Finite Group Databasebeta

← #10589716⋊C3 #13152885

Subgroups — #10589716⋊C3 #13152885

Order 1728.

761 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
13−1+2×#121C211yesmaximal, characteristic, complemented
2#63500⋊C3C211yesmaximal, characteristic, complemented
3#63500⋊C3C212yesmaximal, complemented
4#63497⋊C3C212yesmaximal, complemented
5D4×3−1+2⋊C4C211yesmaximal, characteristic, complemented
6C4×#611⋊C3C211yesmaximal, characteristic, complemented
7#63488⋊C3C211yesmaximal, characteristic, complemented
8C4⋊C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
9#63452⋊C3C211yesmaximal, characteristic, complemented
10(#611⋊C3)⋊C4C212yesmaximal, complemented
11#63441⋊C3C212yesmaximal, complemented
12C3×#5614—31nomaximal
13#10589716C311yesmaximal, characteristic, complemented
14C2×D4×3−1+2C2211yescharacteristic
15C2×D4×3−1+2C2212yes—
16C22×C4×3−1+2C2211yescharacteristic
17C2×#611⋊C3C2211yescharacteristic
18C2×#611⋊C3C2212yes—
19C22×3−1+2⋊C4C2212yes—
20C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
21C2×C4×AGL(1,ℤ/9)C2212yes—
22C4⋊C4×3−1+2C2211yescharacteristic
23C22⋊C4×3−1+2C2212yes—
24#3676⋊C3C2212yes—
25#3676⋊C3C2211yescharacteristic
26#3676⋊C3C2212yes—
27#3676⋊C3C2212yes—
28(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
29C4⋊(3−1+2⋊C4)C2211yescharacteristic
30C4⋊(3−1+2⋊C4)C2212yes—
31C4⋊(3−1+2⋊C4)C2211yescharacteristic
32(3−1+2⋊C4)⋊C4C2212yes—
33(3−1+2⋊C4)⋊C4C2212yes—
34(3−1+2⋊C4)⋊C4C2211yescharacteristic
35C4×3−1+2⋊C4C2211yescharacteristic
36C4×3−1+2⋊C4C2212yes—
37C32×#121D311yescharacteristic
38C3×#938—32no—
39C3×#938—31no—
40C3×#935—32no—
41C3×D4×Dic3—31no—
42C3×C4×#67—31no—
43C3×#926—31no—
44C3×D3×C4⋊C4—31no—
45C3×#890—31no—
46C3×#67⋊C4—32no—
47C3×#879—32no—
48C9×#121—21no—
49C9×#121C2×C311yescharacteristic, complemented
50#63500C2×C311yescharacteristic, complemented
51#63500C2×C312yescomplemented
52#63497C2×C312yescomplemented
53D4×Dic9C2×C311yescharacteristic, complemented
54C4×#611C2×C311yescharacteristic, complemented
55#63488C2×C311yescharacteristic, complemented
56C4⋊C4×D9C2×C311yescharacteristic, complemented
57#63452C2×C311yescharacteristic, complemented
58#611⋊C4C2×C312yescomplemented
59#63441C2×C312yescomplemented
60C23×3−1+2C2312yes—
61C23×3−1+2C2311yescharacteristic
62C22×AGL(1,ℤ/9)C2311yescharacteristic
63D4×3−1+2—22no—
64D4×3−1+2—42no—
65C2×C4×3−1+2—22no—
66C2×C4×3−1+2C2312yes—
67C2×C4×3−1+2C2311yescharacteristic
68C2×C4×3−1+2C2311yescharacteristic
69#611⋊C3—22no—
70#611⋊C3—24no—
71C2×3−1+2⋊C4—42no—
72C2×3−1+2⋊C4C2312yes—
73C2×3−1+2⋊C4C2311yescharacteristic
74C2×3−1+2⋊C4C2311yescharacteristic
75C2×3−1+2⋊C4C2311yescharacteristic
76C2×3−1+2⋊C4C2312yes—
77C4×AGL(1,ℤ/9)—22no—
78C4×AGL(1,ℤ/9)—42no—
79C3×#553—91no—
80#5614—31no—
81C2×C32×D4C2×D311yescharacteristic
82C2×C32×D4C2×D312yes—
83C22×C32×C4C2×D311yescharacteristic
84C2×C3×#67—32no—
85C2×C3×#67—31no—
86C22×C3×Dic3—32no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C22⋊C4C2×D312yes—
91C3×#217—32no—
92C3×#217—32no—
93C3×#217—32no—
94C3×#217—31no—
95C3×(C2×D3)⋊C4—31no—
