184no—
300D4—182no—
301D4—182no—
302D4—362no—
303C2×C4—22no—
304C2×C4—181no—
305C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
306C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
307C8—22no—
308C2×C3—22no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—62no—
313C2×C3—364no—
314C2×C3—721no—
315C2×C3C3×#88211yescharacteristic
316C2×C3C2×C3×D2411yescharacteristic
317C2×C3C3×#90811yescharacteristic
318D3—124no—
319D3—241no—
320C22—21no—
321C22—21no—
322C22—184no—
323C22—368no—
324C22—362no—
325C22—362no—
326C22—361no—
327C22—361no—
328C22—361no—
329C22D72⋊C312yescomplemented
330C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
331C4—21no—
332C4—361no—
333C4D4×AGL(1,ℤ/9)12yes—
334C3—31no—
335C3C3×#473211yescharacteristic
336C2—22no—
337C2—364no—
338C2—721no—
339C2#63444⋊C311yescharacteristic
340C2C2×D72⋊C311yescharacteristic
341C2#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
hypercenterC22⋊C8AGL(1,ℤ/9)
hypocenterC9C3×#447
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
1. C22×C4⋊AGL(1,ℤ/9)C2semidirect
2. #63492⋊C3C2semidirect
3. C2×D72⋊C3C2semidirect
4. 3−1+2×C22⋊C8C2semidirect
5. #63380⋊C3C2semidirect
7. #10588414C3semidirect
14. C2×C4⋊AGL(1,ℤ/9)C22semidirect
15. C2×C8×3−1+2C22semidirect
17. C4⋊(3−1+2⋊C4)C22semidirect
22. C22×D36C2×C3semidirect
25. #63492C2×C3semidirect
26. C2×D72C2×C3semidirect
28. C9×C22⋊C8C2×C3semidirect
29. #63380C2×C3semidirect
44. C4⋊AGL(1,ℤ/9)D4semidirect
66. C2×D36C22×C3semidirect
68. C2×C8×C9C22×C3semidirect
70. C4⋊Dic9C22×C3semidirect
74. C22×3−1+2D8semidirect
130. D36C3×D4semidirect
182. C22×C9C3×D8semidirect
200. C22⋊C8AGL(1,ℤ/9)semidirect
201. 3−1+2#447semidirect
258. C2×C8C2×AGL(1,ℤ/9)semidirect
291. C9C3×#447semidirect
329. C22D72⋊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
#63492⋊C32
C2×D72⋊C32
3−1+2×C22⋊C82
#63380⋊C32
C3×#47323
#105884143