Finite Group Databasebeta

← C3×#10589750 #13152996

Subgroups — C3×#10589750 #13152996

Order 1728.

888 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C211yesmaximal, characteristic
2C22×C3×Dic18C211yesmaximal, characteristic, complemented
3C3×C9×#123C211yesmaximal, characteristic, complemented
4C3×#63497C211yesmaximal, characteristic, complemented
5C3×#63491C211yesmaximal, characteristic, complemented
6C3×#63458C212yesmaximal, complemented
7C3×#63450C212yesmaximal, complemented
8C3×#63448C211yesmaximal, characteristic, complemented
9C3×#63447C212yesmaximal, complemented
10C3×#63440C212yesmaximal, complemented
11C3×#63439C211yesmaximal, characteristic, complemented
12C3×#5648—31nomaximal
13#10589750C311yesmaximal, characteristic, direct product factor
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×C3×#611C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C22×C3×Dic9C2211yescharacteristic
21C3×#3699—24no—
22C2×C3×C4×D9C2211yescharacteristic
23C2×C3×Dic18—24no—
24C2×C3×Dic18C2212yes—
25C2×C3×Dic18C2212yes—
26C2×C3×Dic18C2211yescharacteristic
27C3×C9×C4⋊C4C2212yes—
28C3×C9×C22⋊C4C2211yescharacteristic
29C3×C9×C22⋊C4C2212yes—
30C3×#3676C2211yescharacteristic
31C3×#3676C2212yes—
32C3×(C2×D9)⋊C4C2212yes—
33C3×(C2×D9)⋊C4C2212yes—
34C3×C4⋊Dic9C2212yes—
35C3×Dic9⋊C4C2212yes—
36C3×Dic9⋊C4C2212yes—
37C3×Dic9⋊C4C2212yes—
38C3×C4×Dic9C2212yes—
39C2×C3×#237—31no—
40C22×C3×Dic6—31no—
41C32×#123D311yescharacteristic
42C3×#935—31no—
43C3×#929—31no—
44C3×#896—32no—
45C3×#888—32no—
46C3×#886—31no—
47C3×#885—32no—
48C3×#878—32no—
49C3×#877—31no—
50C2×#3699C2×C311yescharacteristic
51C22×Dic18C2×C311yescharacteristic, complemented
52C9×#123—21no—
53C9×#123C2×C311yescharacteristic, complemented
54#63497C2×C311yescharacteristic, complemented
55#63491C2×C311yescharacteristic, complemented
56#63458C2×C312yescomplemented
57#63450C2×C312yescomplemented
58#63448C2×C311yescharacteristic, complemented
59#63447C2×C312yescomplemented
60#63440C2×C312yescomplemented
61#63439C2×C311yescharacteristic, complemented
62C23×C3×C9C2311yescharacteristic
63C23×C3×C9C2311yescharacteristic
64C22×C3×D9C2311yescharacteristic
65C3×D4×C9—22no—
66C2×C3×C4×C9—22no—
67C2×C3×C4×C9C2312yes—
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3×C4×C9C2312yes—
70C3×#611—22no—
71C3×#611—22no—
72C2×C3×Dic9—22no—
73C2×C3×Dic9—22no—
74C2×C3×Dic9D414yes—
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2312yes—
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80C3×C4×D9—22no—
81C3×Dic18—24no—
82C3×Dic18—24no—
83C3×Dic18—22no—
84C3×#536—91no—
85#5648—31no—
86C2×C32×D4C2×D311yescharacteristic
87C22×C32×C4C2×D311yescharacteristic
88C2×C3×#67—31no—
89C2×C3×#67—31no—
90C22×C3×Dic3—31no—
91C22×C3×Dic3—31no—
92C22×C3×Dic3—31no—
93C3×#237—64no—
94C2×C3×C4×D3—31no—
95C2×C3×Dic6—32no—
96C2×C3×Dic6—32no—
97C2×C3×Dic6—31no—
98C2×C3×Dic6—64no—
99C32×C4⋊C4C2×D312yes—
100C32×C22⋊C4C2×D311yescharacteristic
101C32×C22⋊C4C2×D312yes—
102C3×#217—31no—
103C3×#217—32no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—32no—
106C3×C4⋊Dic3—32no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×C4×Dic3—32no—
111C2×D4×C9—21no—
