Finite Group Databasebeta

← C3×#10589797 #13153157

Subgroups — C3×#10589797 #13153157

Order 1728.

770 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
1C3×C9×#125C211yesmaximal, characteristic, complemented
2C3×#63460C212yesmaximal, complemented
3C3×#63457C211yesmaximal, characteristic, complemented
4C3×#63457C211yesmaximal, characteristic, complemented
5C3×#63457C212yesmaximal, complemented
6C3×#63456C212yesmaximal, complemented
7C3×#63455C212yesmaximal, complemented
8C3×#63454C211yesmaximal, characteristic, complemented
9C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
10C3×C4×D36C211yesmaximal, characteristic, complemented
11C3×C4×D36C211yesmaximal, characteristic, complemented
12C3×#5695—31nomaximal
13#10589797C311yesmaximal, characteristic, direct product factor
14C2×C3×D36C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×C3×D36C2212yes—
17C2×C3×D36C2212yes—
18C2×C3×C4×D9C2211yescharacteristic
19C2×C3×C4×D9C2211yescharacteristic
20C2×C3×C4×D9C2212yes—
21C2×C3×C4×D9C2212yes—
22C3×C9×C4⋊C4C2211yescharacteristic
23C3×C9×C4⋊C4C2211yescharacteristic
24C3×C9×C4⋊C4C2212yes—
25C3×C9×C4⋊C4C2212yes—
26C3×C42×C9C2211yescharacteristic
27C3×(C2×D9)⋊C4C2211yescharacteristic
28C3×(C2×D9)⋊C4C2211yescharacteristic
29C3×(C2×D9)⋊C4C2212yes—
30C3×(C2×D9)⋊C4C2212yes—
31C3×(C2×D9)⋊C4C2212yes—
32C3×(C2×D9)⋊C4C2212yes—
33C3×C4⋊Dic9C2211yescharacteristic
34C3×C4⋊Dic9C2211yescharacteristic
35C3×Dic9⋊C4C2212yes—
36C3×C4×Dic9C2212yes—
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×C3×D9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×Dic9C2312yes—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C3×D36—22no—
72C3×D36—24no—
73C3×D36—22no—
74C3×D36—42no—
75C3×C4×D9—22no—
76C3×C4×D9—22no—
77C3×C4×D9—42no—
78C3×C4×D9—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×C3×C9C2411yescharacteristic
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9—21no—
136C2×C3×D9—22no—
137C2×C3×D9—21no—
138C2×C3×D9—22no—
139C2×C3×D9—22no—
140C2×C3×D9—21no—
141C2×C3×D9#4112yes—
142C3×C4×C9—22no—
143C3×C4×C9—21no—
144C3×C4×C9—21no—
145C3×C4×C9—22no—
146C3×C4×C9#4112yes—
147C3×Dic9—21no—
148C3×Dic9—21no—
149C3×Dic9—22no—
150C3×#126—92no—
151C3×#125—21no—
152C3×#125D911yescharacteristic, complemented
153C3×#125C3×D311yescharacteristic
154C3×#123—92no—
155C3×#121—92no—
156C3×#121—91no—
157C3×#121—91no—
158C3×C4×D4—92no—
159C3×C4×D4—91no—
160C3×C4×D4—91no—
161C3×#117—91no—
162C2×C3×C4⋊C4—91no—
163#898—32no—
164#895—32no—
165#895—31no—
166#895—31no—
167#894—32no—
168#893—32no—
169#892—31no—
170D3×C4⋊C4—31no—
171C4×D12—31no—
172C4×D12—31no—
173C22×C3×D3—31no—
174C22×C3×D3—31no—
175C22×C3×D3—32no—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C3×Dic3—31no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—31no—
184C3×D12—64no—
185C3×D12—62no—
186C3×D12—62no—
187C3×D12—122no—
188C3×C4×D3—62no—
189C3×C4×D3—62no—
190C3×C4×D3—122no—
191C3×C4×D3—122no—
192C22×D9C23×C311yescharacteristic
193C22×D9C23×C312yes—
194C22×D9C23×C311yescharacteristic
195C2×C4×C9—21no—
196C2×C4×C9—21no—
197C2×C4×C9—21no—
198C2×C4×C9—22no—
199C2×C4×C9—22no—
200C2×C4×C9C23×C312yes—
201C2×C4×C9C23×C312yes—
202C2×C4×C9C23×C311yescharacteristic
203C2×C4×C9C23×C311yescharacteristic
204C2×C4×C9C23×C311yescharacteristic
205C2×Dic9C23×C311yescharacteristic
206C2×Dic9C23×C312yes—
207C2×Dic9C23×C311yescharacteristic
