Finite Group Databasebeta

← #10589684⋊C3 #13152743

Subgroups — #10589684⋊C3 #13152743

Order 1728.

683 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Q8×3−1+2C211yesmaximal, characteristic, complemented
2#63509⋊C3C211yesmaximal, characteristic, complemented
3#63508⋊C3C211yesmaximal, characteristic, complemented
4#63458⋊C3C212yesmaximal, complemented
5#63457⋊C3C211yesmaximal, characteristic
6#63456⋊C3C212yesmaximal, complemented
7#63450⋊C3C211yesmaximal, characteristic, complemented
8#11721396C212yesmaximal, complemented
9#63437⋊C3C212yesmaximal, complemented
10C4×C4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
11#63434⋊C3C211yesmaximal, characteristic, complemented
12C3×#5582—31nomaximal
13#10589684C311yesmaximal, characteristic, complemented
14C2×Q8×3−1+2C2211yescharacteristic
15C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
16C2×C4⋊AGL(1,ℤ/9)C2212yes—
17C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
18C2×C4×AGL(1,ℤ/9)C2212yes—
19C2×3−1+2⋊Q8C2212yes—
20C4⋊C4×3−1+2C2211yescharacteristic
21C4⋊C4×3−1+2C2212yes—
22C42×3−1+2C2211yescharacteristic
23C42×3−1+2C2212yes—
24(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
25(C2×AGL(1,ℤ/9))⋊C4C2212yes—
26(C2×AGL(1,ℤ/9))⋊C4C2212yes—
27(C2×AGL(1,ℤ/9))⋊C4C2212yes—
28(C2×AGL(1,ℤ/9))⋊C4C2212yes—
29(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
30(C2×AGL(1,ℤ/9))⋊C4C2212yes—
31C4⋊(3−1+2⋊C4)C2211yescharacteristic
32(3−1+2⋊C4)⋊C4C2212yes—
33(3−1+2⋊C4)⋊C4C2211yescharacteristic
34(3−1+2⋊C4)⋊C4C2212yes—
35(3−1+2⋊C4)⋊C4C2211yescharacteristic
36C4×3−1+2⋊C4C2211yescharacteristic
37C32×C4×Q8D311yescharacteristic
38C3×#947—31no—
39C3×#946—31no—
40C3×#896—32no—
41C3×#895—31no—
42C3×#894—32no—
43C3×#888—31no—
44C3×#876—32no—
45C3×#875—32no—
46C3×C4×D12—31no—
47C3×#872—31no—
48C4×Q8×C9—21no—
49C4×Q8×C9C2×C311yescharacteristic, complemented
50#63509C2×C311yescharacteristic, complemented
51#63508C2×C311yescharacteristic, complemented
52#63458C2×C312yescomplemented
53#63457C2×C311yescharacteristic
54#63456C2×C312yescomplemented
55#63450C2×C311yescharacteristic, complemented
56#63438C2×C312yescomplemented
57#63437C2×C312yescomplemented
58C4×D36C2×C311yescharacteristic, complemented
59#63434C2×C311yescharacteristic, complemented
60C22×AGL(1,ℤ/9)C2311yescharacteristic
61C22×AGL(1,ℤ/9)C2312yes—
62Q8×3−1+2—22no—
63C2×C4×3−1+2C2312yes—
64C2×C4×3−1+2C2311yescharacteristic
65C2×C4×3−1+2C2312yes—
66C2×C4×3−1+2C2311yescharacteristic
67C2×C4×3−1+2C2311yescharacteristic
68C2×3−1+2⋊C4C2311yescharacteristic
69C2×3−1+2⋊C4C2312yes—
70C2×3−1+2⋊C4C2311yescharacteristic
71C2×3−1+2⋊C4C2311yescharacteristic
72C4⋊AGL(1,ℤ/9)—22no—
73C4⋊AGL(1,ℤ/9)—42no—
74C4×AGL(1,ℤ/9)—22no—
75C4×AGL(1,ℤ/9)—42no—
763−1+2⋊Q8—42no—
77C3×#542—91no—
78#5582—31no—
79C2×C32×Q8C2×D311yescharacteristic
80C2×C3×D12—31no—
81C2×C3×D12—32no—
82C2×C3×C4×D3—31no—
83C2×C3×C4×D3—32no—
84C2×C3×Dic6—32no—
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C4⋊C4C2×D312yes—
