Finite Group Databasebeta

← #10589660⋊C3 #13152660

Subgroups — #10589660⋊C3 #13152660

Order 1728.

761 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×3−1+2C211yesmaximal, characteristic, complemented
2#63500⋊C3C211yesmaximal, characteristic, complemented
3#63500⋊C3C211yesmaximal, characteristic, complemented
4#63491⋊C3C212yesmaximal, complemented
5C4×#611⋊C3C212yesmaximal, complemented
6#63454⋊C3C211yesmaximal, characteristic, complemented
7C4⋊C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
8#63446⋊C3C212yesmaximal, complemented
9#63441⋊C3C212yesmaximal, complemented
10C4×C4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
11#63432⋊C3C211yesmaximal, characteristic, complemented
12C3×#5558—31nomaximal
13#10589660C311yesmaximal, characteristic, complemented
14C2×D4×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2212yes—
16C2×#611⋊C3C2212yes—
17C2×#611⋊C3C2212yes—
18C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
19C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
20C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
21C2×C4×AGL(1,ℤ/9)C2212yes—
22C4⋊C4×3−1+2C2211yescharacteristic
23C22⋊C4×3−1+2C2212yes—
24C42×3−1+2C2211yescharacteristic
25#3676⋊C3C2212yes—
26#3676⋊C3C2212yes—
27(C2×AGL(1,ℤ/9))⋊C4C2212yes—
28(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
29(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
30C4⋊(3−1+2⋊C4)C2211yescharacteristic
31C4⋊(3−1+2⋊C4)C2211yescharacteristic
32C4⋊(3−1+2⋊C4)C2211yescharacteristic
33(3−1+2⋊C4)⋊C4C2212yes—
34(3−1+2⋊C4)⋊C4C2212yes—
35(3−1+2⋊C4)⋊C4C2212yes—
36C4×3−1+2⋊C4C2212yes—
37C32×C4×D4D311yescharacteristic
38C3×#938—31no—
39C3×#938—31no—
40C3×#929—32no—
41C3×C4×#67—32no—
42C3×#892—31no—
43C3×D3×C4⋊C4—31no—
44C3×#884—32no—
45C3×#879—32no—
46C3×C4×D12—31no—
47C3×#870—31no—
48C4×D4×C9—21no—
49C4×D4×C9C2×C311yescharacteristic, complemented
50#63500C2×C311yescharacteristic, complemented
51#63500C2×C311yescharacteristic, complemented
52#63491C2×C312yescomplemented
53C4×#611C2×C312yescomplemented
54#63454C2×C311yescharacteristic, complemented
55C4⋊C4×D9C2×C311yescharacteristic, complemented
56#63446C2×C312yescomplemented
57#63441C2×C312yescomplemented
58C4×D36C2×C311yescharacteristic, complemented
59#63432C2×C311yescharacteristic, complemented
60C23×3−1+2C2312yes—
61C22×AGL(1,ℤ/9)C2311yescharacteristic
62C22×AGL(1,ℤ/9)C2311yescharacteristic
63D4×3−1+2—22no—
64C2×C4×3−1+2—42no—
65C2×C4×3−1+2C2312yes—
66C2×C4×3−1+2C2311yescharacteristic
67C2×C4×3−1+2C2311yescharacteristic
68C2×C4×3−1+2C2311yescharacteristic
69#611⋊C3—24no—
70#611⋊C3—42no—
71C2×3−1+2⋊C4C2312yes—
72C2×3−1+2⋊C4C2311yescharacteristic
73C2×3−1+2⋊C4C2311yescharacteristic
74C2×3−1+2⋊C4C2312yes—
75C4⋊AGL(1,ℤ/9)—22no—
76C4×AGL(1,ℤ/9)—22no—
77C4×AGL(1,ℤ/9)—22no—
78C4×AGL(1,ℤ/9)—42no—
79C3×#553—91no—
80#5558—31no—
81C2×C32×D4C2×D311yescharacteristic
82C22×C32×C4C2×D312yes—
83C2×C3×#67—32no—
84C2×C3×#67—32no—
85C2×C3×D12—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C22⋊C4C2×D312yes—
91C32×C42C2×D311yescharacteristic
