Finite Group Databasebeta

← #10589663⋊C3 #13152662

Subgroups — #10589663⋊C3 #13152662

Order 1728.

695 conjugacy classes of subgroups.

The subgroup lattice has 470 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
3#63498⋊C3C211yesmaximal, characteristic, complemented
4#63491⋊C3C212yesmaximal, complemented
5#63486⋊C3C212yesmaximal, complemented
6#63459⋊C3C211yesmaximal, characteristic, complemented
7#63450⋊C3C211yesmaximal, characteristic, complemented
8#63447⋊C3C212yesmaximal, complemented
9#63441⋊C3C212yesmaximal, complemented
10#63434⋊C3C211yesmaximal, characteristic, complemented
11C4×3−1+2⋊Q8C211yesmaximal, characteristic, complemented
12C3×#5561—31nomaximal
13#10589663C311yesmaximal, characteristic, complemented
14C2×D4×3−1+2C2211yescharacteristic
15C22×C4×3−1+2C2212yes—
16C2×#611⋊C3C2212yes—
17C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
18C2×3−1+2⋊Q8C2211yescharacteristic
19C2×3−1+2⋊Q8C2212yes—
20C4⋊C4×3−1+2C2211yescharacteristic
21C22⋊C4×3−1+2C2212yes—
22C42×3−1+2C2211yescharacteristic
23#3676⋊C3C2212yes—
24#3676⋊C3C2212yes—
25#3676⋊C3C2212yes—
26(C2×AGL(1,ℤ/9))⋊C4C2212yes—
27(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
28(C2×AGL(1,ℤ/9))⋊C4C2211yescharacteristic
29C4⋊(3−1+2⋊C4)C2211yescharacteristic
30C4⋊(3−1+2⋊C4)C2212yes—
31(3−1+2⋊C4)⋊C4C2212yes—
32(3−1+2⋊C4)⋊C4C2211yescharacteristic
33(3−1+2⋊C4)⋊C4C2212yes—
34(3−1+2⋊C4)⋊C4C2211yescharacteristic
35C4×3−1+2⋊C4C2211yescharacteristic
36C4×3−1+2⋊C4C2212yes—
37C32×C4×D4D311yescharacteristic
38C3×#938—31no—
39C3×#936—31no—
40C3×#929—32no—
41C3×#924—32no—
42C3×#897—31no—
43C3×#888—31no—
44C3×#885—32no—
45C3×#879—32no—
46C3×#872—31no—
47C3×C4×Dic6—31no—
48C4×D4×C9—21no—
49C4×D4×C9C2×C311yescharacteristic, complemented
50#63500C2×C311yescharacteristic
51#63498C2×C311yescharacteristic, complemented
52#63491C2×C312yescomplemented
53#63486C2×C312yescomplemented
54#63459C2×C311yescharacteristic, complemented
55#63450C2×C311yescharacteristic, complemented
56#63447C2×C312yescomplemented
57#63441C2×C312yescomplemented
58#63434C2×C311yescharacteristic, complemented
59C4×Dic18C2×C311yescharacteristic, complemented
60C23×3−1+2C2312yes—
61C22×AGL(1,ℤ/9)C2311yescharacteristic
62D4×3−1+2—22no—
63C2×C4×3−1+2—42no—
64C2×C4×3−1+2C2312yes—
65C2×C4×3−1+2C2311yescharacteristic
66C2×C4×3−1+2C2311yescharacteristic
67C2×C4×3−1+2C2311yescharacteristic
68#611⋊C3—42no—
69C2×3−1+2⋊C4C2311yescharacteristic
70C2×3−1+2⋊C4C2312yes—
71C2×3−1+2⋊C4C2312yes—
72C2×3−1+2⋊C4C2311yescharacteristic
73C2×3−1+2⋊C4C2311yescharacteristic
74C4×AGL(1,ℤ/9)—22no—
753−1+2⋊Q8—22no—
763−1+2⋊Q8—42no—
77C3×#542—91no—
78#5561—31no—
79C2×C32×D4C2×D311yescharacteristic
80C22×C32×C4C2×D312yes—
81C2×C3×#67—32no—
82C2×C3×C4×D3—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—32no—
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C22⋊C4C2×D312yes—
87C32×C42C2×D311yescharacteristic
