Finite Group Databasebeta

← C3×#10595289 #13174774

Subgroups — C3×#10595289 #13174774

Order 1728.

2134 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×#67C214yesmaximal, complemented
2C3×#64149C211yesmaximal, characteristic, complemented
3C32×#937C212yesmaximal, complemented
4C3×#63971C214yesmaximal, complemented
5C3×#63923C212yesmaximal, complemented
6C3×#63902C212yesmaximal, complemented
7C3×#5590—32nomaximal
8#10595289C311yesmaximal, characteristic, direct product factor
9C23×C32×D3C2212yescomplemented
10C22×C3×D32C2212yescomplemented
11C2×C33×D4C2211yescharacteristic, complemented
12C2×C3×#638C2214yescomplemented
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
14C2×C32×#67C2214yescomplemented
15C3×D3×#67—216no—
16C2×C3×#626C2214yescomplemented
17C2×C3×#625C2212yes—
18C2×C3×D3×Dic3C2214yescomplemented
19C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
20C32×#217C2212yescomplemented
21C32×(C2×D3)⋊C4C2214yescomplemented
22C3×#3724C2212yescomplemented
23C3×#3722C2212yescomplemented
24C22×C3×#67—34no—
25C2×C3×D3×D4—34no—
26C2×D3×#67C2×C314yescomplemented
27C32×#120—32no—
28#64149C2×C311yescharacteristic, complemented
29C3×#940—32no—
30C3×#940—61no—
31C3×#939—34no—
32C3×#937—22no—
33C3×#937D312yescomplemented
34C3×#937C2×C312yescomplemented
35C3×#884—34no—
36C3×#882—34no—
37C3×#67⋊C4—34no—
38#63971C2×C314yescomplemented
39#63923C2×C312yescomplemented
40#63902C2×C312yescomplemented
41C23×C33C2312yes—
42C22×C3×C3⋊D3C2312yes—
43C22×C32×D3—28no—
44C22×C32×D3—42no—
45C22×C32×D3C2314yes—
46C2×C3×D32—28no—
47C2×C3×D32—42no—
48C33×D4—41no—
49C2×C33×C4C2311yescharacteristic
50C3×#638—28no—
51C2×C3×C3⋊Dic3C2312yes—
52C3×C4⋊(C3⋊D3)—41no—
53C32×#67—28no—
54C2×C32×Dic3C2314yes—
55C3×#626—28no—
56C3×#625D418yescomplemented
57C3×D3×Dic3—28no—
58C3×D42—91no—
59#5590—32no—
60C24×C32—32no—
61C23×C3×D3—22no—
62C23×C3×D3—34no—
63C23×C3×D3C2×D312yescomplemented
64C23×C3×D3C22×C312yescomplemented
65C22×D32C22×C312yescomplemented
66C2×C32×D4—21no—
67C2×C32×D4—22no—
68C2×C32×D4—34no—
69C2×C32×D4—41no—
70C2×C32×D4C2×D312yescomplemented
71C2×C32×D4C22×C311yescharacteristic, complemented
72C2×#638C22×C314yescomplemented
73C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
74C2×C3×#67—24no—
75C2×C3×#67—34no—
76C2×C3×#67—34no—
77C2×C3×#67—34no—
78C2×C3×#67—34no—
79C2×C3×#67—616no—
80C2×C3×#67—64no—
81C2×C3×#67C2×D314yescomplemented
82C2×C3×#67C22×C314yescomplemented
83C22×C3×Dic3—34no—
84C3×D3×D4—616no—
85C2×C3×D12—34no—
86C2×C3×D12—32no—
87C2×C3×D12—61no—
88C2×C3×C4×D3—34no—
89D3×#67—216no—
90C2×#626C22×C314yescomplemented
91C2×#625C22×C312yes—
92C2×D3×Dic3C22×C314yescomplemented