87C32×C42C2×D311yescharacteristic
88C32×C42C2×D312yes—
89C3×(C2×D3)⋊C4—31no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4—31no—
95C3×(C2×D3)⋊C4—32no—
96C3×C4⋊Dic3—31no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—31no—
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4—31no—
101C3×C4×Dic3—31no—
102C2×Q8×C9—21no—
103C2×Q8×C9C22×C311yescharacteristic
104C2×D36C22×C311yescharacteristic
105C2×D36C22×C312yes—
106C2×C4×D9C22×C311yescharacteristic
107C2×C4×D9C22×C312yes—
108C2×Dic18C22×C312yes—
109C9×C4⋊C4—21no—
110C9×C4⋊C4—22no—
111C9×C4⋊C4C22×C311yescharacteristic
112C9×C4⋊C4C22×C312yes—
113C42×C9—21no—
114C42×C9—22no—
115C42×C9C22×C311yescharacteristic
116C42×C9C22×C312yes—
117(C2×D9)⋊C4C22×C312yes—
118(C2×D9)⋊C4C22×C312yes—
119(C2×D9)⋊C4C22×C312yes—
120(C2×D9)⋊C4C22×C311yescharacteristic
121(C2×D9)⋊C4C22×C311yescharacteristic
122(C2×D9)⋊C4C22×C312yes—
123(C2×D9)⋊C4C22×C312yes—
124C4⋊Dic9C22×C311yescharacteristic
125Dic9⋊C4C22×C312yes—
126Dic9⋊C4C22×C311yescharacteristic
127Dic9⋊C4C22×C311yescharacteristic
128Dic9⋊C4C22×C312yes—
129C4×Dic9C22×C311yescharacteristic
130C22×3−1+2C2411yescharacteristic
131C2×AGL(1,ℤ/9)—21no—
132C2×AGL(1,ℤ/9)—22no—
133C2×AGL(1,ℤ/9)—22no—
134C2×AGL(1,ℤ/9)—21no—
135C2×AGL(1,ℤ/9)—21no—
136C2×AGL(1,ℤ/9)—22no—
137C4×3−1+2—22no—
138C4×3−1+2—22no—
139C4×3−1+2—21no—
140C4×3−1+2—21no—
141C4×3−1+2#4112yes—
1423−1+2⋊C4—21no—
1433−1+2⋊C4—21no—
1443−1+2⋊C4—22no—
1453−1+2⋊C4—21no—
146C3×C4⋊Q8—91no—
147C3×#126—92no—
148C3×#124—92no—
149C3×#124—91no—
150C3×#123—92no—
151C3×C22⋊Q8—92no—
152C3×C22⋊Q8—91no—
153C3×#121—91no—
154C3×C4×Q8—31no—
155C3×C4×Q8C3×D311yescharacteristic
156C3×C4×D4—91no—
157C3×#117—91no—
158#947—31no—
159#946—31no—
160#896—32no—
161#895—31no—
162#894—32no—
163#888—31no—
164#876—32no—
165#875—32no—
166C4×D12—31no—
167#872—31no—
168C22×C3×D3—31no—
169C22×C3×D3—32no—
170C32×Q8—22no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C3×Dic3—31no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—32no—
180C3×D12—62no—
181C3×D12—122no—
182C3×C4×D3—62no—
183C3×C4×D3—122no—
184C3×Dic6—122no—
185C22×D9C23×C311yescharacteristic
186C22×D9C23×C312yes—
187Q8×C9—24no—
188Q8×C9—22no—
189C2×C4×C9—21no—
190C2×C4×C9—22no—
191C2×C4×C9—21no—
192C2×C4×C9—22no—
193C2×C4×C9—21no—
194C2×C4×C9C23×C311yescharacteristic
195C2×C4×C9C23×C312yes—
196C2×C4×C9C23×C312yes—
197C2×C4×C9C23×C311yescharacteristic
198C2×C4×C9C23×C311yescharacteristic
199C2×Dic9C23×C311yescharacteristic
200C2×Dic9C23×C311yescharacteristic
201C2×Dic9C23×C311yescharacteristic
202C2×Dic9C23×C312yes—
203D36—22no—
204D36—42no—
205C4×D9—22no—
206C4×D9—42no—
207Dic18—42no—
208#542—91noSylow
209C2×3−1+2C2×#4111yescharacteristic
210C2×3−1+2H(5,𝔽2)11yescharacteristic
211C2×3−1+22−1+411yescharacteristic
212AGL(1,ℤ/9)—41no—
213AGL(1,ℤ/9)—42no—
214C2×C3×Q8—31no—
215C2×C3×Q8—92no—
216C2×C3×Q8C2×C3×D311yescharacteristic
217C2×C3×D4—91no—
218C2×C3×D4—92no—
