Finite Group Databasebeta

← #10589797⋊C3 #13153202

Subgroups — #10589797⋊C3 #13153202

Order 1728.

742 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
13−1+2×#125C211yesmaximal, characteristic, complemented
2#63460⋊C3C212yesmaximal, complemented
3#63457⋊C3C211yesmaximal, characteristic, complemented
4#63457⋊C3C211yesmaximal, characteristic, complemented
5#63457⋊C3C212yesmaximal, complemented
6#63456⋊C3C212yesmaximal, complemented
7#63455⋊C3C212yesmaximal, complemented
8#63454⋊C3C211yesmaximal, characteristic, complemented
9C4⋊C4×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
10C4×C4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
11C4×C4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
12C3×#5695—31nomaximal
13#10589797C311yesmaximal, characteristic, complemented
14C2×C4⋊AGL(1,ℤ/9)C2212yes—
15C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
16C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
17C2×C4⋊AGL(1,ℤ/9)C2212yes—
18C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
19C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
20C2×C4×AGL(1,ℤ/9)C2212yes—
21C2×C4×AGL(1,ℤ/9)C2212yes—
22C4⋊C4×3−1+2C2211yescharacteristic
23C4⋊C4×3−1+2C2211yescharacteristic
24C4⋊C4×3−1+2C2212yes—
25C4⋊C4×3−1+2C2212yes—
26C42×3−1+2C2211yescharacteristic
27(C2×AGL(1,ℤ/9))⋊C4C2212yes—
28(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
29(C2×AGL(1,ℤ/9))⋊C4C2212yes—
30(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
31(C2×AGL(1,ℤ/9))⋊C4C2212yes—
32(C2×AGL(1,ℤ/9))⋊C4C2212yes—
33C4⋊(3−1+2⋊C4)C2211yescharacteristic
34C4⋊(3−1+2⋊C4)C2211yescharacteristic
35(3−1+2⋊C4)⋊C4C2212yes—
36C4×3−1+2⋊C4C2212yes—
37C32×#125D311yescharacteristic
38C3×#898—32no—
39C3×#895—32no—
40C3×#895—31no—
41C3×#895—31no—
42C3×#894—32no—
43C3×#893—32no—
44C3×#892—31no—
45C3×D3×C4⋊C4—31no—
46C3×C4×D12—31no—
47C3×C4×D12—31no—
48C9×#125—21no—
49C9×#125C2×C311yescharacteristic, complemented
50#63460C2×C312yescomplemented
51#63457C2×C312yescomplemented
52#63457C2×C311yescharacteristic, complemented
53#63457C2×C311yescharacteristic, complemented
54#63456C2×C312yescomplemented
55#63455C2×C312yescomplemented
56#63454C2×C311yescharacteristic, complemented
57C4⋊C4×D9C2×C311yescharacteristic, complemented
58C4×D36C2×C311yescharacteristic, complemented
59C4×D36C2×C311yescharacteristic, complemented
60C22×AGL(1,ℤ/9)C2311yescharacteristic
61C22×AGL(1,ℤ/9)C2312yes—
62C22×AGL(1,ℤ/9)C2311yescharacteristic
63C2×C4×3−1+2C2311yescharacteristic
64C2×C4×3−1+2C2312yes—
65C2×C4×3−1+2C2312yes—
66C2×C4×3−1+2C2311yescharacteristic
67C2×C4×3−1+2C2311yescharacteristic
68C2×3−1+2⋊C4C2312yes—
69C2×3−1+2⋊C4C2311yescharacteristic
70C2×3−1+2⋊C4C2311yescharacteristic
71C4⋊AGL(1,ℤ/9)—24no—
72C4⋊AGL(1,ℤ/9)—22no—
73C4⋊AGL(1,ℤ/9)—22no—
74C4⋊AGL(1,ℤ/9)—42no—
75C4×AGL(1,ℤ/9)—22no—
76C4×AGL(1,ℤ/9)—22no—
77C4×AGL(1,ℤ/9)—42no—
78C4×AGL(1,ℤ/9)—42no—
79C3×#553—91no—
80#5695—31no—
81C2×C3×D12—32no—
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×D12—32no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C4⋊C4C2×D312yes—
92C32×C4⋊C4C2×D312yes—