112C2×D4×C9C22×C311yescharacteristic
113C22×C4×C9—21no—
114C22×C4×C9C22×C311yescharacteristic
115C2×#611C22×C311yescharacteristic
116C2×#611C22×C311yescharacteristic
117C22×Dic9C22×C311yescharacteristic
118C22×Dic9C22×C311yescharacteristic
119C22×Dic9C22×C311yescharacteristic
120#3699—24no—
121C2×C4×D9C22×C311yescharacteristic
122C2×Dic18—24no—
123C2×Dic18C22×C312yes—
124C2×Dic18C22×C312yes—
125C2×Dic18C22×C311yescharacteristic
126C9×C4⋊C4—22no—
127C9×C4⋊C4C22×C312yes—
128C9×C22⋊C4—21no—
129C9×C22⋊C4—22no—
130C9×C22⋊C4C22×C311yescharacteristic
131C9×C22⋊C4C22×C312yes—
132#3676C22×C311yescharacteristic
133#3676C22×C312yes—
134(C2×D9)⋊C4C22×C312yes—
135(C2×D9)⋊C4C22×C312yes—
136C4⋊Dic9C22×C312yes—
137Dic9⋊C4C22×C312yes—
138Dic9⋊C4C22×C312yes—
139Dic9⋊C4C22×C312yes—
140C4×Dic9C22×C312yes—
141C22×C3×C9—22no—
142C22×C3×C9—21no—
143C22×C3×C9—22no—
144C22×C3×C9C2×D412yes—
145C22×C3×C9C2411yescharacteristic
146C2×C3×D9—21no—
147C2×C3×D9—22no—
148C3×C4×C9—22no—
149C3×C4×C9—21no—
150C3×C4×C9—22no—
151C3×Dic9—22no—
152C3×Dic9—21no—
153C3×Dic9—22no—
154C3×Dic9C2×D412yes—
155C3×Dic9C2×D412yes—
156C2×C3×#41—91no—
157C22×C3×Q8—91no—
158C2×#237—31no—
159C22×Dic6—31no—
160C3×C4⋊Q8—92no—
161C3×#124—92no—
162C3×#123—21no—
163C3×#123—91no—
164C3×#123—91no—
165C3×#123—91no—
166C3×#123D911yescharacteristic, complemented
167C3×#123C3×D311yescharacteristic
168C3×C22⋊Q8—92no—
169C3×C22⋊Q8—92no—
170C3×#117—91no—
171#935—31no—
172#929—31no—
173#896—32no—
174#888—32no—
175#886—31no—
176#885—32no—
177#878—32no—
178#877—31no—
179C23×C32C22×D311yescharacteristic
180C23×C32C22×D311yescharacteristic
181C22×C3×D3—31no—
182C32×D4—22no—
183C2×C32×C4—22no—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D312yes—
187C3×#67—62no—
188C3×#67—62no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—62no—
197C3×C4×D3—62no—
198C3×Dic6—64no—
199C3×Dic6—64no—
200C3×Dic6—62no—
201C23×C9—21no—
202C23×C9—21no—
203C23×C9C23×C311yescharacteristic
204C23×C9C23×C311yescharacteristic
205C22×D9C23×C311yescharacteristic
206D4×C9—22no—
207D4×C9—42no—
208C2×C4×C9—22no—
209C2×C4×C9—22no—
210C2×C4×C9—22no—
211C2×C4×C9—21no—
212C2×C4×C9—42no—
213C2×C4×C9C23×C312yes—
214C2×C4×C9C23×C311yescharacteristic
215C2×C4×C9C23×C312yes—
216#611—22no—
217#611—22no—
218C2×Dic9—22no—
219C2×Dic9—22no—
220C2×Dic9C3×D414yes—
221C2×Dic9C23×C312yes—
222C2×Dic9C23×C311yescharacteristic
223C2×Dic9C23×C312yes—
224C2×Dic9C23×C311yescharacteristic
225C2×Dic9C23×C311yescharacteristic
226C4×D9—22no—
227Dic18—24no—
228Dic18—24no—
229Dic18—22no—
230#536—91noSylow
231C2×C3×C9—22no—
232C2×C3×C9—41no—
233C2×C3×C9C22×D411yescharacteristic
234C2×C3×C92−1+412yes—
235C3×D9—41no—
236C3×#41—184no—
237C2×C3×Q8—92no—
238C2×C3×Q8—92no—
239C2×C3×Q8—91no—
240C2×C3×Q8—184no—
241C2×C3×D4—21no—
242C2×C3×D4—91no—
243C2×C3×D4—91no—
244C2×C3×D4C2×D911yescharacteristic
245C2×C3×D4C2×C3×D311yescharacteristic
246C22×C3×C4—21no—
247C22×C3×C4—91no—
248C22×C3×C4—91no—
249C22×C3×C4—91no—
250C22×C3×C4—91no—
251C22×C3×C4C2×D911yescharacteristic
252C22×C3×C4C2×C3×D311yescharacteristic
253C2×#67—31no—
254C2×#67—31no—
255C22×Dic3—31no—
256C22×Dic3—31no—
257C22×Dic3—31no—
258#237—64no—
