Finite Group Databasebeta

← #10588729⋊C3 #13142894

Subgroups — #10588729⋊C3 #13142894

Order 1728.

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

normal subgroups only

This group has 601 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
23−1+2×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#3675⋊C3C212yesmaximal, complemented
4#63389⋊C3C211yesmaximal, characteristic, complemented
5#63369⋊C3C212yesmaximal, complemented
6C3×#5046—31nomaximal
7#10588729C311yesmaximal, characteristic, complemented
8C23×AGL(1,ℤ/9)—21no—
9C2×Q8×3−1+2C2211yescharacteristic, complemented
10C22×C4×3−1+2C2211yescharacteristic
11C2×C4⋊AGL(1,ℤ/9)—21no—
12C2×C4⋊AGL(1,ℤ/9)—24no—
13C2×C4⋊AGL(1,ℤ/9)C2212yes—
14C4⋊C4×3−1+2—21no—
15C4⋊C4×3−1+2C2211yescharacteristic, complemented
16C22⋊C4×3−1+2—21no—
17#3675⋊C3—24no—
18C2×3−1+2⋊C8C2212yescomplemented
19C22×C3×D12—31no—
20C32×C22⋊Q8D311yescharacteristic
21C2×C3×#215—32no—
22C22×D36C2×C311yescharacteristic
23C3×#830—31no—
24C3×#809—32no—
25C9×C22⋊Q8—21no—
26C9×C22⋊Q8C2×C311yescharacteristic, complemented
27C2×#3675C2×C312yescomplemented
28#63389C2×C311yescharacteristic, complemented
29#63369C2×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—
35Q8×3−1+2—22no—
36C2×C4×3−1+2—22no—
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2D411yescharacteristic
40C2×C4×3−1+2C2311yescharacteristic
41C4⋊AGL(1,ℤ/9)—24no—
42C4⋊AGL(1,ℤ/9)—22no—
43C4⋊AGL(1,ℤ/9)D414yes—
443−1+2⋊C8—22no—
45C3×#450—91no—
46#5046—31no—
47C23×C3×D3—61no—
48C2×C32×Q8C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C2×C3×D12—32no—
51C2×C3×D12—64no—
52C2×C3×D12—61no—
53C23×D9—21no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4C2×D311yescharacteristic
56C32×C22⋊C4—21no—
57C3×#215—64no—
58C2×C3×C3⋊C8—32no—
59C2×Q8×C9—21no—
60C2×Q8×C9C22×C311yescharacteristic, complemented
61C22×C4×C9—21no—
62C22×C4×C9C22×C311yescharacteristic
63C2×D36—21no—
64C2×D36—24no—
65C2×D36C22×C312yes—
66C9×C4⋊C4—22no—
67C9×C4⋊C4—21no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4C22×C311yescharacteristic, complemented
70C9×C22⋊C4—22no—
71C9×C22⋊C4—21no—
72#3675—24no—
73C2×C9⋊C8C22×C312yescomplemented
74C22×3−1+2—21no—
75C22×3−1+2—21no—
76C22×3−1+2SD1612yescomplemented
77C22×3−1+2C2×D411yescharacteristic
78C2×AGL(1,ℤ/9)—24no—
79C2×AGL(1,ℤ/9)—48no—
80C2×AGL(1,ℤ/9)—42no—
81C2×AGL(1,ℤ/9)—42no—
82C4×3−1+2—21no—
83C4×3−1+2—41no—
84C4×3−1+2—41no—
85C4×3−1+2C2×D412yes—
86C22×C3×D4—91no—
87C22×D12—31no—
88C2×C3×SD16—92no—
89C3×C22⋊Q8—31no—
90C3×C22⋊Q8C3×D311yescharacteristic
91C2×#215—32no—
92C3×#102—92no—
93C3×C22⋊C8—91no—
94#830—31no—
95#809—32no—
96C23×C32#6711yescharacteristic
97C22×C3×D3—64no—
98C22×C3×D3—62no—
99C22×C3×D3—122no—
100C22×C3×D3—122no—
101C32×Q8—22no—
102C2×C32×C4—22no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4#6711yescharacteristic
