Finite Group Databasebeta

← C3×#10595300 #13174705

Subgroups — C3×#10595300 #13174705

Order 1728.

2498 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C3×#3811C214yesmaximal, complemented
3C2×C3×#3802C211yesmaximal, characteristic, complemented
4C32×#940C212yesmaximal, complemented
5C3×#63979C214yesmaximal, complemented
6C3×#63911C212yesmaximal, complemented
7C3×C4×#624C211yesmaximal, characteristic, complemented
8C3×D3×#127—32nomaximal
9#10595300C311yesmaximal, characteristic, direct product factor
10C23×C3×C3⋊D3C2212yes—
11C22×C3×D32C2212yes—
12C2×C33×D4C2211yescharacteristic, complemented
13C2×C3×#638C2212yescomplemented
14C3×D4×C3⋊D3—24no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×#67C2218yescomplemented
18C2×C32×D12C2212yescomplemented
19C3×#3811—216no—
20C2×C3×#626C2218yescomplemented
21C2×C3×#625C2212yescomplemented
22C2×C3×#624C2212yes—
23C3×#3802—24no—
24C32×C4×Dic3C2212yescomplemented
25C3×Dic32C2212yescomplemented
26C2×D4×C3⋊D3C2×C311yescharacteristic, complemented
27C2×C3×D3×D4—32no—
28C2×C3×D3×D4—38no—
29C2×C3×D3×D4—32no—
30C2×C3×D3×D4—61no—
31C2×#3811C2×C314yescomplemented
32C2×#3802C2×C311yescharacteristic, complemented
33C32×#127—32no—
34C3×#940—22no—
35C3×#940—38no—
36C3×#940—32no—
37C3×#940D312yescomplemented
38C3×#940C2×C312yescomplemented
39C3×#874—32no—
40C3×C42×D3—32no—
41#63979C2×C314yescomplemented
42#63911C2×C312yescomplemented
43C4×#624C2×C311yescharacteristic, complemented
44C23×C33C2312yes—
45C22×C3×C3⋊D3—24no—
46C22×C3×C3⋊D3—28no—
47C22×C3×C3⋊D3C2312yes—
48C22×C3×C3⋊D3C2311yescharacteristic
49C22×C32×D3C2314yes—
50C2×C3×D32—24no—
51C2×C3×D32—28no—
52C33×D4—22no—
53C2×C33×C4C2311yescharacteristic
54C3×#638—24no—
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C3×C4⋊(C3⋊D3)—22no—
57C3×C4×C3⋊D3D414yescomplemented
58C32×#67—216no—
59C2×C32×Dic3C2314yes—
60C32×D12—24no—
61C3×#626—216no—
62C3×#625—24no—
63C3×#624D418yescomplemented
64C2×C3×#127—91no—
65D3×#127—32no—
66C23×C3⋊D3C22×C312yes—
67C23×C3×D3—34no—
68C23×C3×D3—34no—
69C23×C3×D3—62no—
70C22×D32C22×C312yes—
71C2×C32×D4—21no—
72C2×C32×D4—22no—
73C2×C32×D4—38no—
74C2×C32×D4—32no—
75C2×C32×D4—41no—
76C2×C32×D4C2×D312yescomplemented
77C2×C32×D4C22×C311yescharacteristic, complemented
78C2×#638C22×C312yescomplemented
79D4×C3⋊D3—24no—
80C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
81C2×C4×C3⋊D3C22×C311yescharacteristic
82C2×C3×#67—28no—
83C2×C3×#67—34no—
84C2×C3×#67—34no—
85C2×C3×#67—38no—
86C2×C3×#67—62no—
87C2×C3×#67C2×D318yescomplemented
88C2×C3×#67C22×C318yescomplemented
89C3×D3×D4—68no—
90C3×D3×D4—632no—
91C3×D3×D4—68no—
92C3×D3×D4—68no—
93C2×C3×D12—22no—
94C2×C3×D12—38no—
95C2×C3×D12—32no—
96C2×C3×D12—61no—
97C2×C3×D12C2×D312yescomplemented