88C3×#217—32no—
89C3×#217—32no—
90C3×#217—32no—
91C3×(C2×D3)⋊C4—31no—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—31no—
94C3×C4⋊Dic3—32no—
95C3×C4⋊Dic3—31no—
96C3×Dic3⋊C4—31no—
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—31no—
99C3×Dic3⋊C4—32no—
100C3×C4×Dic3—32no—
101C3×C4×Dic3—31no—
102C2×D4×C9—21no—
103C2×D4×C9C22×C311yescharacteristic
104C22×C4×C9—22no—
105C22×C4×C9C22×C312yes—
106C2×#611C22×C312yes—
107C2×C4×D9C22×C311yescharacteristic
108C2×Dic18C22×C311yescharacteristic
109C2×Dic18C22×C312yes—
110C9×C4⋊C4—21no—
111C9×C4⋊C4C22×C311yescharacteristic
112C9×C22⋊C4—22no—
113C9×C22⋊C4C22×C312yes—
114C42×C9—21no—
115C42×C9C22×C311yescharacteristic
116#3676C22×C312yes—
117#3676C22×C312yes—
118#3676C22×C312yes—
119(C2×D9)⋊C4C22×C311yescharacteristic
120(C2×D9)⋊C4C22×C312yes—
121(C2×D9)⋊C4C22×C311yescharacteristic
122C4⋊Dic9C22×C311yescharacteristic
123C4⋊Dic9C22×C312yes—
124Dic9⋊C4C22×C312yes—
125Dic9⋊C4C22×C312yes—
126Dic9⋊C4C22×C311yescharacteristic
127Dic9⋊C4C22×C311yescharacteristic
128C4×Dic9C22×C311yescharacteristic
129C4×Dic9C22×C312yes—
130C22×3−1+2—22no—
131C22×3−1+2—22no—
132C22×3−1+2—22no—
133C22×3−1+2C2411yescharacteristic
134C2×AGL(1,ℤ/9)—21no—
135C2×AGL(1,ℤ/9)—21no—
136C2×AGL(1,ℤ/9)—21no—
137C4×3−1+2—21no—
138C4×3−1+2—22no—
139C4×3−1+2—21no—
140C4×3−1+2#4112yes—
1413−1+2⋊C4—22no—
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—91no—
152C3×C22⋊Q8—92no—
153C3×#121—91no—
154C3×C4×Q8—91no—
155C3×C4×D4—31no—
156C3×C4×D4C3×D311yescharacteristic
157C3×#117—91no—
158#938—31no—
159#936—31no—
160#929—32no—
161#924—32no—
162#897—31no—
163#888—31no—
164#885—32no—
165#879—32no—
166#872—31no—
167C4×Dic6—31no—
168C23×C32C22×D312yes—
169C22×C3×D3—31no—
170C32×D4—22no—
171C2×C32×C4—42no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C3×#67—122no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C3×C4×D3—62no—
183C3×Dic6—62no—
184C3×Dic6—122no—
185C23×C9—22no—
186C23×C9C23×C312yes—
187C22×D9C23×C311yescharacteristic
188D4×C9—24no—
189D4×C9—22no—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—21no—
193C2×C4×C9—22no—
194C2×C4×C9—42no—
195C2×C4×C9—44no—
196C2×C4×C9C23×C311yescharacteristic
197C2×C4×C9C23×C311yescharacteristic
198C2×C4×C9C23×C312yes—
199C2×C4×C9C23×C311yescharacteristic
200#611—42no—
201C2×Dic9C23×C311yescharacteristic
202C2×Dic9C23×C312yes—
203C2×Dic9C23×C312yes—
204C2×Dic9C23×C311yescharacteristic
205C2×Dic9C23×C311yescharacteristic
206C4×D9—22no—
207Dic18—22no—
208Dic18—42no—
209#542—91noSylow
210C2×3−1+2—42no—
211C2×3−1+2C2×#4111yescharacteristic
212C2×3−1+2H(5,𝔽2)11yescharacteristic
213C2×3−1+22−1+411yescharacteristic
214AGL(1,ℤ/9)—41no—
215C2×C3×Q8—91no—
216C2×C3×Q8—92no—
217C2×C3×D4—31no—
218C2×C3×D4—92no—
219C2×C3×D4C2×C3×D311yescharacteristic
220C22×C3×C4—32no—
221C22×C3×C4—91no—
222C22×C3×C4C2×C3×D312yes—
223C2×#67—32no—
