Finite Group Databasebeta

← #10589634⋊C3 #13152530

Subgroups — #10589634⋊C3 #13152530

Order 1728.

746 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
13−1+2×#117C211yesmaximal, characteristic, complemented
2#63492⋊C3C211yesmaximal, characteristic, complemented
3C4×#611⋊C3C211yesmaximal, characteristic, complemented
4#63457⋊C3C211yesmaximal, characteristic, complemented
5#63455⋊C3C211yesmaximal, characteristic, complemented
6#63454⋊C3C211yesmaximal, characteristic, complemented
7#63450⋊C3C211yesmaximal, characteristic, complemented
8#63447⋊C3C212yesmaximal, complemented
9#63446⋊C3C212yesmaximal, complemented
10#63437⋊C3C212yesmaximal, complemented
11#63434⋊C3C212yesmaximal, complemented
12C3×#5532—31nomaximal
13#10589634C311yesmaximal, characteristic, complemented
14C22×C4×3−1+2C2211yescharacteristic
15C2×#611⋊C3C2211yescharacteristic
16C2×#611⋊C3C2212yes—
17C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
18C2×C4⋊AGL(1,ℤ/9)C2212yes—
19C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
20C2×C4×AGL(1,ℤ/9)C2212yes—
21C2×3−1+2⋊Q8C2212yes—
22C4⋊C4×3−1+2C2211yescharacteristic
23C4⋊C4×3−1+2C2211yescharacteristic
24C22⋊C4×3−1+2C2212yes—
25C42×3−1+2C2212yes—
26#3676⋊C3C2211yescharacteristic
27(C2×AGL(1,ℤ/9))⋊C4C2212yes—
28(C2×AGL(1,ℤ/9))⋊C4C2212yes—
29(C2×AGL(1,ℤ/9))⋊C4C2212yes—
30(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
31(C2×AGL(1,ℤ/9))⋊C4C2212yes—
32C4⋊(3−1+2⋊C4)C2211yescharacteristic
33(3−1+2⋊C4)⋊C4C2212yes—
34(3−1+2⋊C4)⋊C4C2211yescharacteristic
35C4×3−1+2⋊C4C2211yescharacteristic
36C4×3−1+2⋊C4C2212yes—
37C32×#117D311yescharacteristic
38C3×#930—31no—
39C3×C4×#67—31no—
40C3×#895—31no—
41C3×#893—31no—
42C3×#892—31no—
43C3×#888—31no—
44C3×#885—32no—
45C3×#884—32no—
46C3×#875—32no—
47C3×#872—32no—
48C9×#117—21no—
49C9×#117C2×C311yescharacteristic, complemented
50#63492C2×C311yescharacteristic, complemented
51C4×#611C2×C311yescharacteristic, complemented
52#63457C2×C311yescharacteristic, complemented
53#63455C2×C311yescharacteristic, complemented
54#63454C2×C311yescharacteristic, complemented
55#63450C2×C311yescharacteristic, complemented
56#63447C2×C312yescomplemented
57#63446C2×C312yescomplemented
58#63437C2×C312yescomplemented
59#63434C2×C312yescomplemented
60C23×3−1+2C2311yescharacteristic
61C22×AGL(1,ℤ/9)C2312yes—
62C22×AGL(1,ℤ/9)C2311yescharacteristic
63C2×C4×3−1+2—22no—
64C2×C4×3−1+2C2312yes—
65C2×C4×3−1+2C2312yes—
66C2×C4×3−1+2C2311yescharacteristic
67C2×C4×3−1+2C2311yescharacteristic
68#611⋊C3—22no—
69#611⋊C3—42no—
70C2×3−1+2⋊C4C2311yescharacteristic
71C2×3−1+2⋊C4C2312yes—
72C2×3−1+2⋊C4C2311yescharacteristic
73C2×3−1+2⋊C4C2311yescharacteristic
74C4⋊AGL(1,ℤ/9)—22no—
75C4⋊AGL(1,ℤ/9)—42no—
76C4×AGL(1,ℤ/9)—22no—
77C4×AGL(1,ℤ/9)—24no—
783−1+2⋊Q8—42no—
79C3×#555—91no—
80#5532—31no—
81C22×C32×C4C2×D311yescharacteristic
82C2×C3×#67—31no—
83C2×C3×#67—32no—
84C2×C3×D12—31no—
85C2×C3×D12—32no—
86C2×C3×C4×D3—32no—
87C2×C3×C4×D3—31no—
88C2×C3×Dic6—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C22⋊C4C2×D312yes—
92C32×C42C2×D312yes—