92C3×#217—32no—
93C3×#217—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4—31no—
96C3×(C2×D3)⋊C4—31no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—31no—
100C3×Dic3⋊C4—32no—
101C3×Dic3⋊C4—32no—
102C3×Dic3⋊C4—32no—
103C3×C4×Dic3—32no—
104C2×D4×C9—21no—
105C2×D4×C9C22×C311yescharacteristic
106C22×C4×C9—22no—
107C22×C4×C9C22×C312yes—
108C2×#611C22×C312yes—
109C2×#611C22×C312yes—
110C2×D36C22×C311yescharacteristic
111C2×C4×D9C22×C311yescharacteristic
112C2×C4×D9C22×C311yescharacteristic
113C2×C4×D9C22×C312yes—
114C9×C4⋊C4—21no—
115C9×C4⋊C4C22×C311yescharacteristic
116C9×C22⋊C4—22no—
117C9×C22⋊C4C22×C312yes—
118C42×C9—21no—
119C42×C9C22×C311yescharacteristic
120#3676C22×C312yes—
121#3676C22×C312yes—
122(C2×D9)⋊C4C22×C311yescharacteristic
123(C2×D9)⋊C4C22×C311yescharacteristic
124(C2×D9)⋊C4C22×C312yes—
125C4⋊Dic9C22×C311yescharacteristic
126C4⋊Dic9C22×C311yescharacteristic
127C4⋊Dic9C22×C311yescharacteristic
128Dic9⋊C4C22×C312yes—
129Dic9⋊C4C22×C312yes—
130Dic9⋊C4C22×C312yes—
131C4×Dic9C22×C312yes—
132C22×3−1+2—22no—
133C22×3−1+2—22no—
134C22×3−1+2—22no—
135C22×3−1+2C2411yescharacteristic
136C2×AGL(1,ℤ/9)—21no—
137C2×AGL(1,ℤ/9)—21no—
138C2×AGL(1,ℤ/9)—21no—
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—21no—
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—22no—
150C3×#126—92no—
151C3×#125—91no—
152C3×#123—92no—
153C3×#121—92no—
154C3×#121—91no—
155C3×#121—91no—
156C3×C4×D4—31no—
157C3×C4×D4—91no—
158C3×C4×D4—92no—
159C3×C4×D4C3×D311yescharacteristic
160C3×#117—91no—
161C2×C3×C4⋊C4—91no—
162#938—31no—
163#938—31no—
164#929—32no—
165C4×#67—32no—
166#892—31no—
167D3×C4⋊C4—31no—
168#884—32no—
169#879—32no—
170C4×D12—31no—
171#870—31no—
172C23×C32C22×D312yes—
173C22×C3×D3—31no—
174C22×C3×D3—31no—
175C32×D4—22no—
176C2×C32×C4—42no—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C3×#67—64no—
182C3×#67—122no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—31no—
187C3×D12—62no—
188C3×C4×D3—62no—
189C3×C4×D3—62no—
190C3×C4×D3—122no—
191C23×C9—22no—
192C23×C9C23×C312yes—
193C22×D9C23×C311yescharacteristic
194C22×D9C23×C311yescharacteristic
195D4×C9—24no—
196D4×C9—22no—
197C2×C4×C9—21no—
198C2×C4×C9—21no—
199C2×C4×C9—21no—
200C2×C4×C9—22no—
201C2×C4×C9—42no—
202C2×C4×C9—44no—
203C2×C4×C9C23×C312yes—
204C2×C4×C9C23×C311yescharacteristic
205C2×C4×C9C23×C311yescharacteristic
206C2×C4×C9C23×C311yescharacteristic
207#611—24no—
208#611—42no—
209C2×Dic9C23×C311yescharacteristic
210C2×Dic9C23×C312yes—
211C2×Dic9C23×C312yes—
212C2×Dic9C23×C311yescharacteristic
213D36—22no—
214C4×D9—22no—
215C4×D9—22no—
216C4×D9—42no—
217#553—91noSylow
218C2×3−1+2—42no—
219C2×3−1+2C2×#4111yescharacteristic
220C2×3−1+2C2×#4111yescharacteristic
221C2×3−1+2H(5,𝔽2)11yescharacteristic
222AGL(1,ℤ/9)—22no—
223AGL(1,ℤ/9)—41no—
224C2×C3×D4—31no—
225C2×C3×D4—91no—
226C2×C3×D4—92no—
227C2×C3×D4—92no—
228C2×C3×D4C2×C3×D311yescharacteristic
229C22×C3×C4—32no—
230C22×C3×C4—91no—
231C22×C3×C4—91no—
232C22×C3×C4—92no—
233C22×C3×C4C2×C3×D312yes—
234C2×#67—32no—
235C2×#67—32no—