259C2×C4×D3—31no—
260C2×Dic6—32no—
261C2×Dic6—32no—
262C2×Dic6—31no—
263C2×Dic6—64no—
264C3×C4⋊C4—22no—
265C3×C4⋊C4—92no—
266C3×C4⋊C4—92no—
267C3×C4⋊C4—92no—
268C3×C4⋊C4—92no—
269C3×C4⋊C4C2×D912yes—
270C3×C4⋊C4C2×C3×D312yes—
271C3×C22⋊C4—21no—
272C3×C22⋊C4—22no—
273C3×C22⋊C4—91no—
274C3×C22⋊C4—92no—
275C3×C22⋊C4—92no—
276C3×C22⋊C4—92no—
277C3×C22⋊C4C2×D911yescharacteristic
278C3×C22⋊C4C2×D912yes—
279C3×C22⋊C4C2×C3×D311yescharacteristic
280C3×C22⋊C4C2×C3×D312yes—
281C3×C42—92no—
282#217—31no—
283#217—32no—
284(C2×D3)⋊C4—32no—
285(C2×D3)⋊C4—32no—
286C4⋊Dic3—32no—
287Dic3⋊C4—32no—
288Dic3⋊C4—32no—
289Dic3⋊C4—32no—
290C4×Dic3—32no—
291C22×C32—22no—
292C22×C32—21no—
293C22×C32—22no—
294C22×C32D3×D412yes—
295C22×C32C23×D311yescharacteristic
296C2×C3×D3—61no—
297C2×C3×D3—62no—
298C32×C4—22no—
299C32×C4—21no—
300C32×C4—22no—
301C3×Dic3—32no—
302C3×Dic3—32no—
303C3×Dic3—62no—
304C3×Dic3—61no—
305C3×Dic3—62no—
306C22×C9—22no—
307C22×C9—21no—
308C22×C9—22no—
309C22×C9—21no—
310C22×C9—22no—
311C22×C9—42no—
312C22×C9—41no—
313C22×C9—42no—
314C22×C9C2×C3×D412yes—
315C22×C9C24×C311yescharacteristic
316C2×D9—21no—
317C2×D9—22no—
318C4×C9—22no—
319C4×C9—21no—
320C4×C9—22no—
321C4×C9—42no—
322C4×C9—42no—
323C4×C9—41no—
324Dic9—22no—
325Dic9—21no—
326Dic9—22no—
327Dic9C2×C3×D412yes—
328Dic9C2×C3×D412yes—
329C2×#41—91no—
330C22×Q8—91no—
331C4⋊Q8—92no—
332#124—92no—
333#123—91no—
334#123—91no—
335#123—91no—
336#123C3×D911yescharacteristic, complemented
337C22⋊Q8—92no—
338C22⋊Q8—92no—
339#117—91no—
340C3×C9#53611yescharacteristic, complemented, Sylow
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—91no—
344C23×C3C22×D911yescharacteristic
345C23×C3C22×D911yescharacteristic
346C23×C3C22×C3×D311yescharacteristic
347C23×C3C22×C3×D311yescharacteristic
348C22×D3—31no—
349C3×Q8—184no—
350C3×Q8—184no—
351C3×Q8—182no—
352C3×D4—22no—
353C3×D4—22no—
354C3×D4—42no—
355C3×D4—182no—
356C3×D4—182no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—42no—
363C2×C3×C4—94no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—92no—
367C2×C3×C4—92no—
368C2×C3×C4—91no—
369C2×C3×C4—182no—
370C2×C3×C4—182no—
371C2×C3×C4—182no—
372C2×C3×C4C22×D912yes—
373C2×C3×C4C22×D912yes—
374C2×C3×C4C22×D911yescharacteristic
375C2×C3×C4C22×C3×D312yes—
376C2×C3×C4C22×C3×D311yescharacteristic
377C2×C3×C4C22×C3×D312yes—
378#67—62no—
379#67—62no—
380C2×Dic3—31no—
381C2×Dic3—31no—
382C2×Dic3—34no—
383C2×Dic3—32no—
384C2×Dic3—31no—
385C2×Dic3—32no—
386C2×Dic3—62no—
387C2×Dic3—62no—
388C4×D3—62no—
389Dic6—64no—
390Dic6—64no—
391Dic6—62no—
392C2×C32—22no—
393C2×C32—41no—
394C2×C32C2×D3×D411yescharacteristic
395C2×C32#101012yes—
396C3×D3—121no—
397C2×C9—22no—
398C2×C9—21no—
399C2×C9—22no—
400C2×C9—42no—
401C2×C9—41no—
402C2×C9—81no—
403C2×C9C22×C3×D411yescharacteristic
404C2×C9C3×2−1+412yes—
405D9—41no—
406#41—184no—
407C2×Q8—92no—
408C2×Q8—92no—
409C2×Q8—91no—
410C2×Q8—184no—
411C2×D4—91no—
412C2×D4—91no—
413C2×D4C2×C3×D911yescharacteristic
414C22×C4—91no—
415C22×C4—91no—
416C22×C4—91no—
417C22×C4—91no—
418C22×C4C2×C3×D911yescharacteristic
419C4⋊C4—92no—
420C4⋊C4—92no—
421C4⋊C4—92no—
422C4⋊C4—92no—
423C4⋊C4C2×C3×D912yes—