96C3×C4⋊Dic3—32no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—31no—
99C3×Dic3⋊C4—31no—
100C3×Dic3⋊C4—32no—
101C3×Dic3⋊C4—32no—
102C3×C4×Dic3—32no—
103C3×C4×Dic3—31no—
104C2×D4×C9—21no—
105C2×D4×C9—22no—
106C2×D4×C9C22×C311yescharacteristic
107C2×D4×C9C22×C312yes—
108C22×C4×C9—21no—
109C22×C4×C9C22×C311yescharacteristic
110C2×#611C22×C311yescharacteristic
111C2×#611C22×C312yes—
112C22×Dic9C22×C312yes—
113C2×C4×D9C22×C311yescharacteristic
114C2×C4×D9C22×C312yes—
115C9×C4⋊C4—21no—
116C9×C4⋊C4C22×C311yescharacteristic
117C9×C22⋊C4—22no—
118C9×C22⋊C4C22×C312yes—
119#3676C22×C312yes—
120#3676C22×C311yescharacteristic
121#3676C22×C312yes—
122#3676C22×C312yes—
123(C2×D9)⋊C4C22×C311yescharacteristic
124C4⋊Dic9C22×C311yescharacteristic
125C4⋊Dic9C22×C311yescharacteristic
126C4⋊Dic9C22×C312yes—
127Dic9⋊C4C22×C311yescharacteristic
128Dic9⋊C4C22×C312yes—
129Dic9⋊C4C22×C312yes—
130C4×Dic9C22×C312yes—
131C4×Dic9C22×C311yescharacteristic
132C22×3−1+2—21no—
133C22×3−1+2—22no—
134C22×3−1+2—21no—
135C22×3−1+2—22no—
136C22×3−1+2—21no—
137C22×3−1+2—22no—
138C22×3−1+2C2411yescharacteristic
139C2×AGL(1,ℤ/9)—21no—
140C2×AGL(1,ℤ/9)—21no—
141C2×AGL(1,ℤ/9)#4112yes—
142C4×3−1+2—21no—
143C4×3−1+2—22no—
144C4×3−1+2#4112yes—
1453−1+2⋊C4—21no—
1463−1+2⋊C4—21no—
1473−1+2⋊C4—22no—
1483−1+2⋊C4—22no—
1493−1+2⋊C4—21no—
150C3×#126—92no—
151C3×#125—91no—
152C3×#123—92no—
153C3×#121—31no—
154C3×#121—92no—
155C3×#121—91no—
156C3×#121C3×D311yescharacteristic
157C3×C4×D4—92no—
158C3×C4×D4—91no—
159C3×C4×D4—91no—
160C3×#117—91no—
161C2×C3×C4⋊C4—91no—
162#938—32no—
163#938—31no—
164#935—32no—
165D4×Dic3—31no—
166C4×#67—31no—
167#926—31no—
168D3×C4⋊C4—31no—
169#890—31no—
170#67⋊C4—32no—
171#879—32no—
172C23×C32C22×D311yescharacteristic
173C23×C32C22×D312yes—
174C22×C3×D3—31no—
175C32×D4—22no—
176C32×D4—42no—
177C2×C32×C4—22no—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D312yes—
180C2×C32×C4C22×D311yescharacteristic
181C3×#67—64no—
182C3×#67—62no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—122no—
189C3×C4×D3—62no—
190C3×C4×D3—122no—
191C23×C9—21no—
192C23×C9—22no—
193C23×C9C23×C311yescharacteristic
194C23×C9C23×C312yes—
195C22×D9C23×C311yescharacteristic
196D4×C9—22no—
197D4×C9—42no—
198D4×C9—42no—
199D4×C9—44no—
200C2×C4×C9—22no—
201C2×C4×C9—21no—
202C2×C4×C9—21no—
203C2×C4×C9—22no—
204C2×C4×C9—42no—
205C2×C4×C9C23×C311yescharacteristic
206C2×C4×C9C23×C312yes—
207C2×C4×C9C23×C311yescharacteristic
208#611—22no—
209#611—24no—
210C2×Dic9—42no—
211C2×Dic9C23×C311yescharacteristic
212C2×Dic9C23×C312yes—
213C2×Dic9C23×C311yescharacteristic
214C2×Dic9C23×C312yes—
215C2×Dic9C23×C311yescharacteristic
216C4×D9—22no—
217C4×D9—42no—
218#553—91noSylow
219C2×3−1+2—41no—
220C2×3−1+2—42no—
221C2×3−1+2C2×#4111yescharacteristic
222C2×3−1+2C2×#4111yescharacteristic
223C2×3−1+2H(5,𝔽2)11yescharacteristic
224AGL(1,ℤ/9)—22no—
225C2×C3×D4—31no—
226C2×C3×D4—32no—
227C2×C3×D4—92no—
228C2×C3×D4—91no—
229C2×C3×D4C2×C3×D311yescharacteristic
230C2×C3×D4C2×C3×D312yes—
231C22×C3×C4—31no—
232C22×C3×C4—91no—
