Finite Group Databasebeta

← C3×#10589699 #13152788

Subgroups — C3×#10589699 #13152788

Order 1728.

976 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C212yesmaximal, complemented
2C3×C9×#121C211yesmaximal, characteristic, complemented
3C3×#63501C212yesmaximal, complemented
4C3×#63501C212yesmaximal, complemented
5C3×#63492C211yesmaximal, characteristic, complemented
6C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
7C3×#63458C212yesmaximal, complemented
8C3×#63446C212yesmaximal, complemented
9C3×#63440C212yesmaximal, complemented
10C3×#5597—31nomaximal
11#10589699C311yesmaximal, characteristic, direct product factor
12C2×C3×D4×C9C2212yes—
13C2×C3×D4×C9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic
15C2×C3×#611C2212yes—
16C2×C3×#611C2212yes—
17C2×C3×#611C2212yes—
18C22×C3×Dic9C2212yes—
19C22×C3×Dic9C2211yescharacteristic
20C22×C3×Dic9C2211yescharacteristic
21C3×#3699—28no—
22C2×C3×D36C2211yescharacteristic
23C2×C3×C4×D9C2212yes—
24C2×C3×Dic18C2212yes—
25C3×C9×C4⋊C4C2211yescharacteristic
26C3×C9×C22⋊C4C2212yes—
27C3×#3676C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C3×(C2×D9)⋊C4C2212yes—
30C3×C4⋊Dic9C2211yescharacteristic
31C3×Dic9⋊C4C2212yes—
32C3×Dic9⋊C4C2212yes—
33C3×Dic9⋊C4C2212yes—
34C2×C3×#237—32no—
35C32×#121D311yescharacteristic
36C3×#939—32no—
37C3×#939—32no—
38C3×#930—31no—
39C2×C3×Dic3⋊C4—31no—
40C3×#896—32no—
41C3×#884—32no—
42C3×#878—32no—
43C2×#3699C2×C312yescomplemented
44C9×#121—21no—
45C9×#121C2×C311yescharacteristic, complemented
46#63501C2×C312yescomplemented
47#63501C2×C312yescomplemented
48#63492C2×C311yescharacteristic, complemented
49C2×Dic9⋊C4C2×C311yescharacteristic, complemented
50#63458C2×C312yescomplemented
51#63446C2×C312yescomplemented
52#63440C2×C312yescomplemented
53C23×C3×C9C2312yes—
54C23×C3×C9C2311yescharacteristic
55C22×C3×D9C2312yes—
56C3×D4×C9—24no—
57C3×D4×C9—41no—
58C2×C3×C4×C9—41no—
59C2×C3×C4×C9C2312yes—
60C2×C3×C4×C9C2311yescharacteristic
61C2×C3×C4×C9C2311yescharacteristic
62C3×#611—24no—
63C3×#611—24no—
64C3×#611—42no—
65C2×C3×Dic9—24no—
66C2×C3×Dic9—41no—
67C2×C3×Dic9D414yescomplemented
68C2×C3×Dic9C2312yes—
69C2×C3×Dic9C2312yes—
70C2×C3×Dic9C2312yes—
71C3×D36—41no—
72C3×C4×D9—24no—
73C3×Dic18—24no—
74C3×#537—91no—
75#5597—31no—
76C2×C32×D4C2×D312yes—
77C2×C32×D4C2×D311yescharacteristic
78C22×C32×C4C2×D311yescharacteristic
79C2×C3×#67—32no—
80C2×C3×#67—32no—
81C2×C3×#67—32no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—32no—
84C22×C3×Dic3—31no—
85C3×#237—68no—
86C2×C3×D12—31no—
87C2×C3×C4×D3—32no—
88C2×C3×Dic6—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C22⋊C4C2×D312yes—
91C3×#217—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×C4⋊Dic3—31no—
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—32no—
98C2×D4×C9—21no—
99C2×D4×C9—22no—
100C2×D4×C9C22×C312yes—
101C2×D4×C9C22×C311yescharacteristic
102C22×C4×C9—21no—
103C22×C4×C9C22×C311yescharacteristic
104C2×#611C22×C312yes—
105C2×#611C22×C312yes—