93C32×C22⋊C4—32no—
94C32×C22⋊C4—34no—
95C4×C3⋊Dic3C22×C311yescharacteristic, complemented
96C3×#217—22no—
97C3×#217C2×D312yescomplemented
98C3×#217C22×C312yescomplemented
99C3×(C2×D3)⋊C4—24no—
100C3×(C2×D3)⋊C4—34no—
101C3×(C2×D3)⋊C4C2×D314yescomplemented
102C3×(C2×D3)⋊C4C22×C314yescomplemented
103C3×Dic3⋊C4—34no—
104C3×C4×Dic3—32no—
105C3×C4×Dic3—61no—
106#3724C22×C312yescomplemented
107#3722C22×C312yescomplemented
108C22×C33—24no—
109C22×C33—22no—
110C22×C33C2411yescharacteristic
111C2×C3×C3⋊D3—24no—
112C2×C3×C3⋊D3—22no—
113C2×C32×D3—24no—
114C2×C32×D3—24no—
115C2×C32×D3—48no—
116C2×C32×D3C2×D418yescomplemented
117C3×D32—48no—
118C33×C4—21no—
119C3×C3⋊Dic3C2×D414yescomplemented
120C32×Dic3—24no—
121C22×C3×D4—94no—
122C22×#67—34no—
123C2×D3×D4—34no—
124C3×#127—91no—
125C3×#121—94no—
126C3×#120—32no—
127C3×#120—32no—
128C3×#120—62no—
129C3×#120—92no—
130C3×C4×D4—92no—
131#940—32no—
132#940—61no—
133#939—34no—
134#937C3×D312yescomplemented
135#884—34no—
136#882—34no—
137#67⋊C4—34no—
138C23×C32—22no—
139C23×C32—22no—
140C23×C32—22no—
141C23×C32—34no—
142C23×C32—42no—
143C23×C32—68no—
144C23×C32—122no—
145C23×C32C22×D312yes—
146C23×C32C22×D312yes—
147C23×C32C23×C312yes—
148C22×C3⋊D3C23×C312yes—
149C22×C3×D3—28no—
150C22×C3×D3—28no—
151C22×C3×D3—24no—
152C22×C3×D3—34no—
153C22×C3×D3—48no—
154C22×C3×D3—42no—
155C22×C3×D3—42no—
156C22×C3×D3—44no—
157C22×C3×D3—62no—
158C22×C3×D3—68no—
159C22×C3×D3—68no—
160C22×C3×D3—124no—
161C22×C3×D3C22×D314yes—
162C22×C3×D3C23×C314yes—
163C2×D32—28no—
164C2×D32—42no—
165C32×D4—42no—
166C32×D4—41no—
167C32×D4—42no—
168C32×D4—44no—
169C32×D4—44no—
170C32×D4—68no—
171C2×C32×C4—21no—
172C2×C32×C4—22no—
173C2×C32×C4—34no—
174C2×C32×C4—41no—
175C2×C32×C4C22×D312yes—
176C2×C32×C4C23×C311yescharacteristic
177#638—28no—
178C2×C3⋊Dic3C23×C312yes—
179C4⋊(C3⋊D3)—41no—
180C3×#67—28no—
181C3×#67—28no—
182C3×#67—316no—
183C3×#67—48no—
184C3×#67—68no—
185C3×#67—68no—
186C3×#67—68no—
187C3×#67—128no—
188C2×C3×Dic3—24no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—68no—
192C2×C3×Dic3C22×D314yes—
193C2×C3×Dic3C23×C314yes—
194C3×D12—68no—
195C3×D12—122no—
196C3×D12—122no—
197C3×C4×D3—68no—
198#626—28no—
199#625C3×D418yescomplemented
200D3×Dic3—28no—
201D42—91noSylow
202C2×C33—42no—
203C2×C33C22×D412yes—
204C2×C33H(5,𝔽2)11yescharacteristic
205C3×C3⋊D3—42no—
206C32×D3—28no—
207C24×C3—32no—
208C24×C3—32no—
209C24×C3—62no—
210C24×C3—92no—
211C23×D3—34no—
212C23×D3C2×C3×D312yescomplemented
213C2×C3×D4—21no—
214C2×C3×D4—22no—
215C2×C3×D4—34no—
216C2×C3×D4—34no—
217C2×C3×D4—41no—
218C2×C3×D4—64no—
219C2×C3×D4—94no—
220C2×C3×D4—92no—
221C2×C3×D4—94no—