93C32×C42C2×D311yescharacteristic
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4—31no—
96C3×(C2×D3)⋊C4—32no—
97C3×(C2×D3)⋊C4—31no—
98C3×(C2×D3)⋊C4—32no—
99C3×(C2×D3)⋊C4—32no—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3—31no—
102C3×Dic3⋊C4—32no—
103C3×C4×Dic3—32no—
104C2×D36C22×C311yescharacteristic
105C2×D36C22×C311yescharacteristic
106C2×D36C22×C312yes—
107C2×D36C22×C312yes—
108C2×C4×D9C22×C311yescharacteristic
109C2×C4×D9C22×C311yescharacteristic
110C2×C4×D9C22×C312yes—
111C2×C4×D9C22×C312yes—
112C9×C4⋊C4—21no—
113C9×C4⋊C4—21no—
114C9×C4⋊C4—22no—
115C9×C4⋊C4—22no—
116C9×C4⋊C4C22×C311yescharacteristic
117C9×C4⋊C4C22×C311yescharacteristic
118C9×C4⋊C4C22×C312yes—
119C9×C4⋊C4C22×C312yes—
120C42×C9—21no—
121C42×C9C22×C311yescharacteristic
122(C2×D9)⋊C4C22×C311yescharacteristic
123(C2×D9)⋊C4C22×C311yescharacteristic
124(C2×D9)⋊C4C22×C312yes—
125(C2×D9)⋊C4C22×C312yes—
126(C2×D9)⋊C4C22×C312yes—
127(C2×D9)⋊C4C22×C312yes—
128C4⋊Dic9C22×C311yescharacteristic
129C4⋊Dic9C22×C311yescharacteristic
130Dic9⋊C4C22×C312yes—
131C4×Dic9C22×C312yes—
132C22×3−1+2C2411yescharacteristic
133C2×AGL(1,ℤ/9)—21no—
134C2×AGL(1,ℤ/9)—21no—
135C2×AGL(1,ℤ/9)—21no—
136C2×AGL(1,ℤ/9)—22no—
137C2×AGL(1,ℤ/9)—21no—
138C2×AGL(1,ℤ/9)—22no—
139C2×AGL(1,ℤ/9)—22no—
140C2×AGL(1,ℤ/9)—21no—
141C2×AGL(1,ℤ/9)#4112yes—
142C4×3−1+2—22no—
143C4×3−1+2—21no—
144C4×3−1+2—21no—
145C4×3−1+2—22no—
146C4×3−1+2#4112yes—
1473−1+2⋊C4—21no—
1483−1+2⋊C4—21no—
1493−1+2⋊C4—22no—
150C3×#126—92no—
151C3×#125—31no—
152C3×#125C3×D311yescharacteristic
153C3×#123—92no—
154C3×#121—92no—
155C3×#121—91no—
156C3×#121—91no—
157C3×C4×D4—91no—
158C3×C4×D4—91no—
159C3×C4×D4—92no—
160C3×#117—91no—
161C2×C3×C4⋊C4—91no—
162#898—32no—
163#895—32no—
164#895—31no—
165#895—31no—
166#894—32no—
167#893—32no—
168#892—31no—
169D3×C4⋊C4—31no—
170C4×D12—31no—
171C4×D12—31no—
172C22×C3×D3—31no—
173C22×C3×D3—31no—
174C22×C3×D3—32no—
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C3×Dic3—31no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—31no—
183C3×D12—64no—
184C3×D12—62no—
185C3×D12—62no—
186C3×D12—122no—
187C3×C4×D3—62no—
188C3×C4×D3—62no—
189C3×C4×D3—122no—
190C3×C4×D3—122no—
191C22×D9C23×C311yescharacteristic
192C22×D9C23×C312yes—
193C22×D9C23×C311yescharacteristic
194C2×C4×C9—21no—
195C2×C4×C9—21no—
196C2×C4×C9—21no—
197C2×C4×C9—22no—
198C2×C4×C9—22no—
199C2×C4×C9C23×C312yes—
200C2×C4×C9C23×C312yes—
201C2×C4×C9C23×C311yescharacteristic
202C2×C4×C9C23×C311yescharacteristic
203C2×C4×C9C23×C311yescharacteristic
204C2×Dic9C23×C311yescharacteristic
205C2×Dic9C23×C312yes—
206C2×Dic9C23×C311yescharacteristic
207D36—22no—
208D36—24no—
209D36—22no—
210D36—42no—
211C4×D9—22no—
212C4×D9—22no—
213C4×D9—42no—
214C4×D9—42no—
215#553—91noSylow
216C2×3−1+2C2×#4111yescharacteristic
217C2×3−1+2C2×#4111yescharacteristic
218C2×3−1+2H(5,𝔽2)11yescharacteristic
219AGL(1,ℤ/9)—22no—
220AGL(1,ℤ/9)—41no—
221AGL(1,ℤ/9)—42no—
222C2×C3×D4—91no—
223C2×C3×D4—91no—
224C2×C3×D4—92no—
225C2×C3×D4—92no—
226C22×C3×C4—91no—
227C22×C3×C4—91no—
228C22×C3×C4—92no—
229C22×C3×C4—92no—
230C2×D12—31no—
231C2×D12—31no—