106C2×#611C22×C312yes—
107C22×Dic9C22×C312yes—
108C22×Dic9C22×C311yescharacteristic
109C22×Dic9C22×C311yescharacteristic
110#3699—28no—
111C2×D36C22×C311yescharacteristic
112C2×C4×D9C22×C312yes—
113C2×Dic18C22×C312yes—
114C9×C4⋊C4—21no—
115C9×C4⋊C4C22×C311yescharacteristic
116C9×C22⋊C4—22no—
117C9×C22⋊C4C22×C312yes—
118#3676C22×C312yes—
119(C2×D9)⋊C4C22×C312yes—
120(C2×D9)⋊C4C22×C312yes—
121C4⋊Dic9C22×C311yescharacteristic
122Dic9⋊C4C22×C312yes—
123Dic9⋊C4C22×C312yes—
124Dic9⋊C4C22×C312yes—
125C22×C3×C9—21no—
126C22×C3×C9—21no—
127C22×C3×C9—22no—
128C22×C3×C9—22no—
129C22×C3×C9—22no—
130C22×C3×C9C2×D412yes—
131C22×C3×C9C2411yescharacteristic
132C2×C3×D9—22no—
133C2×C3×D9—22no—
134C2×C3×D9—22no—
135C3×C4×C9—22no—
136C3×C4×C9—21no—
137C3×C4×C9—21no—
138C3×Dic9—22no—
139C3×Dic9—22no—
140C3×Dic9C2×D414yes—
141C2×C3×#41—92no—
142C2×#237—32no—
143C3×C22⋊Q8—92no—
144C3×C22⋊Q8—92no—
145C3×#121—21no—
146C3×#121—92no—
147C3×#121—92no—
148C3×#121—91no—
149C3×#121—92no—
150C3×#121D911yescharacteristic, complemented
151C3×#121C3×D311yescharacteristic
152C2×C3×C4⋊C4—91no—
153#939—32no—
154#939—32no—
155#930—31no—
156C2×Dic3⋊C4—31no—
157#896—32no—
158#884—32no—
159#878—32no—
160C23×C32C22×D312yes—
161C23×C32C22×D311yescharacteristic
162C22×C3×D3—32no—
163C32×D4—24no—
164C32×D4—41no—
165C2×C32×C4—41no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D312yes—
169C3×#67—64no—
170C3×#67—64no—
171C3×#67—122no—
172C2×C3×Dic3—34no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—64no—
177C2×C3×Dic3—121no—
178C3×D12—121no—
179C3×C4×D3—64no—
180C3×Dic6—64no—
181C23×C9—21no—
182C23×C9—22no—
183C23×C9C23×C312yes—
184C23×C9C23×C311yescharacteristic
185C22×D9C23×C312yes—
186D4×C9—24no—
187D4×C9—44no—
188D4×C9—41no—
189D4×C9—42no—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—22no—
193C2×C4×C9—41no—
194C2×C4×C9—42no—
195C2×C4×C9C23×C311yescharacteristic
196C2×C4×C9C23×C311yescharacteristic
197C2×C4×C9C23×C312yes—
198#611—24no—
199#611—24no—
200#611—42no—
201C2×Dic9—24no—
202C2×Dic9—41no—
203C2×Dic9C3×D414yescomplemented
204C2×Dic9C23×C312yes—
205C2×Dic9C23×C312yes—
206C2×Dic9C23×C312yes—
207D36—41no—
208C4×D9—24no—
209Dic18—24no—
210#537—91noSylow
211C2×C3×C9—22no—
212C2×C3×C9—42no—
213C2×C3×C9C22×D411yescharacteristic
214C2×C3×C9H(5,𝔽2)11yescharacteristic
215C2×C3×C92−1+411yescharacteristic
216C3×D9—42no—
217C3×#41—188no—
218C2×C3×Q8—92no—
219C2×C3×D4—22no—
220C2×C3×D4—21no—
221C2×C3×D4—92no—
222C2×C3×D4—92no—
223C2×C3×D4—92no—
224C2×C3×D4—91no—
225C2×C3×D4C2×D912yes—
226C2×C3×D4C2×D911yescharacteristic
227C2×C3×D4C2×C3×D312yes—
228C2×C3×D4C2×C3×D311yescharacteristic
229C22×C3×C4—21no—
230C22×C3×C4—91no—
231C22×C3×C4—92no—
232C22×C3×C4—91no—
233C22×C3×C4—92no—
234C22×C3×C4C2×D911yescharacteristic
235C22×C3×C4C2×C3×D311yescharacteristic
236C2×#67—32no—
237C2×#67—32no—
238C2×#67—32no—
239C22×Dic3—31no—
240C22×Dic3—32no—
241C22×Dic3—31no—
242#237—68no—
243C2×D12—31no—
244C2×C4×D3—32no—