219C22×C3×C4—91no—
220C22×C3×C4—92no—
221C2×D12—31no—
222C2×D12—32no—
223C2×C4×D3—31no—
224C2×C4×D3—32no—
225C2×Dic6—32no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—32no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—92no—
231C3×C4⋊C4—92no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4C2×C3×D311yescharacteristic
234C3×C4⋊C4C2×C3×D312yes—
235C3×C22⋊C4—91no—
236C3×C22⋊C4—91no—
237C3×C22⋊C4—92no—
238C3×C22⋊C4—92no—
239C3×C22⋊C4—92no—
240C3×C22⋊C4—92no—
241C3×C22⋊C4—92no—
242C3×C42—31no—
243C3×C42—32no—
244C3×C42—91no—
245C3×C42C2×C3×D311yescharacteristic
246C3×C42C2×C3×D312yes—
247(C2×D3)⋊C4—31no—
248(C2×D3)⋊C4—31no—
249(C2×D3)⋊C4—32no—
250(C2×D3)⋊C4—32no—
251(C2×D3)⋊C4—32no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254C4⋊Dic3—31no—
255Dic3⋊C4—31no—
256Dic3⋊C4—31no—
257Dic3⋊C4—32no—
258Dic3⋊C4—32no—
259C4×Dic3—31no—
260C22×C32C23×D311yescharacteristic
261C2×C3×D3—61no—
262C2×C3×D3—62no—
263C2×C3×D3—61no—
264C2×C3×D3—62no—
265C2×C3×D3—62no—
266C2×C3×D3—61no—
267C32×C4—22no—
268C32×C4—21no—
269C32×C4—22no—
270C32×C4—21no—
271C32×C4#23512yes—
272C3×Dic3—61no—
273C3×Dic3—61no—
274C3×Dic3—61no—
275C3×Dic3—62no—
276C22×C9—21no—
277C22×C9C24×C311yescharacteristic
278C2×D9—21no—
279C2×D9—22no—
280C2×D9—22no—
281C2×D9—21no—
282C2×D9—21no—
283C2×D9—22no—
284C4×C9—22no—
285C4×C9—24no—
286C4×C9—22no—
287C4×C9—22no—
288C4×C9—21no—
289C4×C9—21no—
290C4×C9—22no—
291C4×C9—42no—
292C4×C9—41no—
293C4×C9C3×#4112yes—
294Dic9—21no—
295Dic9—21no—
296Dic9—22no—
297Dic9—21no—
298C4⋊Q8—91no—
299#126—92no—
300#124—91no—
301#124—92no—
302#123—92no—
303C22⋊Q8—91no—
304C22⋊Q8—92no—
305#121—91no—
306C4×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
307C4×D4—91no—
308#117—91no—
3093−1+2#54211yescharacteristic, complemented, Sylow
310C23×C3—91no—
311C23×C3—92no—
312C22×D3—31no—
313C22×D3—32no—
314C3×Q8—22no—
315C3×Q8—62no—
316C3×Q8—362no—
317C3×D4—182no—
318C3×D4—362no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—32no—
322C2×C3×C4—31no—
323C2×C3×C4—32no—
324C2×C3×C4—91no—
325C2×C3×C4—91no—
326C2×C3×C4—91no—
327C2×C3×C4—92no—
328C2×C3×C4—182no—
329C2×C3×C4—362no—
330C2×C3×C4C22×C3×D311yescharacteristic
331C2×C3×C4C22×C3×D312yes—
332C2×C3×C4C22×C3×D312yes—
333C2×C3×C4C22×C3×D311yescharacteristic
334C2×C3×C4C22×C3×D311yescharacteristic
335C2×Dic3—31no—
336C2×Dic3—31no—
337C2×Dic3—31no—
338C2×Dic3—32no—
339D12—62no—
340D12—122no—
341C4×D3—62no—
342C4×D3—122no—
343Dic6—122no—
344C2×C32C2×#23511yescharacteristic
345C2×C32#100711yescharacteristic
346C2×C32#100911yescharacteristic
347C3×D3—121no—
348C3×D3—122no—
349C2×C9—21no—
350C2×C9—21no—
351C2×C9—21no—
352C2×C9C2×C3×#4111yescharacteristic
353C2×C9C3×H(5,𝔽2)11yescharacteristic
354C2×C9C3×2−1+411yescharacteristic
355D9—41no—
356D9—42no—
357C2×Q8—92no—
358C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
359C2×D4—91no—
360C2×D4—92no—
361C22×C4—91no—
362C22×C4—92no—
363C4⋊C4—91no—