98C2×C3×D12C22×C312yescomplemented
99C2×C3×C4×D3—32no—
100C2×C3×C4×D3—34no—
101C2×C3×C4×D3—61no—
102#3811—216no—
103C2×#626C22×C318yescomplemented
104C2×#625C22×C312yescomplemented
105C2×#624C22×C312yes—
106#3802—24no—
107C32×C42—32no—
108C3×C4×Dic3—22no—
109C3×C4×Dic3—34no—
110C3×C4×Dic3C2×D312yescomplemented
111C3×C4×Dic3C22×C312yescomplemented
112Dic32C22×C312yescomplemented
113C22×C33—22no—
114C22×C33—24no—
115C22×C33C2411yescharacteristic
116C2×C3×C3⋊D3—22no—
117C2×C3×C3⋊D3—24no—
118C2×C3×C3⋊D3—48no—
119C2×C3×C3⋊D3C2×D414yes—
120C2×C3×C3⋊D3C2×D412yes—
121C2×C32×D3—24no—
122C2×C32×D3—28no—
123C3×D32—48no—
124C33×C4C2×D412yescomplemented
125C3×C3⋊Dic3C2×D412yescomplemented
126C32×Dic3C2×D418yescomplemented
127C22×C3×D4—94no—
128C22×C3×D4—91no—
129C22×C3×D4—91no—
130C2×D3×D4—32no—
131C2×D3×D4—38no—
132C2×D3×D4—32no—
133C2×D3×D4—61no—
134C3×#127—32no—
135C3×#127—32no—
136C3×#127—62no—
137C3×#127—92no—
138C3×#127—94no—
139C2×C3×C42—91no—
140#940—38no—
141#940—32no—
142#940C3×D312yescomplemented
143#874—32no—
144C42×D3—32no—
145C23×C32—22no—
146C23×C32—24no—
147C23×C32—34no—
148C23×C32—42no—
149C23×C32C22×D314yes—
150C23×C32C23×C312yes—
151C22×C3⋊D3—24no—
152C22×C3⋊D3—28no—
153C22×C3⋊D3C23×C312yes—
154C22×C3⋊D3C23×C311yescharacteristic
155C22×C3×D3—24no—
156C22×C3×D3—32no—
157C22×C3×D3—34no—
158C22×C3×D3—61no—
159C22×C3×D3—62no—
160C22×C3×D3—68no—
161C22×C3×D3—616no—
162C22×C3×D3—68no—
163C22×C3×D3—616no—
164C22×C3×D3—68no—
165C22×C3×D3—128no—
166C22×C3×D3C22×D314yes—
167C22×C3×D3C23×C314yes—
168C2×D32—24no—
169C2×D32—28no—
170C32×D4—22no—
171C32×D4—24no—
172C32×D4—24no—
173C32×D4—44no—
174C32×D4—44no—
175C32×D4—616no—
176C32×D4—64no—
177C2×C32×C4—21no—
178C2×C32×C4—22no—
179C2×C32×C4—34no—
180C2×C32×C4—41no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C23×C311yescharacteristic
183#638—24no—
184C2×C3⋊Dic3C23×C311yescharacteristic
185C4⋊(C3⋊D3)—22no—
186C4×C3⋊D3C3×D414yescomplemented
187C3×#67—216no—
188C3×#67—216no—
189C3×#67—416no—
190C3×#67—68no—
191C3×#67—68no—
192C3×#67—68no—
193C3×#67—616no—
194C2×C3×Dic3—24no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3C22×D314yes—
198C2×C3×Dic3C23×C314yes—
199C3×D12—24no—
200C3×D12—24no—
201C3×D12—44no—
202C3×D12—64no—
203C3×D12—616no—
204C3×D12—64no—
205C3×C4×D3—38no—
206C3×C4×D3—316no—
207C3×C4×D3—64no—
208#626—216no—
209#625—24no—
210#624C3×D418yescomplemented
211C2×#127—91noSylow
212C2×C33—42no—
213C2×C33C22×D412yes—
214C2×C33C22×D411yescharacteristic
215C3×C3⋊D3—42no—
216C3×C3⋊D3#12714yescomplemented
217C32×D3—44no—
218C24×C3—92no—
219C24×C3—92no—
220C23×D3—34no—
221C23×D3—34no—
222C23×D3—62no—
223C2×C3×D4—21no—
224C2×C3×D4—22no—
225C2×C3×D4—38no—
226C2×C3×D4—38no—
227C2×C3×D4—32no—
228C2×C3×D4—32no—
229C2×C3×D4—41no—