232C2×D12—32no—
233C2×D12—32no—
234C2×C4×D3—31no—
235C2×C4×D3—31no—
236C2×C4×D3—32no—
237C2×C4×D3—32no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—32no—
241C3×C4⋊C4—32no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4C2×C3×D311yescharacteristic
246C3×C4⋊C4C2×C3×D311yescharacteristic
247C3×C4⋊C4C2×C3×D312yes—
248C3×C4⋊C4C2×C3×D312yes—
249C3×C22⋊C4—92no—
250C3×C22⋊C4—91no—
251C3×C22⋊C4—92no—
252C3×C22⋊C4—92no—
253C3×C22⋊C4—91no—
254C3×C22⋊C4—92no—
255C3×C42—31no—
256C3×C42—92no—
257C3×C42C2×C3×D311yescharacteristic
258(C2×D3)⋊C4—31no—
259(C2×D3)⋊C4—31no—
260(C2×D3)⋊C4—32no—
261(C2×D3)⋊C4—32no—
262(C2×D3)⋊C4—32no—
263(C2×D3)⋊C4—32no—
264C4⋊Dic3—31no—
265C4⋊Dic3—31no—
266Dic3⋊C4—32no—
267C4×Dic3—32no—
268C22×C32C23×D311yescharacteristic
269C2×C3×D3—32no—
270C2×C3×D3—61no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C2×C3×D3—62no—
274C2×C3×D3—61no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C2×C3×D3—61no—
278C32×C4—22no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—22no—
282C32×C4#23912yes—
283C3×Dic3—61no—
284C3×Dic3—61no—
285C3×Dic3—62no—
286C22×C9—21no—
287C22×C9C24×C311yescharacteristic
288C2×D9—21no—
289C2×D9—21no—
290C2×D9—21no—
291C2×D9—22no—
292C2×D9—21no—
293C2×D9—22no—
294C2×D9—22no—
295C2×D9—21no—
296C2×D9C3×#4112yes—
297C4×C9—22no—
298C4×C9—21no—
299C4×C9—21no—
300C4×C9—22no—
301C4×C9—22no—
302C4×C9—41no—
303C4×C9—42no—
304C4×C9—41no—
305C4×C9—42no—
306C4×C9C3×#4112yes—
307Dic9—21no—
308Dic9—21no—
309Dic9—22no—
310#126—92no—
311#125AGL(1,ℤ/9)11yescharacteristic, complemented
312#123—92no—
313#121—92no—
314#121—91no—
315#121—91no—
316C4×D4—92no—
317C4×D4—91no—
318C4×D4—91no—
319#117—91no—
320C2×C4⋊C4—91no—
3213−1+2#55311yescharacteristic, complemented, Sylow
322C23×C3—91no—
323C23×C3—91no—
324C23×C3—92no—
325C22×D3—31no—
326C22×D3—31no—
327C22×D3—32no—
328C3×D4—182no—
329C3×D4—182no—
330C3×D4—184no—
331C3×D4—362no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—32no—
336C2×C3×C4—32no—
337C2×C3×C4—91no—
338C2×C3×C4—92no—
339C2×C3×C4—91no—
340C2×C3×C4—182no—
341C2×C3×C4—182no—
342C2×C3×C4—362no—
343C2×C3×C4—362no—
344C2×C3×C4C22×C3×D312yes—
345C2×C3×C4C22×C3×D312yes—
346C2×C3×C4C22×C3×D311yescharacteristic
347C2×C3×C4C22×C3×D311yescharacteristic
348C2×C3×C4C22×C3×D311yescharacteristic
349C2×Dic3—31no—
350C2×Dic3—32no—
351C2×Dic3—31no—
352D12—62no—
353D12—64no—
354D12—62no—
355D12—122no—
356C4×D3—62no—
357C4×D3—62no—
358C4×D3—122no—
359C4×D3—122no—
360C2×C32C2×#23911yescharacteristic
361C2×C32D3×#4111yescharacteristic
362C2×C32#100911yescharacteristic
363C3×D3—62no—
364C3×D3—121no—
365C3×D3—122no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9—21no—
369C2×C9C2×C3×#4111yescharacteristic
370C2×C9C2×C3×#4111yescharacteristic
371C2×C9C3×H(5,𝔽2)11yescharacteristic
372D9—22no—
373D9—41no—
374D9—42no—
375C2×D4—92no—
376C2×D4—91no—
377C2×D4—91no—
378C2×D4—92no—
379C22×C4—91no—
380C22×C4—91no—
381C22×C4—92no—
382C22×C4—92no—
383C4⋊C4—91no—
384C4⋊C4—92no—
385C4⋊C4—91no—
386C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
387C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