245C2×Dic6—32no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4C2×D911yescharacteristic
252C3×C4⋊C4C2×C3×D311yescharacteristic
253C3×C22⋊C4—22no—
254C3×C22⋊C4—92no—
255C3×C22⋊C4—92no—
256C3×C22⋊C4—92no—
257C3×C22⋊C4C2×D912yes—
258C3×C22⋊C4C2×C3×D312yes—
259#217—32no—
260(C2×D3)⋊C4—32no—
261(C2×D3)⋊C4—32no—
262C4⋊Dic3—31no—
263Dic3⋊C4—32no—
264Dic3⋊C4—32no—
265Dic3⋊C4—32no—
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—22no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32D3×D412yes—
272C22×C32C23×D311yescharacteristic
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C32×C4—21no—
277C32×C4—22no—
278C32×C4—21no—
279C3×Dic3—34no—
280C3×Dic3—62no—
281C3×Dic3—62no—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—22no—
285C22×C9—22no—
286C22×C9—22no—
287C22×C9—21no—
288C22×C9—22no—
289C22×C9—41no—
290C22×C9—41no—
291C22×C9—42no—
292C22×C9—42no—
293C22×C9—42no—
294C22×C9C2×C3×D412yes—
295C22×C9C24×C311yescharacteristic
296C2×D9—22no—
297C2×D9—22no—
298C2×D9—22no—
299C4×C9—22no—
300C4×C9—22no—
301C4×C9—21no—
302C4×C9—21no—
303C4×C9—41no—
304C4×C9—42no—
305Dic9—22no—
306Dic9—22no—
307Dic9C2×C3×D414yes—
308C2×#41—92no—
309C22⋊Q8—92no—
310C22⋊Q8—92no—
311#121—92no—
312#121—92no—
313#121—91no—
314#121—92no—
315#121C3×D911yescharacteristic, complemented
316C2×C4⋊C4—91no—
317C3×C9#53711yescharacteristic, complemented, Sylow
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—92no—
321C23×C3C22×D912yes—
322C23×C3C22×D911yescharacteristic
323C23×C3C22×C3×D312yes—
324C23×C3C22×C3×D311yescharacteristic
325C22×D3—32no—
326C3×Q8—184no—
327C3×D4—24no—
328C3×D4—24no—
329C3×D4—44no—
330C3×D4—41no—
331C3×D4—41no—
332C3×D4—42no—
333C3×D4—184no—
334C3×D4—184no—
335C3×D4—361no—
336C3×D4—362no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—94no—
344C2×C3×C4—92no—
345C2×C3×C4—92no—
346C2×C3×C4—92no—
347C2×C3×C4—184no—
348C2×C3×C4—184no—
349C2×C3×C4—361no—
350C2×C3×C4C22×D911yescharacteristic
351C2×C3×C4C22×D912yes—
352C2×C3×C4C22×D911yescharacteristic
353C2×C3×C4C22×C3×D311yescharacteristic
354C2×C3×C4C22×C3×D311yescharacteristic
355C2×C3×C4C22×C3×D312yes—
356#67—64no—
357#67—64no—
358#67—122no—
359C2×Dic3—32no—
360C2×Dic3—34no—
361C2×Dic3—32no—
362C2×Dic3—32no—
363C2×Dic3—64no—
364C2×Dic3—121no—
365D12—121no—
366C4×D3—64no—
367Dic6—64no—
368C2×C32—22no—
369C2×C32—42no—
370C2×C32C2×D3×D411yescharacteristic
371C2×C32#100411yescharacteristic
372C2×C32#101011yescharacteristic
373C3×D3—122no—
374C2×C9—22no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—42no—
379C2×C9—42no—
380C2×C9—82no—
381C2×C9C22×C3×D411yescharacteristic
382C2×C9C3×H(5,𝔽2)11yescharacteristic
383C2×C9C3×2−1+411yescharacteristic
384D9—42no—
385#41—188no—
386C2×Q8—92no—
387C2×D4—92no—
388C2×D4—92no—
389C2×D4—92no—
390C2×D4—91no—
391C2×D4C2×C3×D912yes—
392C2×D4C2×C3×D911yescharacteristic
393C22×C4—91no—
394C22×C4—92no—
395C22×C4—91no—
396C22×C4—92no—
397C22×C4C2×C3×D911yescharacteristic
398C4⋊C4—92no—
399C4⋊C4—92no—