424C22⋊C4—91no—
425C22⋊C4—92no—
426C22⋊C4—92no—
427C22⋊C4—92no—
428C22⋊C4C2×C3×D911yescharacteristic
429C22⋊C4C2×C3×D912yes—
430C42—92no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—21no—
434C22×C3—22no—
435C22×C3—21no—
436C22×C3—22no—
437C22×C3—21no—
438C22×C3—22no—
439C22×C3—42no—
440C22×C3—41no—
441C22×C3—42no—
442C22×C3—181no—
443C22×C3—182no—
444C22×C3D4×D912yes—
445C22×C3C23×D911yescharacteristic
446C22×C3C3×D3×D412yes—
447C22×C3C23×C3×D311yescharacteristic
448C2×D3—61no—
449C2×D3—62no—
450C3×C4—22no—
451C3×C4—21no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—21no—
455C3×C4—22no—
456C3×C4—42no—
457C3×C4—41no—
458C3×C4—42no—
459C3×C4—92no—
460C3×C4—92no—
461C3×C4—181no—
462C3×C4—182no—
463C3×C4—182no—
464Dic3—32no—
465Dic3—32no—
466Dic3—62no—
467Dic3—61no—
468Dic3—62no—
469C32#564811yescharacteristic
470C9—21no—
471C9C3×#53611yescharacteristic, complemented
472C23—91no—
473C23C22×C3×D911yescharacteristic
474C23C22×C3×D911yescharacteristic
475Q8—184no—
476Q8—184no—
477Q8—182no—
478D4—22no—
479D4—182no—
480D4—182no—
481C2×C4—22no—
482C2×C4—91no—
483C2×C4—91no—
484C2×C4—94no—
485C2×C4—92no—
486C2×C4—92no—
487C2×C4—91no—
488C2×C4—182no—
489C2×C4—182no—
490C2×C4—182no—
491C2×C4C22×C3×D912yes—
492C2×C4C22×C3×D912yes—
493C2×C4C22×C3×D911yescharacteristic
494C2×C3—22no—
495C2×C3—22no—
496C2×C3—21no—
497C2×C3—22no—
498C2×C3—42no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—81no—
502C2×C3—361no—
503C2×C3C2×D4×D911yescharacteristic
504C2×C3#6371612yes—
505C2×C3C2×C3×D3×D411yescharacteristic
506C2×C3C3×#101012yes—
507D3—121no—
508C22—22no—
509C22—21no—
510C22—22no—
511C22—181no—
512C22—182no—
513C22C3×D4×D912yes—
514C22C23×C3×D911yescharacteristic
515C4—22no—
516C4—21no—
517C4—22no—
518C4—92no—
519C4—92no—
520C4—181no—
521C4—182no—
522C4—182no—
523C3—21no—
524C3#1058975011yescharacteristic, direct product factor
525C3C3×#564811yescharacteristic
526C2—22no—
527C2—41no—
528C2—361no—
529C2C2×C3×D4×D911yescharacteristic
530C2C3×#6371612yes—
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#123C2
socleC22×C32C23×D3
prime radicalC2×C3×#3699C2
hypercenterC3×#123D9
hypocenterC9C3×#536
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#123

Complements

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

NH ≅ G/Ntype
2. C22×C3×Dic18C2semidirect
3. C3×C9×#123C2semidirect
4. C3×#63497C2semidirect
5. C3×#63491C2semidirect
6. C3×#63458C2semidirect
7. C3×#63450C2semidirect
8. C3×#63448C2semidirect
9. C3×#63447C2semidirect
10. C3×#63440C2semidirect
11. C3×#63439C2semidirect
13. #10589750C3direct (N × H)
51. C22×Dic18C2×C3semidirect
53. C9×#123C2×C3semidirect
54. #63497C2×C3semidirect
55. #63491C2×C3semidirect
56. #63458C2×C3semidirect
57. #63450C2×C3semidirect
58. #63448C2×C3semidirect
59. #63447C2×C3semidirect
60. #63440C2×C3semidirect
61. #63439C2×C3semidirect
166. C3×#123D9semidirect
336. #123C3×D9semidirect
340. C3×C9#536semidirect
471. C9C3×#536semidirect
524. C3#10589750direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C22×C3×Dic182
C3×C9×#1232
C3×#634972
C3×#634912
C3×#634582
C3×#634502
C3×#634482
C3×#634472
C3×#634402
C3×#634392
C3×#56483
#105897503