230C2×C3×D4—62no—
231C2×C3×D4—68no—
232C2×C3×D4—98no—
233C2×C3×D4—92no—
234C2×C3×D4—92no—
235C2×C3×D4—91no—
236C2×C3×D4—184no—
237C2×C3×D4—1816no—
238C2×C3×D4—184no—
239C2×C3×D4D3211yescharacteristic, complemented
240C2×C3×D4C2×C3×D312yescomplemented
241C22×C3×C4—92no—
242C22×C3×C4—91no—
243C2×#67—34no—
244C2×#67—34no—
245C2×#67—38no—
246C2×#67—62no—
247C2×#67C2×C3×D318yescomplemented
248D3×D4—68no—
249D3×D4—632no—
250D3×D4—68no—
251D3×D4—68no—
252C2×D12—38no—
253C2×D12—32no—
254C2×D12—61no—
255C2×D12C2×C3×D312yescomplemented
256C2×C4×D3—32no—
257C2×C4×D3—34no—
258C2×C4×D3—61no—
259C3×C42—32no—
260C3×C42—32no—
261C3×C42—62no—
262C3×C42—92no—
263C4×Dic3—34no—
264C4×Dic3C2×C3×D312yescomplemented
265C22×C32—22no—
266C22×C32—24no—
267C22×C32—28no—
268C22×C32—24no—
269C22×C32—24no—
270C22×C32—21no—
271C22×C32—22no—
272C22×C32—41no—
273C22×C32—44no—
274C22×C32—44no—
275C22×C32—44no—
276C22×C32—48no—
277C22×C32—68no—
278C22×C32—64no—
279C22×C32—84no—
280C22×C32C23×D312yes—
281C22×C32C24×C311yescharacteristic
282C2×C3⋊D3—22no—
283C2×C3⋊D3—24no—
284C2×C3⋊D3—48no—
285C2×C3⋊D3C2×C3×D414yes—
286C2×C3⋊D3C2×C3×D412yes—
287C2×C3×D3—24no—
288C2×C3×D3—28no—
289C2×C3×D3—24no—
290C2×C3×D3—28no—
291C2×C3×D3—38no—
292C2×C3×D3—34no—
293C2×C3×D3—44no—
294C2×C3×D3—48no—
295C2×C3×D3—62no—
296C2×C3×D3—64no—
297C2×C3×D3—64no—
298C2×C3×D3—68no—
299C2×C3×D3—64no—
300C2×C3×D3—124no—
301C2×C3×D3—1216no—
302C2×C3×D3—1216no—
303C2×C3×D3—1216no—
304D32—48no—
305C32×C4—22no—
306C32×C4—24no—
307C32×C4—38no—
308C32×C4—42no—
309C32×C4D3×D414yescomplemented
310C32×C4C2×C3×D412yescomplemented
311C3⋊Dic3C2×C3×D412yescomplemented
312C3×Dic3—28no—
313C3×Dic3—34no—
314C3×Dic3—62no—
315C3×Dic3D3×D418yescomplemented
316C3×Dic3C2×C3×D418yescomplemented
317C22×D4—94no—
318C22×D4—91no—
319C22×D4—91no—
320#127—32no—
321#127—92no—
322#127—94no—
323C2×C42—91no—
324C33C2×#12711yescharacteristic, complemented, Sylow
325C23×C3—22no—
326C23×C3—24no—
327C23×C3—34no—
328C23×C3—34no—
329C23×C3—42no—
330C23×C3—64no—
331C23×C3—91no—
332C23×C3—92no—
333C23×C3—184no—
334C23×C3—188no—
335C23×C3—188no—
336C23×C3—184no—
337C23×C3C2×D3212yes—
338C23×C3C22×C3×D314yes—
339C22×D3—32no—
340C22×D3—34no—
341C22×D3—61no—
342C22×D3—62no—
343C22×D3—68no—
344C22×D3—616no—
345C22×D3—68no—
346C22×D3—616no—
347C22×D3—68no—
348C22×D3—128no—
349C22×D3C22×C3×D314yes—
350C3×D4—24no—
351C3×D4—22no—
352C3×D4—24no—
353C3×D4—44no—
354C3×D4—44no—
355C3×D4—616no—
356C3×D4—616no—
357C3×D4—64no—
358C3×D4—64no—
359C3×D4—124no—
360C3×D4—1216no—
361C3×D4—1816no—
362C3×D4—184no—
363C3×D4—184no—
364C3×D4—182no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—34no—
368C2×C3×C4—34no—
369C2×C3×C4—41no—
370C2×C3×C4—64no—
371C2×C3×C4—98no—
372C2×C3×C4—91no—