93C3×#217—31no—
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4—32no—
96C3×(C2×D3)⋊C4—32no—
97C3×(C2×D3)⋊C4—31no—
98C3×(C2×D3)⋊C4—32no—
99C3×C4⋊Dic3—31no—
100C3×Dic3⋊C4—32no—
101C3×Dic3⋊C4—31no—
102C3×C4×Dic3—32no—
103C3×C4×Dic3—31no—
104C22×C4×C9—21no—
105C22×C4×C9C22×C311yescharacteristic
106C2×#611C22×C311yescharacteristic
107C2×#611C22×C312yes—
108C2×D36C22×C311yescharacteristic
109C2×D36C22×C312yes—
110C2×C4×D9C22×C311yescharacteristic
111C2×C4×D9C22×C312yes—
112C2×Dic18C22×C312yes—
113C9×C4⋊C4—21no—
114C9×C4⋊C4—21no—
115C9×C4⋊C4C22×C311yescharacteristic
116C9×C4⋊C4C22×C311yescharacteristic
117C9×C22⋊C4—22no—
118C9×C22⋊C4C22×C312yes—
119C42×C9—22no—
120C42×C9C22×C312yes—
121#3676C22×C311yescharacteristic
122(C2×D9)⋊C4C22×C311yescharacteristic
123(C2×D9)⋊C4C22×C312yes—
124(C2×D9)⋊C4C22×C312yes—
125(C2×D9)⋊C4C22×C312yes—
126(C2×D9)⋊C4C22×C312yes—
127C4⋊Dic9C22×C311yescharacteristic
128Dic9⋊C4C22×C311yescharacteristic
129Dic9⋊C4C22×C312yes—
130C4×Dic9C22×C311yescharacteristic
131C4×Dic9C22×C312yes—
132C22×3−1+2—21no—
133C22×3−1+2—21no—
134C22×3−1+2—21no—
135C22×3−1+2C2411yescharacteristic
136C2×AGL(1,ℤ/9)—21no—
137C2×AGL(1,ℤ/9)—22no—
138C2×AGL(1,ℤ/9)—22no—
139C2×AGL(1,ℤ/9)—21no—
140C2×AGL(1,ℤ/9)—21no—
141C2×AGL(1,ℤ/9)—22no—
142C4×3−1+2—21no—
143C4×3−1+2—22no—
144C4×3−1+2—22no—
145C4×3−1+2#4112yes—
1463−1+2⋊C4—21no—
1473−1+2⋊C4—21no—
1483−1+2⋊C4—22no—
1493−1+2⋊C4#4112yes—
150C3×C4⋊Q8—91no—
151C3×#124—92no—
152C3×#124—92no—
153C3×#121—92no—
154C3×#121—91no—
155C3×#121—91no—
156C3×C4×D4—91no—
157C3×C4×D4—91no—
158C3×#117—31no—
159C3×#117—92no—
160C3×#117—91no—
161C3×#117C3×D311yescharacteristic
162#930—31no—
163C4×#67—31no—
164#895—31no—
165#893—31no—
166#892—31no—
167#888—31no—
168#885—32no—
169#884—32no—
170#875—32no—
171#872—32no—
172C23×C32C22×D311yescharacteristic
173C22×C3×D3—31no—
174C22×C3×D3—32no—
175C2×C32×C4—22no—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D311yescharacteristic
180C3×#67—62no—
181C3×#67—122no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—31no—
186C3×D12—62no—
187C3×D12—122no—
188C3×C4×D3—64no—
189C3×C4×D3—62no—
190C3×Dic6—122no—
191C23×C9—21no—
192C23×C9C23×C311yescharacteristic
193C22×D9C23×C312yes—
194C22×D9C23×C311yescharacteristic
195C2×C4×C9—24no—
196C2×C4×C9—22no—
197C2×C4×C9—21no—
198C2×C4×C9—21no—
199C2×C4×C9—22no—
200C2×C4×C9—22no—
201C2×C4×C9C23×C312yes—
202C2×C4×C9C23×C311yescharacteristic
203C2×C4×C9C23×C312yes—
204C2×C4×C9C23×C311yescharacteristic
205#611—22no—
206#611—42no—
207C2×Dic9C23×C311yescharacteristic
208C2×Dic9C23×C312yes—
209C2×Dic9C23×C311yescharacteristic
210C2×Dic9C23×C311yescharacteristic
211D36—22no—
212D36—42no—
213C4×D9—22no—
214C4×D9—24no—
215Dic18—42no—
216#555—91noSylow
217C2×3−1+2—41no—
218C2×3−1+2C2×#4111yescharacteristic
219C2×3−1+2C2×#4111yescharacteristic
220C2×3−1+2H(5,𝔽2)11yescharacteristic
221AGL(1,ℤ/9)—42no—
222AGL(1,ℤ/9)—41no—
223C2×C3×Q8—92no—
224C2×C3×D4—91no—
225C2×C3×D4—92no—
226C2×C3×D4—91no—
227C2×C3×D4—92no—
228C22×C3×C4—31no—
229C22×C3×C4—91no—
230C22×C3×C4—92no—