106C2×C32×C4C22×D311yescharacteristic
107C3×D12—34no—
108C3×D12—64no—
109C3×D12—62no—
110C23×C9—21no—
111C23×C9C3×D411yescharacteristic
112C22×D9—24no—
113C22×D9—22no—
114C22×D9—42no—
115C22×D9—42no—
116C3×C3⋊C8—62no—
117Q8×C9—22no—
118Q8×C9—42no—
119C2×C4×C9—22no—
120C2×C4×C9—22no—
121C2×C4×C9—22no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—21no—
125C2×C4×C9—21no—
126C2×C4×C9—42no—
127C2×C4×C9C3×D411yescharacteristic
128C2×C4×C9C23×C311yescharacteristic
129D36—24no—
130D36—22no—
131D36C3×D414yes—
132C9⋊C8—22no—
133#450—91noSylow
134C2×3−1+2—22no—
135C2×3−1+2#12011yescharacteristic
136C2×3−1+2C2×SD1611yescharacteristic
137C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
138AGL(1,ℤ/9)—44no—
139C24×C3—181no—
140C23×D3—61no—
141C2×C3×Q8—31no—
142C2×C3×Q8C2×C3×D311yescharacteristic
143C2×C3×D4—92no—
144C2×C3×D4—181no—
145C2×C3×D4—184no—
146C22×C3×C4—31no—
147C22×C3×C4C2×C3×D311yescharacteristic
148C2×D12—32no—
149C2×D12—64no—
150C2×D12—61no—
151C3×SD16—184no—
152C2×C3×C8—92no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4C2×C3×D311yescharacteristic
157C3×C22⋊C4—21no—
158C3×C22⋊C4—61no—
159#215—64no—
160C2×C3⋊C8—32no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32#21512yes—
164C22×C32C2×#6711yescharacteristic
165C2×C3×D3—64no—
166C2×C3×D3—128no—
167C2×C3×D3—122no—
168C2×C3×D3—122no—
169C32×C4—21no—
170C32×C4—41no—
171C32×C4—41no—
172C32×C4D3×D412yes—
173C22×C9—21no—
174C22×C9—21no—
175C22×C9—22no—
176C22×C9—21no—
177C22×C9—41no—
178C22×C9—41no—
179C22×C9C3×SD1612yescomplemented
180C22×C9C2×C3×D411yescharacteristic
181C2×D9—24no—
182C2×D9—48no—
183C2×D9—42no—
184C2×D9—42no—
185C4×C9—21no—
186C4×C9—22no—
187C4×C9—41no—
188C4×C9—42no—
189C4×C9—42no—
190C4×C9—41no—
191C4×C9—41no—
192C4×C9C2×C3×D412yes—
193C22×D4—91no—
194C2×SD16—92no—
195C22⋊Q8AGL(1,ℤ/9)11yescharacteristic, complemented
196#102—92no—
197C22⋊C8—91no—
1983−1+2#45011yescharacteristic, complemented, Sylow
199C23×C3—31no—
200C23×C3—182no—
201C23×C3—184no—
202C23×C3—362no—
203C23×C3—362no—
204C23×C3C3×#6711yescharacteristic
205C22×D3—64no—
206C22×D3—62no—
207C22×D3—122no—
208C22×D3—122no—
209C3×Q8—22no—
210C3×Q8—62no—
211C3×D4—94no—
212C3×D4—182no—
213C3×D4—184no—
214C2×C3×C4—22no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—62no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4C3×#6711yescharacteristic
223C2×C3×C4C22×C3×D311yescharacteristic
224D12—34no—
225D12—64no—
226D12—62no—
227C3×C8—182no—
228C3⋊C8—62no—
229C2×C32—22no—
230C2×C32#93711yescharacteristic
231C2×C32C2×#21511yescharacteristic
232C2×C32#94911yescharacteristic
233C3×D3—124no—
234C2×C9—22no—
235C2×C9—21no—
236C2×C9—21no—
237C2×C9—21no—
238C2×C9—42no—
239C2×C9C3×#12011yescharacteristic
240C2×C9C2×C3×SD1611yescharacteristic