373C2×C3×C4—94no—
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C22×C3×D312yes—
376#67—216no—
377#67—68no—
378#67—68no—
379#67—68no—
380#67—616no—
381C2×Dic3—32no—
382C2×Dic3—61no—
383C2×Dic3C22×C3×D314yes—
384D12—24no—
385D12—64no—
386D12—616no—
387D12—64no—
388C4×D3—38no—
389C4×D3—316no—
390C4×D3—64no—
391C2×C32—22no—
392C2×C32—21no—
393C2×C32—24no—
394C2×C32—22no—
395C2×C32—41no—
396C2×C32—42no—
397C2×C32—42no—
398C2×C32—44no—
399C2×C32—44no—
400C2×C32—84no—
401C2×C32—84no—
402C2×C32—124no—
403C2×C32C2×D3×D414yes—
404C2×C32C2×D3×D412yes—
405C2×C32C22×C3×D412yes—
406C2×C32C22×C3×D411yescharacteristic
407C3⋊D3—42no—
408C3⋊D3C3×#12714yescomplemented
409C3×D3—38no—
410C3×D3—44no—
411C3×D3—44no—
412C3×D3—64no—
413C3×D3—84no—
414C3×D3—124no—
415C3×D3—124no—
416C24—92no—
417C24—92no—
418C2×D4—38no—
419C2×D4—32no—
420C2×D4—98no—
421C2×D4—92no—
422C2×D4—92no—
423C2×D4—91no—
424C2×D4—184no—
425C2×D4—1816no—
426C2×D4—184no—
427C2×D4C3×D3211yescharacteristic, complemented
428C22×C4—92no—
429C22×C4—91no—
430C42—32no—
431C42—92no—
432C22×C3—24no—
433C22×C3—24no—
434C22×C3—22no—
435C22×C3—28no—
436C22×C3—24no—
437C22×C3—21no—
438C22×C3—22no—
439C22×C3—44no—
440C22×C3—44no—
441C22×C3—48no—
442C22×C3—44no—
443C22×C3—41no—
444C22×C3—68no—
445C22×C3—64no—
446C22×C3—64no—
447C22×C3—68no—
448C22×C3—84no—
449C22×C3—94no—
450C22×C3—92no—
451C22×C3—124no—
452C22×C3—128no—
453C22×C3—184no—
454C22×C3—182no—
455C22×C3—368no—
456C22×C3—368no—
457C22×C3C22×D3211yescharacteristic
458C22×C3C23×C3×D312yes—
459C2×D3—24no—
460C2×D3—28no—
461C2×D3—34no—
462C2×D3—38no—
463C2×D3—62no—
464C2×D3—64no—
465C2×D3—64no—
466C2×D3—64no—
467C2×D3—68no—
468C2×D3—124no—
469C2×D3—1216no—
470C2×D3—1216no—
471C2×D3—1216no—
472C3×C4—22no—
473C3×C4—24no—
474C3×C4—38no—
475C3×C4—38no—
476C3×C4—42no—
477C3×C4—68no—
478C3×C4—92no—
479C3×C4#381112yescomplemented
480C3×C4C3×D3×D414yescomplemented
481Dic3—34no—
482Dic3—62no—
483Dic3C3×D3×D418yescomplemented
484C32—22no—
485C32—21no—
486C32—41no—
487C32D3×#12712yescomplemented
488C32C2×C3×#12711yescharacteristic, complemented
489C23—34no—
490C23—91no—
491C23—92no—
492C23—184no—
493C23—188no—
494C23—184no—
495C23—188no—
496C23C2×C3×D3212yes—
497D4—22no—
498D4—616no—
499D4—64no—
500D4—1816no—
501D4—184no—
502D4—184no—
503D4—182no—
504C2×C4—34no—
505C2×C4—98no—
506C2×C4—91no—
507C2×C4—94no—
508C2×C4C2×C3×D3211yescharacteristic
509C2×C3—21no—
510C2×C3—22no—
511C2×C3—24no—
512C2×C3—22no—
513C2×C3—44no—
514C2×C3—44no—
515C2×C3—42no—
516C2×C3—41no—
517C2×C3—42no—
518C2×C3—84no—
519C2×C3—84no—
520C2×C3—94no—
521C2×C3—124no—
522C2×C3—124no—
523C2×C3—244no—
524C2×C3—362no—
525C2×C3D32×D412yes—
526C2×C3C2×#381111yescharacteristic
527C2×C3C2×C3×D3×D412yes—
528C2×C3C2×C3×D3×D414yes—