233C22×C3×C4—92no—
234C22×C3×C4—92no—
235C22×C3×C4C2×C3×D311yescharacteristic
236C2×#67—32no—
237C2×#67—31no—
238C22×Dic3—32no—
239C2×C4×D3—31no—
240C2×C4×D3—32no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4C2×C3×D311yescharacteristic
249C3×C22⋊C4—32no—
250C3×C22⋊C4—91no—
251C3×C22⋊C4—92no—
252C3×C22⋊C4—92no—
253C3×C22⋊C4—91no—
254C3×C22⋊C4—92no—
255C3×C22⋊C4C2×C3×D312yes—
256C3×C42—92no—
257C3×C42—91no—
258#217—32no—
259#217—32no—
260#217—32no—
261#217—31no—
262(C2×D3)⋊C4—31no—
263C4⋊Dic3—32no—
264C4⋊Dic3—31no—
265C4⋊Dic3—31no—
266Dic3⋊C4—31no—
267Dic3⋊C4—32no—
268Dic3⋊C4—32no—
269C4×Dic3—32no—
270C4×Dic3—31no—
271C22×C32—21no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—22no—
277C22×C32C23×D311yescharacteristic
278C2×C3×D3—32no—
279C2×C3×D3—61no—
280C2×C3×D3—61no—
281C32×C4—21no—
282C32×C4—22no—
283C32×C4#23712yes—
284C3×Dic3—61no—
285C3×Dic3—62no—
286C3×Dic3—61no—
287C3×Dic3—62no—
288C3×Dic3—61no—
289C22×C9—22no—
290C22×C9—21no—
291C22×C9—22no—
292C22×C9—21no—
293C22×C9—22no—
294C22×C9—22no—
295C22×C9—21no—
296C22×C9—21no—
297C22×C9—41no—
298C22×C9—41no—
299C22×C9—42no—
300C22×C9—42no—
301C22×C9—42no—
302C22×C9C24×C311yescharacteristic
303C2×D9—21no—
304C2×D9—21no—
305C2×D9C3×#4112yes—
306C4×C9—21no—
307C4×C9—22no—
308C4×C9—22no—
309C4×C9—41no—
310C4×C9—42no—
311C4×C9C3×#4112yes—
312Dic9—21no—
313Dic9—22no—
314Dic9—21no—
315Dic9—22no—
316Dic9—21no—
317#126—92no—
318#125—91no—
319#123—92no—
320#121—92no—
321#121—91no—
322#121AGL(1,ℤ/9)11yescharacteristic, complemented
323C4×D4—91no—
324C4×D4—91no—
325C4×D4—92no—
326#117—91no—
327C2×C4⋊C4—91no—
3283−1+2#55311yescharacteristic, complemented, Sylow
329C23×C3—32no—
330C23×C3—31no—
331C23×C3—91no—
332C23×C3C22×C3×D311yescharacteristic
333C23×C3C22×C3×D312yes—
334C22×D3—31no—
335C3×D4—22no—
336C3×D4—42no—
337C3×D4—62no—
338C3×D4—122no—
339C3×D4—184no—
340C3×D4—182no—
341C2×C3×C4—22no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—32no—
345C2×C3×C4—62no—
346C2×C3×C4—92no—
347C2×C3×C4—91no—
348C2×C3×C4—91no—
349C2×C3×C4—92no—
350C2×C3×C4—91no—
351C2×C3×C4—182no—
352C2×C3×C4—362no—
353C2×C3×C4—362no—
354C2×C3×C4C22×C3×D311yescharacteristic
355C2×C3×C4C22×C3×D312yes—
356C2×C3×C4C22×C3×D311yescharacteristic
357#67—64no—
358#67—62no—
359C2×Dic3—31no—
360C2×Dic3—32no—
361C2×Dic3—31no—
362C2×Dic3—32no—
363C2×Dic3—31no—
364C2×Dic3—122no—
365C4×D3—62no—
366C4×D3—122no—
367C2×C32—41no—
368C2×C32—42no—
369C2×C32C2×#23711yescharacteristic
370C2×C32#100411yescharacteristic
371C2×C32D3×#4111yescharacteristic
372C3×D3—62no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—42no—
377C2×C9—41no—
378C2×C9—42no—
379C2×C9—82no—
380C2×C9C2×C3×#4111yescharacteristic
381C2×C9C2×C3×#4111yescharacteristic
382C2×C9C3×H(5,𝔽2)11yescharacteristic
383D9—22no—
384C2×D4—91no—
385C2×D4—92no—
386C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
387C2×D4C2×AGL(1,ℤ/9)12yes—
388C22×C4—92no—
389C22×C4—91no—
390C22×C4—92no—
391C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
392C4⋊C4—91no—
393C4⋊C4—92no—
394C4⋊C4—91no—