208D36—22no—
209D36—24no—
210D36—22no—
211D36—42no—
212C4×D9—22no—
213C4×D9—22no—
214C4×D9—42no—
215C4×D9—42no—
216#553—91noSylow
217C2×C3×C9C2×#4111yescharacteristic
218C2×C3×C9C2×#4111yescharacteristic
219C2×C3×C9H(5,𝔽2)11yescharacteristic
220C3×D9—22no—
221C3×D9—41no—
222C3×D9—42no—
223C2×C3×D4—92no—
224C2×C3×D4—91no—
225C2×C3×D4—91no—
226C2×C3×D4—92no—
227C22×C3×C4—91no—
228C22×C3×C4—91no—
229C22×C3×C4—92no—
230C22×C3×C4—92no—
231C2×D12—31no—
232C2×D12—31no—
233C2×D12—32no—
234C2×D12—32no—
235C2×C4×D3—31no—
236C2×C4×D3—31no—
237C2×C4×D3—32no—
238C2×C4×D3—32no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4C2×D911yescharacteristic
247C3×C4⋊C4C2×D911yescharacteristic
248C3×C4⋊C4C2×D912yes—
249C3×C4⋊C4C2×D912yes—
250C3×C4⋊C4C2×C3×D311yescharacteristic
251C3×C4⋊C4C2×C3×D311yescharacteristic
252C3×C4⋊C4C2×C3×D312yes—
253C3×C4⋊C4C2×C3×D312yes—
254C3×C22⋊C4—92no—
255C3×C22⋊C4—91no—
256C3×C22⋊C4—92no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4—91no—
259C3×C22⋊C4—92no—
260C3×C42—21no—
261C3×C42—92no—
262C3×C42C2×D911yescharacteristic
263C3×C42C2×C3×D311yescharacteristic
264(C2×D3)⋊C4—31no—
265(C2×D3)⋊C4—31no—
266(C2×D3)⋊C4—32no—
267(C2×D3)⋊C4—32no—
268(C2×D3)⋊C4—32no—
269(C2×D3)⋊C4—32no—
270C4⋊Dic3—31no—
271C4⋊Dic3—31no—
272Dic3⋊C4—32no—
273C4×Dic3—32no—
274C22×C32C23×D311yescharacteristic
275C2×C3×D3—32no—
276C2×C3×D3—61no—
277C2×C3×D3—61no—
278C2×C3×D3—61no—
279C2×C3×D3—62no—
280C2×C3×D3—61no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3—61no—
284C32×C4—22no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—22no—
288C32×C4#23912yes—
289C3×Dic3—61no—
290C3×Dic3—61no—
291C3×Dic3—62no—
292C22×C9—21no—
293C22×C9C24×C311yescharacteristic
294C2×D9—21no—
295C2×D9—21no—
296C2×D9—21no—
297C2×D9—22no—
298C2×D9—21no—
299C2×D9—22no—
300C2×D9—22no—
301C2×D9—21no—
302C2×D9C3×#4112yes—
303C4×C9—22no—
304C4×C9—21no—
305C4×C9—21no—
306C4×C9—22no—
307C4×C9—22no—
308C4×C9—41no—
309C4×C9—42no—
310C4×C9—41no—
311C4×C9—42no—
312C4×C9C3×#4112yes—
313Dic9—21no—
314Dic9—21no—
315Dic9—22no—
316#126—92no—
317#125C3×D911yescharacteristic, complemented
318#123—92no—
319#121—92no—
320#121—91no—
321#121—91no—
322C4×D4—92no—
323C4×D4—91no—
324C4×D4—91no—
325#117—91no—
326C2×C4⋊C4—91no—
327C3×C9#55311yescharacteristic, complemented, Sylow
328C23×C3—91no—
329C23×C3—91no—
330C23×C3—92no—
331C22×D3—31no—
332C22×D3—31no—
333C22×D3—32no—
334C3×D4—184no—
335C3×D4—182no—
336C3×D4—182no—
337C3×D4—362no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—91no—
344C2×C3×C4—92no—
345C2×C3×C4—91no—
346C2×C3×C4—182no—
347C2×C3×C4—182no—
348C2×C3×C4—362no—
349C2×C3×C4—362no—
350C2×C3×C4C22×D912yes—
351C2×C3×C4C22×D911yescharacteristic
352C2×C3×C4C22×D911yescharacteristic
353C2×C3×C4C22×D912yes—
354C2×C3×C4C22×D911yescharacteristic
355C2×C3×C4C22×C3×D312yes—
356C2×C3×C4C22×C3×D312yes—
357C2×C3×C4C22×C3×D311yescharacteristic
358C2×C3×C4C22×C3×D311yescharacteristic
359C2×C3×C4C22×C3×D311yescharacteristic
360C2×Dic3—31no—
361C2×Dic3—32no—
362C2×Dic3—31no—
363D12—62no—
364D12—64no—
365D12—62no—
366D12—122no—
367C4×D3—62no—
368C4×D3—62no—
369C4×D3—122no—
370C4×D3—122no—
371C2×C32C2×#23911yescharacteristic
372C2×C32D3×#4111yescharacteristic
373C2×C32#100911yescharacteristic
374C3×D3—62no—
375C3×D3—121no—
376C3×D3—122no—
377C2×C9—21no—
378C2×C9—21no—
379C2×C9—21no—