529D3—38no—
530D3—44no—
531D3—64no—
532D3—124no—
533D3—124no—
534C22—22no—
535C22—24no—
536C22—64no—
537C22—68no—
538C22—92no—
539C22—94no—
540C22—182no—
541C22—184no—
542C22—368no—
543C22—368no—
544C22C22×C3×D3211yescharacteristic
545C4—38no—
546C4—92no—
547C4C3×#381112yescomplemented
548C3—21no—
549C3—22no—
550C3—41no—
551C3#1059530011yescharacteristic, direct product factor
552C3C3×D3×#12712yescomplemented
553C2—42no—
554C2—94no—
555C2—124no—
556C2—362no—
557C2C3×D32×D412yes—
558C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C24
hypercenterC2×C3×D4D32
hypocenterC32C2×C3×#127
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×D4×C3⋊D3C2semidirect
2. C2×C3×#3811C2semidirect
3. C2×C3×#3802C2semidirect
4. C32×#940C2semidirect
5. C3×#63979C2semidirect
6. C3×#63911C2semidirect
7. C3×C4×#624C2semidirect
9. #10595300C3direct (N × H)
12. C2×C33×D4C22semidirect
13. C2×C3×#638C22semidirect
15. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C2×C32×#67C22semidirect
18. C2×C32×D12C22semidirect
20. C2×C3×#626C22semidirect
21. C2×C3×#625C22semidirect
24. C32×C4×Dic3C22semidirect
25. C3×Dic32C22semidirect
26. C2×D4×C3⋊D3C2×C3semidirect
31. C2×#3811C2×C3semidirect
32. C2×#3802C2×C3semidirect
37. C3×#940D3semidirect
38. C3×#940C2×C3semidirect
41. #63979C2×C3semidirect
42. #63911C2×C3semidirect
43. C4×#624C2×C3semidirect
57. C3×C4×C3⋊D3D4semidirect
63. C3×#624D4semidirect
76. C2×C32×D4C2×D3semidirect
77. C2×C32×D4C22×C3semidirect
78. C2×#638C22×C3semidirect
80. C2×C4⋊(C3⋊D3)C22×C3semidirect
87. C2×C3×#67C2×D3semidirect
88. C2×C3×#67C22×C3semidirect
97. C2×C3×D12C2×D3semidirect
98. C2×C3×D12C22×C3semidirect
103. C2×#626C22×C3semidirect
104. C2×#625C22×C3semidirect
110. C3×C4×Dic3C2×D3semidirect
111. C3×C4×Dic3C22×C3semidirect
112. Dic32C22×C3semidirect
124. C33×C4C2×D4semidirect
125. C3×C3⋊Dic3C2×D4semidirect
126. C32×Dic3C2×D4semidirect
142. #940C3×D3semidirect
186. C4×C3⋊D3C3×D4semidirect
210. #624C3×D4semidirect
216. C3×C3⋊D3#127semidirect
239. C2×C3×D4D32semidirect
240. C2×C3×D4C2×C3×D3semidirect
247. C2×#67C2×C3×D3semidirect
255. C2×D12C2×C3×D3semidirect
264. C4×Dic3C2×C3×D3semidirect
309. C32×C4D3×D4semidirect
310. C32×C4C2×C3×D4semidirect
311. C3⋊Dic3C2×C3×D4semidirect
315. C3×Dic3D3×D4semidirect
316. C3×Dic3C2×C3×D4semidirect
324. C33C2×#127semidirect
408. C3⋊D3C3×#127semidirect
427. C2×D4C3×D32semidirect
479. C3×C4#3811semidirect
480. C3×C4C3×D3×D4semidirect
483. Dic3C3×D3×D4semidirect
487. C32D3×#127semidirect
488. C32C2×C3×#127semidirect
547. C4C3×#3811semidirect
551. C3#10595300direct (N × H)
552. C3C3×D3×#127semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×C3×#38112
C2×C3×#38022
C32×#9402
C3×#639792
C3×#639112
C3×C4×#6242
C3×D3×#1273
#105953003