241C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
242D9—44no—
243C24—181no—
244C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic, complemented
245C2×D4—92no—
246C2×D4—184no—
247C2×D4—181no—
248C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
249SD16—184no—
250C2×C8—92no—
251C4⋊C4—21no—
252C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic, complemented
253C22⋊C4—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—31no—
257C22×C3—32no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3—184no—
261C22×C3—362no—
262C22×C3—362no—
263C22×C3—368no—
264C22×C3C3×#21512yes—
265C22×C3C2×C3×#6711yescharacteristic
266C2×D3—64no—
267C2×D3—128no—
268C2×D3—122no—
269C2×D3—122no—
270C3×C4—21no—
271C3×C4—32no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—61no—
275C3×C4—121no—
276C3×C4—121no—
277C3×C4C3×D3×D412yes—
278C32#504611yescharacteristic
279C9—21no—
280C9C3×#45011yescharacteristic, complemented
281C23—184no—
282C23—182no—
283C23—362no—
284C23—362no—
285C23#611⋊C311yescharacteristic
286Q8—22no—
287D4—94no—
288D4—184no—
289D4—182no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4#611⋊C311yescharacteristic
294C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
295C8—182no—
296C2×C3—22no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—62no—
301C2×C3—364no—
302C2×C3C3×#93711yescharacteristic
303C2×C3C2×C3×#21511yescharacteristic
304C2×C3C3×#94911yescharacteristic
305D3—124no—
306C22—21no—
307C22—21no—
308C22—184no—
309C22—368no—
310C22—362no—
311C22—362no—
312C22#3675⋊C312yescomplemented
313C22C2×#611⋊C311yescharacteristic
314C4—21no—
315C4—41no—
316C4—41no—
317C4D4×AGL(1,ℤ/9)12yes—
318C3—31no—
319C3C3×#504611yescharacteristic
320C2—22no—
321C2—364no—
322C2#63499⋊C311yescharacteristic
323C2C2×#3675⋊C311yescharacteristic
324C2#63512⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC22×C4⋊AGL(1,ℤ/9)C2
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#450
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. 3−1+2×C22⋊Q8C2semidirect
3. C2×#3675⋊C3C2semidirect
4. #63389⋊C3C2semidirect
5. #63369⋊C3C2semidirect
7. #10588729C3semidirect
9. C2×Q8×3−1+2C22semidirect
15. C4⋊C4×3−1+2C22semidirect
18. C2×3−1+2⋊C8C22semidirect
26. C9×C22⋊Q8C2×C3semidirect
27. C2×#3675C2×C3semidirect
28. #63389C2×C3semidirect
29. #63369C2×C3semidirect
60. C2×Q8×C9C22×C3semidirect
69. C9×C4⋊C4C22×C3semidirect
73. C2×C9⋊C8C22×C3semidirect
76. C22×3−1+2SD16semidirect
179. C22×C9C3×SD16semidirect
195. C22⋊Q8AGL(1,ℤ/9)semidirect
198. 3−1+2#450semidirect
244. C2×Q8C2×AGL(1,ℤ/9)semidirect
252. C4⋊C4C2×AGL(1,ℤ/9)semidirect
280. C9C3×#450semidirect
312. C22#3675⋊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
3−1+2×C22⋊Q82
C2×#3675⋊C32
#63389⋊C32
#63369⋊C32
C3×#50463
#105887293