388C4⋊C4C2×AGL(1,ℤ/9)12yes—
389C4⋊C4C2×AGL(1,ℤ/9)12yes—
390C22⋊C4—92no—
391C22⋊C4—91no—
392C22⋊C4—92no—
393C22⋊C4—92no—
394C22⋊C4—91no—
395C22⋊C4—92no—
396C42—92no—
397C42C2×AGL(1,ℤ/9)11yescharacteristic
398C22×C3—31no—
399C22×C3—92no—
400C22×C3—181no—
401C22×C3—181no—
402C22×C3—181no—
403C22×C3—182no—
404C22×C3—182no—
405C22×C3—181no—
406C22×C3—182no—
407C22×C3—181no—
408C22×C3C23×C3×D311yescharacteristic
409C2×D3—32no—
410C2×D3—61no—
411C2×D3—61no—
412C2×D3—61no—
413C2×D3—62no—
414C2×D3—61no—
415C2×D3—62no—
416C2×D3—62no—
417C2×D3—61no—
418C3×C4—22no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—22no—
422C3×C4—32no—
423C3×C4—61no—
424C3×C4—62no—
425C3×C4—61no—
426C3×C4—62no—
427C3×C4—181no—
428C3×C4—181no—
429C3×C4—182no—
430C3×C4C3×#23912yes—
431Dic3—61no—
432Dic3—61no—
433Dic3—62no—
434C32#569511yescharacteristic
435C9—21no—
436C9C3×#55311yescharacteristic, complemented
437C23—91no—
438C23—91no—
439C23—92no—
440D4—184no—
441D4—182no—
442D4—182no—
443D4—362no—
444C2×C4—91no—
445C2×C4—92no—
446C2×C4—91no—
447C2×C4—182no—
448C2×C4—182no—
449C2×C4—362no—
450C2×C4—362no—
451C2×C4C22×AGL(1,ℤ/9)12yes—
452C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
453C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
454C2×C4C22×AGL(1,ℤ/9)12yes—
455C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
456C2×C3—31no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—182no—
460C2×C3—361no—
461C2×C3—362no—
462C2×C3C2×C3×#23911yescharacteristic
463C2×C3C3×D3×#4111yescharacteristic
464C2×C3C3×#100911yescharacteristic
465D3—62no—
466D3—121no—
467D3—122no—
468C22—92no—
469C22—181no—
470C22—181no—
471C22—181no—
472C22—182no—
473C22—181no—
474C22—182no—
475C22—182no—
476C22—181no—
477C22C23×AGL(1,ℤ/9)11yescharacteristic
478C4—22no—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—181no—
483C4—181no—
484C4—182no—
485C4#8872412yes—
486C3—31no—
487C3C3×#569511yescharacteristic
488C2—182no—
489C2—361no—
490C2—362no—
491C2C2×#8872411yescharacteristic
492C2#41×AGL(1,ℤ/9)11yescharacteristic
493C2#1172264111yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#125C2
hypercenter#125AGL(1,ℤ/9)
hypocenterC9C3×#553
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#125C2semidirect
2. #63460⋊C3C2semidirect
3. #63457⋊C3C2semidirect
4. #63457⋊C3C2semidirect
5. #63457⋊C3C2semidirect
6. #63456⋊C3C2semidirect
7. #63455⋊C3C2semidirect
8. #63454⋊C3C2semidirect
9. C4⋊C4×AGL(1,ℤ/9)C2semidirect
10. C4×C4⋊AGL(1,ℤ/9)C2semidirect
11. C4×C4⋊AGL(1,ℤ/9)C2semidirect
13. #10589797C3semidirect
49. C9×#125C2×C3semidirect
50. #63460C2×C3semidirect
51. #63457C2×C3semidirect
52. #63457C2×C3semidirect
53. #63457C2×C3semidirect
54. #63456C2×C3semidirect
55. #63455C2×C3semidirect
56. #63454C2×C3semidirect
57. C4⋊C4×D9C2×C3semidirect
58. C4×D36C2×C3semidirect
59. C4×D36C2×C3semidirect
311. #125AGL(1,ℤ/9)semidirect
321. 3−1+2#553semidirect
436. 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×#1252
#63460⋊C32
#63457⋊C32
#63456⋊C32
#63455⋊C32
#63454⋊C32
C4⋊C4×AGL(1,ℤ/9)2
C4×C4⋊AGL(1,ℤ/9)2
C3×#56953
#105897973