364C4⋊C4—91no—
365C4⋊C4—91no—
366C4⋊C4—92no—
367C4⋊C4—92no—
368C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
369C4⋊C4C2×AGL(1,ℤ/9)12yes—
370C22⋊C4—91no—
371C22⋊C4—91no—
372C22⋊C4—92no—
373C22⋊C4—92no—
374C22⋊C4—92no—
375C22⋊C4—92no—
376C22⋊C4—92no—
377C42—91no—
378C42C2×AGL(1,ℤ/9)11yescharacteristic
379C42C2×AGL(1,ℤ/9)12yes—
380C22×C3—31no—
381C22×C3—181no—
382C22×C3—182no—
383C22×C3—182no—
384C22×C3—181no—
385C22×C3—181no—
386C22×C3—182no—
387C22×C3C23×C3×D311yescharacteristic
388C2×D3—61no—
389C2×D3—62no—
390C2×D3—61no—
391C2×D3—62no—
392C2×D3—62no—
393C2×D3—61no—
394C3×C4—22no—
395C3×C4—21no—
396C3×C4—22no—
397C3×C4—21no—
398C3×C4—32no—
399C3×C4—61no—
400C3×C4—62no—
401C3×C4—62no—
402C3×C4—61no—
403C3×C4—181no—
404C3×C4—182no—
405C3×C4—181no—
406C3×C4—181no—
407C3×C4C3×#23512yes—
408Dic3—61no—
409Dic3—61no—
410Dic3—61no—
411Dic3—62no—
412C32#558211yescharacteristic
413C9—21no—
414C9C3×#54211yescharacteristic, complemented
415C23—91no—
416C23—92no—
417Q8—22no—
418Q8—362no—
419D4—182no—
420D4—362no—
421C2×C4—91no—
422C2×C4—91no—
423C2×C4—91no—
424C2×C4—92no—
425C2×C4—182no—
426C2×C4—362no—
427C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
428C2×C4C22×AGL(1,ℤ/9)12yes—
429C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
430C2×C4C22×AGL(1,ℤ/9)12yes—
431C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—361no—
436C2×C3—362no—
437C2×C3C2×C3×#23511yescharacteristic
438C2×C3C3×#100711yescharacteristic
439C2×C3C3×#100911yescharacteristic
440D3—121no—
441D3—122no—
442C22—181no—
443C22—182no—
444C22—181no—
445C22—182no—
446C22—182no—
447C22—181no—
448C22C23×AGL(1,ℤ/9)11yescharacteristic
449C4—22no—
450C4—21no—
451C4—21no—
452C4—22no—
453C4—181no—
454C4—181no—
455C4—182no—
456C4—181no—
457C4#3697⋊C312yes—
458C3—31no—
459C3C3×#558211yescharacteristic
460C2—361no—
461C2—362no—
462C2C2×#3697⋊C311yescharacteristic
463C2#1172263011yescharacteristic
464C2#1172264111yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC4×Q8×3−1+2C2
prime radical#63457⋊C3C2
hypercenterC4×Q8AGL(1,ℤ/9)
hypocenterC9C3×#542
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×3−1+2C2semidirect
2. #63509⋊C3C2semidirect
3. #63508⋊C3C2semidirect
4. #63458⋊C3C2semidirect
6. #63456⋊C3C2semidirect
7. #63450⋊C3C2semidirect
8. #11721396C2semidirect
9. #63437⋊C3C2semidirect
10. C4×C4⋊AGL(1,ℤ/9)C2semidirect
11. #63434⋊C3C2semidirect
13. #10589684C3semidirect
49. C4×Q8×C9C2×C3semidirect
50. #63509C2×C3semidirect
51. #63508C2×C3semidirect
52. #63458C2×C3semidirect
54. #63456C2×C3semidirect
55. #63450C2×C3semidirect
56. #63438C2×C3semidirect
57. #63437C2×C3semidirect
58. C4×D36C2×C3semidirect
59. #63434C2×C3semidirect
306. C4×Q8AGL(1,ℤ/9)semidirect
309. 3−1+2#542semidirect
414. C9C3×#542semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×3−1+22
#63509⋊C32
#63508⋊C32
#63458⋊C32
#63457⋊C32
#63456⋊C32
#63450⋊C32
#117213962
#63437⋊C32
C4×C4⋊AGL(1,ℤ/9)2
#63434⋊C32
C3×#55823
#105896843