380C2×C9C2×C3×#4111yescharacteristic
381C2×C9C2×C3×#4111yescharacteristic
382C2×C9C3×H(5,𝔽2)11yescharacteristic
383D9—22no—
384D9—41no—
385D9—42no—
386C2×D4—92no—
387C2×D4—91no—
388C2×D4—91no—
389C2×D4—92no—
390C22×C4—91no—
391C22×C4—91no—
392C22×C4—92no—
393C22×C4—92no—
394C4⋊C4—91no—
395C4⋊C4—92no—
396C4⋊C4—91no—
397C4⋊C4C2×C3×D911yescharacteristic
398C4⋊C4C2×C3×D911yescharacteristic
399C4⋊C4C2×C3×D912yes—
400C4⋊C4C2×C3×D912yes—
401C22⋊C4—92no—
402C22⋊C4—91no—
403C22⋊C4—92no—
404C22⋊C4—92no—
405C22⋊C4—91no—
406C22⋊C4—92no—
407C42—92no—
408C42C2×C3×D911yescharacteristic
409C22×C3—21no—
410C22×C3—92no—
411C22×C3—181no—
412C22×C3—181no—
413C22×C3—181no—
414C22×C3—182no—
415C22×C3—181no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3—181no—
419C22×C3C23×D911yescharacteristic
420C22×C3C23×C3×D311yescharacteristic
421C2×D3—32no—
422C2×D3—61no—
423C2×D3—61no—
424C2×D3—61no—
425C2×D3—62no—
426C2×D3—61no—
427C2×D3—62no—
428C2×D3—62no—
429C2×D3—61no—
430C3×C4—22no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—41no—
440C3×C4—42no—
441C3×C4—41no—
442C3×C4—42no—
443C3×C4—181no—
444C3×C4—181no—
445C3×C4—182no—
446C3×C4#370112yes—
447C3×C4C3×#23912yes—
448Dic3—61no—
449Dic3—61no—
450Dic3—62no—
451C32#569511yescharacteristic
452C9—21no—
453C9C3×#55311yescharacteristic, complemented
454C23—91no—
455C23—91no—
456C23—92no—
457D4—184no—
458D4—182no—
459D4—182no—
460D4—362no—
461C2×C4—91no—
462C2×C4—92no—
463C2×C4—91no—
464C2×C4—182no—
465C2×C4—182no—
466C2×C4—362no—
467C2×C4—362no—
468C2×C4C22×C3×D912yes—
469C2×C4C22×C3×D911yescharacteristic
470C2×C4C22×C3×D911yescharacteristic
471C2×C4C22×C3×D912yes—
472C2×C4C22×C3×D911yescharacteristic
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—182no—
477C2×C3—361no—
478C2×C3—362no—
479C2×C3C2×#370111yescharacteristic
480C2×C3#41×D911yescharacteristic
481C2×C3#6371511yescharacteristic
482C2×C3C2×C3×#23911yescharacteristic
483C2×C3C3×D3×#4111yescharacteristic
484C2×C3C3×#100911yescharacteristic
485D3—62no—
486D3—121no—
487D3—122no—
488C22—92no—
489C22—181no—
490C22—181no—
491C22—181no—
492C22—182no—
493C22—181no—
494C22—182no—
495C22—182no—
496C22—181no—
497C22C23×C3×D911yescharacteristic
498C4—22no—
499C4—21no—
500C4—21no—
501C4—22no—
502C4—181no—
503C4—181no—
504C4—182no—
505C4C3×#370112yes—
506C3—21no—
507C3#1058979711yescharacteristic, direct product factor
508C3C3×#569511yescharacteristic
509C2—182no—
510C2—361no—
511C2—362no—
512C2C2×C3×#370111yescharacteristic
513C2C3×#41×D911yescharacteristic
514C2C3×#6371511yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#125C2
socleC22×C32C23×D3
hypercenterC3×#125D9
hypocenterC9C3×#553
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. C3×C9×#125C2semidirect
2. C3×#63460C2semidirect
3. C3×#63457C2semidirect
4. C3×#63457C2semidirect
5. C3×#63457C2semidirect
6. C3×#63456C2semidirect
7. C3×#63455C2semidirect
8. C3×#63454C2semidirect
9. C3×C4⋊C4×D9C2semidirect
10. C3×C4×D36C2semidirect
11. C3×C4×D36C2semidirect
13. #10589797C3direct (N × H)
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
152. C3×#125D9semidirect
317. #125C3×D9semidirect
327. C3×C9#553semidirect
453. C9C3×#553semidirect
507. C3#10589797direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×#1252
C3×#634602
C3×#634572
C3×#634562
C3×#634552
C3×#634542
C3×C4⋊C4×D92
C3×C4×D362
C3×#56953
#105897973