236C2×D12—31no—
237C2×C4×D3—31no—
238C2×C4×D3—31no—
239C2×C4×D3—32no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4C2×C3×D311yescharacteristic
248C3×C22⋊C4—32no—
249C3×C22⋊C4—92no—
250C3×C22⋊C4—91no—
251C3×C22⋊C4—91no—
252C3×C22⋊C4—92no—
253C3×C22⋊C4—92no—
254C3×C22⋊C4C2×C3×D312yes—
255C3×C42—31no—
256C3×C42—92no—
257C3×C42C2×C3×D311yescharacteristic
258#217—32no—
259#217—32no—
260(C2×D3)⋊C4—31no—
261(C2×D3)⋊C4—31no—
262(C2×D3)⋊C4—32no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—31no—
266Dic3⋊C4—32no—
267Dic3⋊C4—32no—
268Dic3⋊C4—32no—
269C4×Dic3—32no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—22no—
273C22×C32C23×D311yescharacteristic
274C2×C3×D3—32no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C2×C3×D3—61no—
278C2×C3×D3—61no—
279C2×C3×D3—61no—
280C32×C4—22no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4#23512yes—
284C3×Dic3—61no—
285C3×Dic3—61no—
286C3×Dic3—62no—
287C3×Dic3—62no—
288C22×C9—24no—
289C22×C9—22no—
290C22×C9—22no—
291C22×C9—22no—
292C22×C9—21no—
293C22×C9—42no—
294C22×C9—42no—
295C22×C9C24×C311yescharacteristic
296C2×D9—21no—
297C2×D9—21no—
298C2×D9—21no—
299C2×D9—21no—
300C2×D9—21no—
301C2×D9C3×#4112yes—
302C4×C9—22no—
303C4×C9—21no—
304C4×C9—22no—
305C4×C9—21no—
306C4×C9—22no—
307C4×C9—41no—
308C4×C9—42no—
309C4×C9C3×#4112yes—
310Dic9—21no—
311Dic9—21no—
312Dic9—22no—
313Dic9—22no—
314#126—92no—
315#125—91no—
316#123—92no—
317#121—91no—
318#121—91no—
319#121—92no—
320C4×D4—92no—
321C4×D4—91no—
322C4×D4AGL(1,ℤ/9)11yescharacteristic, complemented
323#117—91no—
324C2×C4⋊C4—91no—
3253−1+2#55311yescharacteristic, complemented, Sylow
326C23×C3—32no—
327C23×C3—91no—
328C23×C3—91no—
329C23×C3C22×C3×D312yes—
330C22×D3—31no—
331C22×D3—31no—
332C3×D4—22no—
333C3×D4—62no—
334C3×D4—182no—
335C3×D4—184no—
336C3×D4—362no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—31no—
340C2×C3×C4—32no—
341C2×C3×C4—42no—
342C2×C3×C4—91no—
343C2×C3×C4—92no—
344C2×C3×C4—91no—
345C2×C3×C4—92no—
346C2×C3×C4—122no—
347C2×C3×C4—182no—
348C2×C3×C4—182no—
349C2×C3×C4—362no—
350C2×C3×C4C22×C3×D312yes—
351C2×C3×C4C22×C3×D311yescharacteristic
352C2×C3×C4C22×C3×D311yescharacteristic
353C2×C3×C4C22×C3×D311yescharacteristic
354#67—64no—
355#67—122no—
356C2×Dic3—31no—
357C2×Dic3—32no—
358C2×Dic3—32no—
359C2×Dic3—31no—
360D12—62no—
361C4×D3—62no—
362C4×D3—62no—
363C4×D3—122no—
364C2×C32—42no—
365C2×C32C2×#23511yescharacteristic
366C2×C32#100411yescharacteristic
367C2×C32D3×#4111yescharacteristic
368C3×D3—62no—
369C3×D3—121no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—44no—
374C2×C9—42no—
375C2×C9C2×C3×#4111yescharacteristic
376C2×C9C2×C3×#4111yescharacteristic
377C2×C9C3×H(5,𝔽2)11yescharacteristic
378D9—22no—
379D9—41no—
380C2×D4—92no—
381C2×D4—92no—
382C2×D4—91no—
383C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
384C22×C4—91no—
385C22×C4—91no—
386C22×C4—92no—
387C22×C4C2×AGL(1,ℤ/9)12yes—
388C4⋊C4—91no—
389C4⋊C4—92no—
390C4⋊C4—91no—
391C4⋊C4—92no—
392C4⋊C4—91no—
393C4⋊C4—92no—
394C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