400C4⋊C4—92no—
401C4⋊C4—91no—
402C4⋊C4C2×C3×D911yescharacteristic
403C22⋊C4—92no—
404C22⋊C4—92no—
405C22⋊C4—92no—
406C22⋊C4C2×C3×D912yes—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—182no—
425C22×C3—182no—
426C22×C3—182no—
427C22×C3D4×D912yes—
428C22×C3C23×D911yescharacteristic
429C22×C3C3×D3×D412yes—
430C22×C3C23×C3×D311yescharacteristic
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—62no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—22no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—41no—
442C3×C4—42no—
443C3×C4—94no—
444C3×C4—182no—
445C3×C4—182no—
446Dic3—34no—
447Dic3—62no—
448Dic3—62no—
449C32#559711yescharacteristic
450C9—21no—
451C9C3×#53711yescharacteristic, complemented
452C23—92no—
453C23C22×C3×D912yes—
454C23C22×C3×D911yescharacteristic
455Q8—184no—
456D4—24no—
457D4—41no—
458D4—184no—
459D4—184no—
460D4—361no—
461D4—362no—
462C2×C4—41no—
463C2×C4—92no—
464C2×C4—94no—
465C2×C4—92no—
466C2×C4—92no—
467C2×C4—184no—
468C2×C4—184no—
469C2×C4—361no—
470C2×C4C22×C3×D911yescharacteristic
471C2×C4C22×C3×D912yes—
472C2×C4C22×C3×D911yescharacteristic
473C2×C3—22no—
474C2×C3—22no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—42no—
479C2×C3—42no—
480C2×C3—42no—
481C2×C3—82no—
482C2×C3—362no—
483C2×C3C2×D4×D911yescharacteristic
484C2×C3#6371011yescharacteristic
485C2×C3#6371611yescharacteristic
486C2×C3C2×C3×D3×D411yescharacteristic
487C2×C3C3×#100411yescharacteristic
488C2×C3C3×#101011yescharacteristic
489D3—122no—
490C22—21no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—182no—
496C22—182no—
497C22—182no—
498C22C3×D4×D912yes—
499C22C23×C3×D911yescharacteristic
500C4—22no—
501C4—21no—
502C4—21no—
503C4—94no—
504C4—182no—
505C4—182no—
506C3—21no—
507C3#1058969911yescharacteristic, direct product factor
508C3C3×#559711yescharacteristic
509C2—22no—
510C2—42no—
511C2—362no—
512C2C2×C3×D4×D911yescharacteristic
513C2C3×#6371011yescharacteristic
514C2C3×#6371611yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#121C2
socleC22×C32C23×D3
hypercenterC3×#121D9
hypocenterC9C3×#537
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3699C2semidirect
2. C3×C9×#121C2semidirect
3. C3×#63501C2semidirect
4. C3×#63501C2semidirect
5. C3×#63492C2semidirect
6. C2×C3×Dic9⋊C4C2semidirect
7. C3×#63458C2semidirect
8. C3×#63446C2semidirect
9. C3×#63440C2semidirect
11. #10589699C3direct (N × H)
43. C2×#3699C2×C3semidirect
45. C9×#121C2×C3semidirect
46. #63501C2×C3semidirect
47. #63501C2×C3semidirect
48. #63492C2×C3semidirect
49. C2×Dic9⋊C4C2×C3semidirect
50. #63458C2×C3semidirect
51. #63446C2×C3semidirect
52. #63440C2×C3semidirect
67. C2×C3×Dic9D4semidirect
150. C3×#121D9semidirect
203. C2×Dic9C3×D4semidirect
315. #121C3×D9semidirect
317. C3×C9#537semidirect
451. C9C3×#537semidirect
507. C3#10589699direct (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
C3×C9×#1212
C3×#635012
C3×#634922
C2×C3×Dic9⋊C42
C3×#634582
C3×#634462
C3×#634402
C3×#55973
#105896993