231C22×C3×C4C2×C3×D311yescharacteristic
232C2×#67—31no—
233C2×#67—32no—
234C2×D12—31no—
235C2×D12—32no—
236C2×C4×D3—32no—
237C2×C4×D3—31no—
238C2×Dic6—32no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4C2×C3×D311yescharacteristic
245C3×C4⋊C4C2×C3×D311yescharacteristic
246C3×C22⋊C4—32no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249C3×C22⋊C4—92no—
250C3×C22⋊C4—91no—
251C3×C22⋊C4—92no—
252C3×C22⋊C4—91no—
253C3×C22⋊C4C2×C3×D312yes—
254C3×C42—32no—
255C3×C42—92no—
256C3×C42—91no—
257C3×C42C2×C3×D312yes—
258#217—31no—
259(C2×D3)⋊C4—32no—
260(C2×D3)⋊C4—32no—
261(C2×D3)⋊C4—31no—
262(C2×D3)⋊C4—32no—
263(C2×D3)⋊C4—32no—
264C4⋊Dic3—31no—
265Dic3⋊C4—31no—
266Dic3⋊C4—32no—
267C4×Dic3—32no—
268C4×Dic3—31no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32C23×D311yescharacteristic
273C2×C3×D3—61no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—61no—
277C2×C3×D3—61no—
278C2×C3×D3—62no—
279C32×C4—21no—
280C32×C4—22no—
281C32×C4—22no—
282C32×C4#23512yes—
283C3×Dic3—32no—
284C3×Dic3—61no—
285C3×Dic3—61no—
286C3×Dic3—62no—
287C22×C9—22no—
288C22×C9—21no—
289C22×C9—21no—
290C22×C9—21no—
291C22×C9—21no—
292C22×C9—41no—
293C22×C9—41no—
294C22×C9C24×C311yescharacteristic
295C2×D9—21no—
296C2×D9—22no—
297C2×D9—22no—
298C2×D9—21no—
299C2×D9—21no—
300C2×D9—22no—
301C4×C9—22no—
302C4×C9—22no—
303C4×C9—21no—
304C4×C9—22no—
305C4×C9—22no—
306C4×C9—42no—
307C4×C9—42no—
308C4×C9C3×#4112yes—
309Dic9—21no—
310Dic9—21no—
311Dic9—22no—
312Dic9C3×#4112yes—
313C4⋊Q8—91no—
314#124—92no—
315#124—92no—
316#121—91no—
317#121—91no—
318#121—92no—
319C4×D4—91no—
320C4×D4—91no—
321#117—91no—
322#117—92no—
323#117AGL(1,ℤ/9)11yescharacteristic, complemented
3243−1+2#55511yescharacteristic, complemented, Sylow
325C23×C3—31no—
326C23×C3—92no—
327C23×C3—91no—
328C23×C3C22×C3×D311yescharacteristic
329C22×D3—31no—
330C22×D3—32no—
331C3×Q8—362no—
332C3×D4—182no—
333C3×D4—182no—
334C3×D4—362no—
335C3×D4—362no—
336C2×C3×C4—22no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—32no—
340C2×C3×C4—32no—
341C2×C3×C4—62no—
342C2×C3×C4—91no—
343C2×C3×C4—91no—
344C2×C3×C4—91no—
345C2×C3×C4—92no—
346C2×C3×C4—182no—
347C2×C3×C4—184no—
348C2×C3×C4C22×C3×D312yes—
349C2×C3×C4C22×C3×D311yescharacteristic
350C2×C3×C4C22×C3×D312yes—
351C2×C3×C4C22×C3×D311yescharacteristic
352#67—62no—
353#67—122no—
354C2×Dic3—31no—
355C2×Dic3—31no—
356C2×Dic3—32no—
357C2×Dic3—31no—
358D12—62no—
359D12—122no—
360C4×D3—64no—
361C4×D3—62no—
362Dic6—122no—
363C2×C32—41no—
364C2×C32C2×#23511yescharacteristic
365C2×C32D3×#4111yescharacteristic
366C2×C32#100911yescharacteristic
367C3×D3—121no—
368C3×D3—122no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—42no—
373C2×C9—41no—
374C2×C9C2×C3×#4111yescharacteristic
375C2×C9C2×C3×#4111yescharacteristic
376C2×C9C3×H(5,𝔽2)11yescharacteristic
377D9—42no—
378D9—41no—
379C2×Q8—92no—
380C2×D4—91no—
381C2×D4—92no—
382C2×D4—91no—
383C2×D4—92no—
384C22×C4—92no—
385C22×C4—91no—
386C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
387C4⋊C4—91no—
388C4⋊C4—92no—
389C4⋊C4—91no—