222C2×C3×D4—91no—
223C2×C3×D4—1816no—
224C2×C3×D4D3211yescharacteristic, complemented
225C2×C3×D4C2×C3×D312yescomplemented
226C22×C3×C4—94no—
227C2×#67—34no—
228C2×#67—34no—
229C2×#67—34no—
230C2×#67—34no—
231C2×#67—64no—
232C2×#67—616no—
233C2×#67C2×C3×D314yescomplemented
234C22×Dic3—34no—
235D3×D4—616no—
236C2×D12—34no—
237C2×D12—32no—
238C2×D12—61no—
239C2×C4×D3—34no—
240C3×C4⋊C4—92no—
241C3×C22⋊C4—32no—
242C3×C22⋊C4—32no—
243C3×C22⋊C4—34no—
244C3×C22⋊C4—34no—
245C3×C22⋊C4—64no—
246C3×C22⋊C4—62no—
247C3×C22⋊C4—92no—
248C3×C42—91no—
249#217C2×C3×D312yescomplemented
250(C2×D3)⋊C4—34no—
251(C2×D3)⋊C4C2×C3×D314yescomplemented
252Dic3⋊C4—34no—
253C4×Dic3—32no—
254C4×Dic3—61no—
255C22×C32—22no—
256C22×C32—24no—
257C22×C32—24no—
258C22×C32—24no—
259C22×C32—22no—
260C22×C32—24no—
261C22×C32—22no—
262C22×C32—21no—
263C22×C32—22no—
264C22×C32—38no—
265C22×C32—41no—
266C22×C32—42no—
267C22×C32—42no—
268C22×C32—44no—
269C22×C32—44no—
270C22×C32—44no—
271C22×C32—44no—
272C22×C32—64no—
273C22×C32—64no—
274C22×C32—84no—
275C22×C32—128no—
276C22×C32C23×D312yes—
277C22×C32C24×C311yescharacteristic
278C2×C3⋊D3—24no—
279C2×C3⋊D3—22no—
280C2×C3×D3—24no—
281C2×C3×D3—24no—
282C2×C3×D3—24no—
283C2×C3×D3—24no—
284C2×C3×D3—28no—
285C2×C3×D3—44no—
286C2×C3×D3—44no—
287C2×C3×D3—48no—
288C2×C3×D3—48no—
289C2×C3×D3—416no—
290C2×C3×D3—64no—
291C2×C3×D3—64no—
292C2×C3×D3—64no—
293C2×C3×D3—122no—
294C2×C3×D3—124no—
295C2×C3×D3—1216no—
296C2×C3×D3D3×D418yescomplemented
297C2×C3×D3C2×C3×D418yescomplemented
298D32—48no—
299C32×C4—21no—
300C32×C4—22no—
301C32×C4—22no—
302C32×C4—42no—
303C32×C4—42no—
304C32×C4—64no—
305C3⋊Dic3C2×C3×D414yescomplemented
306C3×Dic3—24no—
307C3×Dic3—24no—
308C3×Dic3—38no—
309C3×Dic3—44no—
310C3×Dic3—64no—
311C22×D4—94no—
312#127—91no—
313#121—94no—
314#120—32no—
315#120—92no—
316C4×D4—92no—
317C33D4211yescharacteristic, complemented, Sylow
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—34no—
322C23×C3—34no—
323C23×C3—42no—
324C23×C3—64no—
325C23×C3—68no—
326C23×C3—68no—
327C23×C3—92no—
328C23×C3—128no—
329C23×C3—122no—
330C23×C3—122no—
331C23×C3—124no—
332C23×C3—188no—
333C23×C3—362no—
334C23×C3C2×D3212yes—
335C23×C3C22×C3×D312yes—
336C23×C3C22×C3×D312yes—
337C22×D3—28no—
338C22×D3—34no—
339C22×D3—42no—
340C22×D3—62no—
341C22×D3—68no—
342C22×D3—68no—
343C22×D3—124no—
344C22×D3C22×C3×D314yes—
345C3×D4—41no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—44no—
349C3×D4—44no—
350C3×D4—68no—
351C3×D4—68no—
352C3×D4—98no—
353C3×D4—128no—
354C3×D4—188no—
355C3×D4—188no—
356C3×D4—361no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—34no—
360C2×C3×C4—34no—