224C2×C4×D3—31no—
225C2×Dic6—31no—
226C2×Dic6—32no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—92no—
231C3×C4⋊C4—92no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4C2×C3×D311yescharacteristic
235C3×C22⋊C4—32no—
236C3×C22⋊C4—91no—
237C3×C22⋊C4—91no—
238C3×C22⋊C4—92no—
239C3×C22⋊C4—92no—
240C3×C22⋊C4—92no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4C2×C3×D312yes—
243C3×C42—31no—
244C3×C42—91no—
245C3×C42—92no—
246C3×C42C2×C3×D311yescharacteristic
247#217—32no—
248#217—32no—
249#217—32no—
250(C2×D3)⋊C4—31no—
251(C2×D3)⋊C4—31no—
252(C2×D3)⋊C4—32no—
253C4⋊Dic3—31no—
254C4⋊Dic3—32no—
255Dic3⋊C4—32no—
256Dic3⋊C4—31no—
257Dic3⋊C4—31no—
258Dic3⋊C4—32no—
259C4×Dic3—32no—
260C4×Dic3—31no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32C23×D311yescharacteristic
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C32×C4—22no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4#23512yes—
272C3×Dic3—62no—
273C3×Dic3—61no—
274C3×Dic3—61no—
275C3×Dic3—62no—
276C3×Dic3—61no—
277C22×C9—24no—
278C22×C9—22no—
279C22×C9—22no—
280C22×C9—22no—
281C22×C9—21no—
282C22×C9—42no—
283C22×C9—42no—
284C22×C9C24×C311yescharacteristic
285C2×D9—21no—
286C2×D9—21no—
287C2×D9—21no—
288C4×C9—22no—
289C4×C9—22no—
290C4×C9—21no—
291C4×C9—21no—
292C4×C9—22no—
293C4×C9—41no—
294C4×C9—42no—
295C4×C9C3×#4112yes—
296Dic9—22no—
297Dic9—21no—
298Dic9—21no—
299Dic9—22no—
300Dic9—21no—
301C4⋊Q8—91no—
302#126—92no—
303#124—91no—
304#124—92no—
305#123—92no—
306C22⋊Q8—92no—
307C22⋊Q8—91no—
308#121—91no—
309C4×Q8—91no—
310C4×D4AGL(1,ℤ/9)11yescharacteristic, complemented
311#117—91no—
3123−1+2#54211yescharacteristic, complemented, Sylow
313C23×C3—32no—
314C23×C3—91no—
315C23×C3C22×C3×D312yes—
316C22×D3—31no—
317C3×Q8—182no—
318C3×Q8—362no—
319C3×D4—22no—
320C3×D4—62no—
321C3×D4—362no—
322C2×C3×C4—31no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—32no—
326C2×C3×C4—42no—
327C2×C3×C4—91no—
328C2×C3×C4—92no—
329C2×C3×C4—91no—
330C2×C3×C4—92no—
331C2×C3×C4—91no—
332C2×C3×C4—122no—
333C2×C3×C4—182no—
334C2×C3×C4C22×C3×D311yescharacteristic
335C2×C3×C4C22×C3×D311yescharacteristic
336C2×C3×C4C22×C3×D312yes—
337C2×C3×C4C22×C3×D311yescharacteristic
338#67—122no—
339C2×Dic3—31no—
340C2×Dic3—32no—
341C2×Dic3—32no—
342C2×Dic3—31no—
343C2×Dic3—31no—
344C4×D3—62no—
345Dic6—62no—
346Dic6—122no—
347C2×C32—42no—
348C2×C32C2×#23511yescharacteristic
349C2×C32#100411yescharacteristic
350C2×C32#101011yescharacteristic
351C3×D3—121no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—44no—
356C2×C9—42no—
357C2×C9C2×C3×#4111yescharacteristic
358C2×C9C3×H(5,𝔽2)11yescharacteristic
359C2×C9C3×2−1+411yescharacteristic
360D9—41no—
361C2×Q8—91no—
362C2×Q8—92no—
363C2×D4—92no—
364C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
365C22×C4—91no—
366C22×C4C2×AGL(1,ℤ/9)12yes—
367C4⋊C4—91no—
368C4⋊C4—91no—
369C4⋊C4—91no—
370C4⋊C4—92no—