390C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
391C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
392C22⋊C4—91no—
393C22⋊C4—92no—
394C22⋊C4—92no—
395C22⋊C4—92no—
396C22⋊C4—92no—
397C22⋊C4—91no—
398C22⋊C4C2×AGL(1,ℤ/9)12yes—
399C42—92no—
400C42—91no—
401C42C2×AGL(1,ℤ/9)12yes—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—31no—
406C22×C3—61no—
407C22×C3—61no—
408C22×C3—61no—
409C22×C3—181no—
410C22×C3—182no—
411C22×C3—182no—
412C22×C3—181no—
413C22×C3—181no—
414C22×C3—182no—
415C22×C3C23×C3×D311yescharacteristic
416C2×D3—61no—
417C2×D3—62no—
418C2×D3—62no—
419C2×D3—61no—
420C2×D3—61no—
421C2×D3—62no—
422C3×C4—21no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—32no—
426C3×C4—61no—
427C3×C4—62no—
428C3×C4—62no—
429C3×C4—92no—
430C3×C4—181no—
431C3×C4—182no—
432C3×C4—181no—
433C3×C4C3×#23512yes—
434Dic3—32no—
435Dic3—61no—
436Dic3—61no—
437Dic3—62no—
438C32#553211yescharacteristic
439C9—21no—
440C9C3×#55511yescharacteristic, complemented
441C23—91no—
442C23—92no—
443C23C22×AGL(1,ℤ/9)11yescharacteristic
444Q8—362no—
445D4—182no—
446D4—182no—
447D4—362no—
448D4—362no—
449C2×C4—22no—
450C2×C4—91no—
451C2×C4—91no—
452C2×C4—91no—
453C2×C4—92no—
454C2×C4—184no—
455C2×C4—182no—
456C2×C4C22×AGL(1,ℤ/9)12yes—
457C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
458C2×C4C22×AGL(1,ℤ/9)12yes—
459C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—41no—
464C2×C3—121no—
465C2×C3—361no—
466C2×C3—362no—
467C2×C3C2×C3×#23511yescharacteristic
468C2×C3C3×D3×#4111yescharacteristic
469C2×C3C3×#100911yescharacteristic
470D3—121no—
471D3—122no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—181no—
476C22—182no—
477C22—182no—
478C22—181no—
479C22—181no—
480C22—182no—
481C22C23×AGL(1,ℤ/9)11yescharacteristic
482C4—21no—
483C4—22no—
484C4—22no—
485C4—92no—
486C4—181no—
487C4—182no—
488C4—181no—
489C4#3697⋊C312yes—
490C3—31no—
491C3C3×#553211yescharacteristic
492C2—41no—
493C2—361no—
494C2—362no—
495C2C2×#3697⋊C311yescharacteristic
496C2#41×AGL(1,ℤ/9)11yescharacteristic
497C2#1172264111yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#117C2
hypercenter#117AGL(1,ℤ/9)
hypocenterC9C3×#555
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#117C2semidirect
2. #63492⋊C3C2semidirect
3. C4×#611⋊C3C2semidirect
4. #63457⋊C3C2semidirect
5. #63455⋊C3C2semidirect
6. #63454⋊C3C2semidirect
7. #63450⋊C3C2semidirect
8. #63447⋊C3C2semidirect
9. #63446⋊C3C2semidirect
10. #63437⋊C3C2semidirect
11. #63434⋊C3C2semidirect
13. #10589634C3semidirect
49. C9×#117C2×C3semidirect
50. #63492C2×C3semidirect
51. C4×#611C2×C3semidirect
52. #63457C2×C3semidirect
53. #63455C2×C3semidirect
54. #63454C2×C3semidirect
55. #63450C2×C3semidirect
56. #63447C2×C3semidirect
57. #63446C2×C3semidirect
58. #63437C2×C3semidirect
59. #63434C2×C3semidirect
323. #117AGL(1,ℤ/9)semidirect
324. 3−1+2#555semidirect
440. C9C3×#555semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1172
#63492⋊C32
C4×#611⋊C32
#63457⋊C32
#63455⋊C32
#63454⋊C32
#63450⋊C32
#63447⋊C32
#63446⋊C32
#63437⋊C32
#63434⋊C32
C3×#55323
#105896343