361C2×C3×C4—41no—
362C2×C3×C4—64no—
363C2×C3×C4—92no—
364C2×C3×C4—188no—
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C22×C3×D312yes—
367#67—28no—
368#67—316no—
369#67—68no—
370#67—68no—
371#67—68no—
372#67—128no—
373C2×Dic3—34no—
374C2×Dic3—62no—
375C2×Dic3—68no—
376C2×Dic3C22×C3×D314yes—
377D12—68no—
378D12—122no—
379D12—122no—
380C4×D3—68no—
381C2×C32—22no—
382C2×C32—21no—
383C2×C32—24no—
384C2×C32—22no—
385C2×C32—42no—
386C2×C32—41no—
387C2×C32—42no—
388C2×C32—44no—
389C2×C32—42no—
390C2×C32—42no—
391C2×C32—68no—
392C2×C32—82no—
393C2×C32—82no—
394C2×C32—84no—
395C2×C32C2×D3×D414yes—
396C2×C32#100412yes—
397C2×C32C22×C3×D412yes—
398C2×C32C3×H(5,𝔽2)11yescharacteristic
399C3⋊D3—42no—
400C3×D3—28no—
401C3×D3—28no—
402C3×D3—48no—
403C3×D3—124no—
404C3×D3—242no—
405C24—32no—
406C24—92no—
407C2×D4—34no—
408C2×D4—94no—
409C2×D4—92no—
410C2×D4—94no—
411C2×D4—91no—
412C2×D4—1816no—
413C2×D4C3×D3211yescharacteristic, complemented
414C22×C4—94no—
415C4⋊C4—92no—
416C22⋊C4—32no—
417C22⋊C4—34no—
418C22⋊C4—92no—
419C42—91no—
420C22×C3—24no—
421C22×C3—24no—
422C22×C3—22no—
423C22×C3—24no—
424C22×C3—24no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—21no—
428C22×C3—22no—
429C22×C3—38no—
430C22×C3—38no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—41no—
438C22×C3—68no—
439C22×C3—64no—
440C22×C3—64no—
441C22×C3—64no—
442C22×C3—64no—
443C22×C3—84no—
444C22×C3—124no—
445C22×C3—124no—
446C22×C3—128no—
447C22×C3—128no—
448C22×C3—1216no—
449C22×C3—184no—
450C22×C3—182no—
451C22×C3—368no—
452C22×C3C22×D3211yescharacteristic
453C22×C3C23×C3×D312yes—
454C2×D3—24no—
455C2×D3—24no—
456C2×D3—48no—
457C2×D3—64no—
458C2×D3—64no—
459C2×D3—64no—
460C2×D3—122no—
461C2×D3—124no—
462C2×D3—1216no—
463C2×D3C3×D3×D418yescomplemented
464C3×C4—22no—
465C3×C4—21no—
466C3×C4—22no—
467C3×C4—42no—
468C3×C4—42no—
469C3×C4—64no—
470C3×C4—64no—
471C3×C4—94no—
472C3×C4—124no—
473Dic3—24no—
474Dic3—38no—
475Dic3—64no—
476C32—22no—
477C32—21no—
478C32—41no—
479C32#559012yescomplemented
480C32C3×D4211yescharacteristic, complemented
481C23—34no—
482C23—68no—
483C23—92no—
484C23—122no—
485C23—188no—
486C23—362no—
487C23C2×C3×D3212yes—
488D4—41no—
489D4—68no—
490D4—98no—
491D4—188no—
492D4—188no—
493D4—361no—
494C2×C4—34no—
495C2×C4—92no—
496C2×C4—188no—
497C2×C4C2×C3×D3211yescharacteristic
498C2×C3—21no—
499C2×C3—22no—
500C2×C3—22no—
501C2×C3—24no—
502C2×C3—44no—
503C2×C3—42no—
504C2×C3—42no—
505C2×C3—42no—
506C2×C3—41no—
507C2×C3—42no—
508C2×C3—68no—
509C2×C3—68no—
510C2×C3—82no—
511C2×C3—82no—
512C2×C3—84no—
513C2×C3—128no—
514C2×C3—362no—
515C2×C3D32×D412yes—
516C2×C3#6433011yescharacteristic