371C4⋊C4—92no—
372C4⋊C4—92no—
373C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
374C22⋊C4—92no—
375C22⋊C4—92no—
376C22⋊C4—92no—
377C22⋊C4—91no—
378C22⋊C4—91no—
379C22⋊C4—92no—
380C22⋊C4C2×AGL(1,ℤ/9)12yes—
381C42—91no—
382C42—92no—
383C42C2×AGL(1,ℤ/9)11yescharacteristic
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—31no—
388C22×C3—62no—
389C22×C3—62no—
390C22×C3—62no—
391C22×C3—181no—
392C22×C3—181no—
393C22×C3—181no—
394C22×C3C23×C3×D311yescharacteristic
395C2×D3—61no—
396C2×D3—61no—
397C2×D3—61no—
398C3×C4—22no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—32no—
402C3×C4—61no—
403C3×C4—62no—
404C3×C4—61no—
405C3×C4—181no—
406C3×C4—182no—
407C3×C4—182no—
408C3×C4—181no—
409C3×C4—181no—
410C3×C4C3×#23512yes—
411Dic3—62no—
412Dic3—61no—
413Dic3—61no—
414Dic3—62no—
415Dic3—61no—
416C32#556111yescharacteristic
417C9—21no—
418C9C3×#54211yescharacteristic, complemented
419C23—91no—
420C23C22×AGL(1,ℤ/9)12yes—
421Q8—182no—
422Q8—362no—
423D4—22no—
424D4—362no—
425C2×C4—42no—
426C2×C4—91no—
427C2×C4—92no—
428C2×C4—91no—
429C2×C4—92no—
430C2×C4—91no—
431C2×C4—182no—
432C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
433C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
434C2×C4C22×AGL(1,ℤ/9)12yes—
435C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—42no—
440C2×C3—122no—
441C2×C3—361no—
442C2×C3C2×C3×#23511yescharacteristic
443C2×C3C3×#100411yescharacteristic
444C2×C3C3×#101011yescharacteristic
445D3—121no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—181no—
450C22—181no—
451C22—181no—
452C22C23×AGL(1,ℤ/9)11yescharacteristic
453C4—21no—
454C4—22no—
455C4—21no—
456C4—181no—
457C4—182no—
458C4—182no—
459C4—181no—
460C4—181no—
461C4#3697⋊C312yes—
462C3—31no—
463C3C3×#556111yescharacteristic
464C2—42no—
465C2—361no—
466C2C2×#3697⋊C311yescharacteristic
467C2#63710⋊C311yescharacteristic
468C2#1172264211yescharacteristic
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC4×D4×3−1+2C2
prime radical#63500⋊C3C2
hypercenterC4×D4AGL(1,ℤ/9)
hypocenterC9C3×#542
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
3. #63498⋊C3C2semidirect
4. #63491⋊C3C2semidirect
5. #63486⋊C3C2semidirect
6. #63459⋊C3C2semidirect
7. #63450⋊C3C2semidirect
8. #63447⋊C3C2semidirect
9. #63441⋊C3C2semidirect
10. #63434⋊C3C2semidirect
11. C4×3−1+2⋊Q8C2semidirect
13. #10589663C3semidirect
49. C4×D4×C9C2×C3semidirect
51. #63498C2×C3semidirect
52. #63491C2×C3semidirect
53. #63486C2×C3semidirect
54. #63459C2×C3semidirect
55. #63450C2×C3semidirect
56. #63447C2×C3semidirect
57. #63441C2×C3semidirect
58. #63434C2×C3semidirect
59. C4×Dic18C2×C3semidirect
310. C4×D4AGL(1,ℤ/9)semidirect
312. 3−1+2#542semidirect
418. C9C3×#542semidirect

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
#63498⋊C32
#63491⋊C32
#63486⋊C32
#63459⋊C32
#63450⋊C32
#63447⋊C32
#63441⋊C32
#63434⋊C32
C4×3−1+2⋊Q82
C3×#55613
#105896633