395C22⋊C4—92no—
396C22⋊C4—92no—
397C22⋊C4—92no—
398C22⋊C4—91no—
399C22⋊C4—91no—
400C22⋊C4C2×AGL(1,ℤ/9)12yes—
401C42—92no—
402C42C2×AGL(1,ℤ/9)11yescharacteristic
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—31no—
407C22×C3—62no—
408C22×C3—62no—
409C22×C3—62no—
410C22×C3—92no—
411C22×C3—181no—
412C22×C3—181no—
413C22×C3—181no—
414C22×C3—181no—
415C22×C3—181no—
416C22×C3C23×C3×D311yescharacteristic
417C2×D3—32no—
418C2×D3—61no—
419C2×D3—61no—
420C2×D3—61no—
421C2×D3—61no—
422C2×D3—61no—
423C3×C4—22no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—32no—
427C3×C4—61no—
428C3×C4—62no—
429C3×C4—61no—
430C3×C4—181no—
431C3×C4—181no—
432C3×C4—182no—
433C3×C4—182no—
434C3×C4C3×#23512yes—
435Dic3—61no—
436Dic3—61no—
437Dic3—62no—
438Dic3—62no—
439C32#555811yescharacteristic
440C9—21no—
441C9C3×#55311yescharacteristic, complemented
442C23—91no—
443C23—91no—
444C23C22×AGL(1,ℤ/9)12yes—
445D4—22no—
446D4—182no—
447D4—184no—
448D4—362no—
449C2×C4—42no—
450C2×C4—91no—
451C2×C4—92no—
452C2×C4—91no—
453C2×C4—92no—
454C2×C4—182no—
455C2×C4—182no—
456C2×C4—362no—
457C2×C4C22×AGL(1,ℤ/9)12yes—
458C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
459C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
460C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—42no—
465C2×C3—122no—
466C2×C3—182no—
467C2×C3—361no—
468C2×C3C2×C3×#23511yescharacteristic
469C2×C3C3×#100411yescharacteristic
470C2×C3C3×D3×#4111yescharacteristic
471D3—62no—
472D3—121no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—92no—
477C22—181no—
478C22—181no—
479C22—181no—
480C22—181no—
481C22—181no—
482C22C23×AGL(1,ℤ/9)11yescharacteristic
483C4—21no—
484C4—22no—
485C4—21no—
486C4—181no—
487C4—181no—
488C4—182no—
489C4—182no—
490C4#3697⋊C312yes—
491C3—31no—
492C3C3×#555811yescharacteristic
493C2—42no—
494C2—182no—
495C2—361no—
496C2C2×#3697⋊C311yescharacteristic
497C2#63710⋊C311yescharacteristic
498C2#41×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC4×D4×3−1+2C2
hypercenterC4×D4AGL(1,ℤ/9)
hypocenterC9C3×#553
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C4×D4×3−1+2C2semidirect
2. #63500⋊C3C2semidirect
3. #63500⋊C3C2semidirect
4. #63491⋊C3C2semidirect
5. C4×#611⋊C3C2semidirect
6. #63454⋊C3C2semidirect
7. C4⋊C4×AGL(1,ℤ/9)C2semidirect
8. #63446⋊C3C2semidirect
9. #63441⋊C3C2semidirect
10. C4×C4⋊AGL(1,ℤ/9)C2semidirect
11. #63432⋊C3C2semidirect
13. #10589660C3semidirect
49. C4×D4×C9C2×C3semidirect
50. #63500C2×C3semidirect
51. #63500C2×C3semidirect
52. #63491C2×C3semidirect
53. C4×#611C2×C3semidirect
54. #63454C2×C3semidirect
55. C4⋊C4×D9C2×C3semidirect
56. #63446C2×C3semidirect
57. #63441C2×C3semidirect
58. C4×D36C2×C3semidirect
59. #63432C2×C3semidirect
322. C4×D4AGL(1,ℤ/9)semidirect
325. 3−1+2#553semidirect
441. C9C3×#553semidirect

Maximal subgroups

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

subgroupindex
C4×D4×3−1+22
#63500⋊C32
#63491⋊C32
C4×#611⋊C32
#63454⋊C32
C4⋊C4×AGL(1,ℤ/9)2
#63446⋊C32
#63441⋊C32
C4×C4⋊AGL(1,ℤ/9)2
#63432⋊C32
C3×#55583
#105896603