517C2×C3C2×C3×D3×D414yes—
518C2×C3C3×#100412yes—
519D3—28no—
520D3—124no—
521D3—242no—
522C22—22no—
523C22—24no—
524C22—38no—
525C22—64no—
526C22—64no—
527C22—128no—
528C22—182no—
529C22—184no—
530C22—368no—
531C22C22×C3×D3211yescharacteristic
532C4—21no—
533C4—64no—
534C4—94no—
535C3—21no—
536C3—22no—
537C3—41no—
538C3#1059528911yescharacteristic, direct product factor
539C3C3×#559012yescomplemented
540C2—42no—
541C2—68no—
542C2—362no—
543C2C3×D32×D412yes—
544C2C3×#6433011yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C24
hypercenterC2×C3×D4D32
hypocenterC32C3×D42
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×#67C2semidirect
2. C3×#64149C2semidirect
3. C32×#937C2semidirect
4. C3×#63971C2semidirect
5. C3×#63923C2semidirect
6. C3×#63902C2semidirect
8. #10595289C3direct (N × H)
9. C23×C32×D3C22semidirect
10. C22×C3×D32C22semidirect
11. C2×C33×D4C22semidirect
12. C2×C3×#638C22semidirect
13. C2×C3×C4⋊(C3⋊D3)C22semidirect
14. C2×C32×#67C22semidirect
16. C2×C3×#626C22semidirect
18. C2×C3×D3×Dic3C22semidirect
19. C3×C4×C3⋊Dic3C22semidirect
20. C32×#217C22semidirect
21. C32×(C2×D3)⋊C4C22semidirect
22. C3×#3724C22semidirect
23. C3×#3722C22semidirect
26. C2×D3×#67C2×C3semidirect
28. #64149C2×C3semidirect
33. C3×#937D3semidirect
34. C3×#937C2×C3semidirect
38. #63971C2×C3semidirect
39. #63923C2×C3semidirect
40. #63902C2×C3semidirect
56. C3×#625D4semidirect
63. C23×C3×D3C2×D3semidirect
64. C23×C3×D3C22×C3semidirect
65. C22×D32C22×C3semidirect
70. C2×C32×D4C2×D3semidirect
71. C2×C32×D4C22×C3semidirect
72. C2×#638C22×C3semidirect
73. C2×C4⋊(C3⋊D3)C22×C3semidirect
81. C2×C3×#67C2×D3semidirect
82. C2×C3×#67C22×C3semidirect
90. C2×#626C22×C3semidirect
92. C2×D3×Dic3C22×C3semidirect
95. C4×C3⋊Dic3C22×C3semidirect
97. C3×#217C2×D3semidirect
98. C3×#217C22×C3semidirect
101. C3×(C2×D3)⋊C4C2×D3semidirect
102. C3×(C2×D3)⋊C4C22×C3semidirect
106. #3724C22×C3semidirect
107. #3722C22×C3semidirect
116. C2×C32×D3C2×D4semidirect
119. C3×C3⋊Dic3C2×D4semidirect
134. #937C3×D3semidirect
199. #625C3×D4semidirect
212. C23×D3C2×C3×D3semidirect
224. C2×C3×D4D32semidirect
225. C2×C3×D4C2×C3×D3semidirect
233. C2×#67C2×C3×D3semidirect
249. #217C2×C3×D3semidirect
251. (C2×D3)⋊C4C2×C3×D3semidirect
296. C2×C3×D3D3×D4semidirect
297. C2×C3×D3C2×C3×D4semidirect
305. C3⋊Dic3C2×C3×D4semidirect
317. C33D42semidirect
413. C2×D4C3×D32semidirect
463. C2×D3C3×D3×D4semidirect
479. C32#5590semidirect
480. C32C3×D42semidirect
538. C3#10595289direct (N × H)
539. C3C3×#5590semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×#672
C3×#641492
C32×#9372
C3×#639712
C3×#639232
C3×#639022
C3×#55903
#105952893