395C4⋊C4—92no—
396C4⋊C4—91no—
397C4⋊C4—92no—
398C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
399C22⋊C4—92no—
400C22⋊C4—92no—
401C22⋊C4—91no—
402C22⋊C4—92no—
403C22⋊C4—91no—
404C22⋊C4C2×AGL(1,ℤ/9)12yes—
405C42—92no—
406C42—91no—
407C22×C3—21no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—22no—
413C22×C3—31no—
414C22×C3—61no—
415C22×C3—62no—
416C22×C3—62no—
417C22×C3—61no—
418C22×C3—62no—
419C22×C3—61no—
420C22×C3—92no—
421C22×C3—181no—
422C22×C3—181no—
423C22×C3C23×C3×D311yescharacteristic
424C2×D3—32no—
425C2×D3—61no—
426C2×D3—61no—
427C3×C4—21no—
428C3×C4—22no—
429C3×C4—32no—
430C3×C4—61no—
431C3×C4—62no—
432C3×C4—181no—
433C3×C4—182no—
434C3×C4—181no—
435C3×C4—182no—
436C3×C4—181no—
437C3×C4C3×#23712yes—
438Dic3—61no—
439Dic3—62no—
440Dic3—61no—
441Dic3—62no—
442Dic3—61no—
443C32#561411yescharacteristic
444C9—21no—
445C9C3×#55311yescharacteristic, complemented
446C23—91no—
447C23C22×AGL(1,ℤ/9)11yescharacteristic
448C23C22×AGL(1,ℤ/9)12yes—
449D4—22no—
450D4—42no—
451D4—182no—
452D4—184no—
453C2×C4—22no—
454C2×C4—92no—
455C2×C4—91no—
456C2×C4—92no—
457C2×C4—91no—
458C2×C4—91no—
459C2×C4—182no—
460C2×C4—362no—
461C2×C4—362no—
462C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
463C2×C4C22×AGL(1,ℤ/9)12yes—
464C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
465C2×C3—31no—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—41no—
469C2×C3—42no—
470C2×C3—121no—
471C2×C3—122no—
472C2×C3—182no—
473C2×C3C2×C3×#23711yescharacteristic
474C2×C3C3×#100411yescharacteristic
475C2×C3C3×D3×#4111yescharacteristic
476D3—62no—
477C22—21no—
478C22—22no—
479C22—22no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—92no—
484C22—181no—
485C22—181no—
486C22C23×AGL(1,ℤ/9)11yescharacteristic
487C4—21no—
488C4—22no—
489C4—181no—
490C4—182no—
491C4—181no—
492C4—182no—
493C4—181no—
494C4#3699⋊C312yes—
495C3—31no—
496C3C3×#561411yescharacteristic
497C2—41no—
498C2—42no—
499C2—182no—
500C2C2×#3699⋊C311yescharacteristic
501C2#63710⋊C311yescharacteristic
502C2#41×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#121C2
hypercenter#121AGL(1,ℤ/9)
hypocenterC9C3×#553
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#121C2semidirect
2. #63500⋊C3C2semidirect
3. #63500⋊C3C2semidirect
4. #63497⋊C3C2semidirect
5. D4×3−1+2⋊C4C2semidirect
6. C4×#611⋊C3C2semidirect
7. #63488⋊C3C2semidirect
8. C4⋊C4×AGL(1,ℤ/9)C2semidirect
9. #63452⋊C3C2semidirect
10. (#611⋊C3)⋊C4C2semidirect
11. #63441⋊C3C2semidirect
13. #10589716C3semidirect
49. C9×#121C2×C3semidirect
50. #63500C2×C3semidirect
51. #63500C2×C3semidirect
52. #63497C2×C3semidirect
53. D4×Dic9C2×C3semidirect
54. C4×#611C2×C3semidirect
55. #63488C2×C3semidirect
56. C4⋊C4×D9C2×C3semidirect
57. #63452C2×C3semidirect
58. #611⋊C4C2×C3semidirect
59. #63441C2×C3semidirect
322. #121AGL(1,ℤ/9)semidirect
328. 3−1+2#553semidirect
445. C9C3×#553semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1212
#63500⋊C32
#63497⋊C32
D4×3−1+2⋊C42
C4×#611⋊C32
#63488⋊C32
C4⋊C4×AGL(1,ℤ/9)2
#63452⋊C32
(#611⋊C3)⋊C42
#63441⋊C32
C3×#56143
#105897163