Finite Group Databasebeta

← C2×C3×#63926 #13174802

Subgroups — C2×C3×#63926 #13174802

Order 1728.

1794 conjugacy classes of subgroups.

The subgroup lattice has 572 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×#3802C211yesmaximal, characteristic, complemented
3C2×C32×#213C212yesmaximal, complemented
4C3×#63926C218yesmaximal, direct product factor
5C2×C3×#3714C212yesmaximal, complemented
6C2×C3×#3710C211yesmaximal, characteristic, complemented
7C2×C3×D3×D8—32nomaximal
8C2×#63926C311yesmaximal, characteristic, direct product factor
9C23×C3×C3⋊D3—21no—
10C22×C3×D32—21no—
11C2×C33×D4C2211yescharacteristic, complemented
12C2×C3×#638—21no—
13C3×D4×C3⋊D3—22no—
14C3×D4×C3⋊D3C2214yescomplemented
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×D12C2212yescomplemented
18C2×C3×#625—21no—
19C3×#3802—22no—
20C3×#3802C2214yescomplemented
21C32×#213C2218yescomplemented
22C2×C32×C3⋊C8C2212yescomplemented
23C3×#3714C2218yescomplemented
24C3×#3710C2214yescomplemented
25C2×D4×C3⋊D3C2×C311yescharacteristic, complemented
26C2×C3×D3×D4—32no—
27C2×C3×D3×D4—32no—
28C2×C3×D3×D4—61no—
29C2×#3802C2×C311yescharacteristic, complemented
30C2×C32×D8—32no—
31C2×C3×#213—22no—
32C2×C3×#213—32no—
33C2×C3×#213D312yescomplemented
34C2×C3×#213C2×C312yescomplemented
35C3×D3×D8—316no—
36C2×C3×D24—32no—
37C2×C3×D3×C8—32no—
38#63926C2×C318yesdirect product factor
39C2×#3714C2×C312yescomplemented
40C2×#3710C2×C311yescharacteristic, complemented
41C23×C33—21no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3—24no—
44C22×C3×C3⋊D3—44no—
45C22×C3×C3⋊D3D411yescharacteristic
46C22×C32×D3—22no—
47C2×C3×D32—24no—
48C2×C3×D32—44no—
49C33×D4—21no—
50C33×D4C2312yescomplemented
51C2×C33×C4C2311yescharacteristic
52C3×#638—24no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)C2312yescomplemented
56C3×C4×C3⋊D3D412yes—
57C3×C4×C3⋊D3C2312yes—
58C32×D12—22no—
59C32×D12C2314yescomplemented
60C3×#625—24no—
61C32×C3⋊C8C2314yescomplemented
62C22×C3×D8—91no—
63C2×D3×D8—32no—
64C23×C3⋊D3—21no—
65C23×C3×D3—62no—
66C23×C3×D3—62no—
67C23×C3×D3—62no—
68C22×D32—21no—
69C2×C32×D4—21no—
70C2×C32×D4—22no—
71C2×C32×D4—32no—
72C2×C32×D4—41no—
73C2×C32×D4C2×D312yescomplemented
74C2×C32×D4C22×C311yescharacteristic, complemented
75C2×#638—21no—
76D4×C3⋊D3—22no—
77D4×C3⋊D3C22×C314yescomplemented
78C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
79C2×C4×C3⋊D3C22×C311yescharacteristic
80C2×C3×#67—62no—
81C2×C3×#67—62no—
82C2×C3×#67—62no—
83C3×D3×D4—38no—
84C3×D3×D4—38no—
85C3×D3×D4—64no—
86C3×D3×D4—64no—
87C3×D3×D4—64no—
88C3×D3×D4—64no—
89C2×C3×D12—22no—
90C2×C3×D12—32no—
91C2×C3×D12—61no—
92C2×C3×D12C2×D312yescomplemented
93C2×C3×D12C22×C312yescomplemented
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—61no—
96C2×#625—21no—
97#3802—22no—
98#3802C22×C314yescomplemented
99C32×D8—38no—
100C2×C32×C8—32no—
101C3×#213—28no—
102C3×#213—38no—
103C3×#213C2×D318yescomplemented
104C3×#213C22×C318yescomplemented
105C2×C3×C3⋊C8—22no—
106C2×C3×C3⋊C8C2×D312yescomplemented
107C2×C3×C3⋊C8C22×C312yescomplemented
108C3×D24—38no—
109C3×D3×C8—38no—
110#3714C22×C318yescomplemented
111#3710C22×C314yescomplemented
112C22×C33—22no—
113C22×C33—42no—
114C22×C33C2×D411yescharacteristic
115C2×C3×C3⋊D3—22no—
116C2×C3×C3⋊D3—42no—
117C2×C3×C3⋊D3—48no—
118C2×C3×C3⋊D3D814yescomplemented
119C2×C3×C3⋊D3C2×D412yes—
120C2×C32×D3—24no—
121C2×C32×D3—44no—
122C3×D32—48no—
123C33×C4C2×D411yescharacteristic
124C33×C4C2411yescharacteristic
125C3×C3⋊Dic3C2×D412yes—
126C22×C3×D4—91no—
127C22×C3×D4—91no—
128C2×D3×D4—32no—
129C2×D3×D4—32no—
130C2×D3×D4—61no—
131C2×C3×D8—32no—
132C2×C3×D8—32no—
133C2×C3×D8—62no—
134C2×C3×D8—92no—
135C2×C3×D8—98no—
136C22×C3×C8—91no—
137C2×#213—32no—
138C2×#213C3×D312yescomplemented
139D3×D8—316no—
140C2×D24—32no—
141C2×D3×C8—32no—
142C23×C32—21no—
143C23×C32—22no—
144C23×C32—22no—
145C23×C32—42no—
146C23×C32—42no—
147C23×C32—62no—
148C22×C3⋊D3—21no—
149C22×C3⋊D3—24no—
150C22×C3⋊D3—44no—
151C22×C3⋊D3C3×D411yescharacteristic
152C22×C3×D3—22no—
153C22×C3×D3—22no—
154C22×C3×D3—32no—
155C22×C3×D3—42no—
156C22×C3×D3—61no—
157C22×C3×D3—68no—
158C22×C3×D3—68no—
159C22×C3×D3—62no—
160C22×C3×D3—68no—
161C22×C3×D3—62no—
162C22×C3×D3—128no—
163C22×C3×D3—128no—
164C22×C3×D3—128no—
165C2×D32—24no—
166C2×D32—44no—
167C32×D4—21no—
168C32×D4—22no—
169C32×D4—22no—
170C32×D4—22no—
171C32×D4—24no—
172C32×D4—34no—
173C32×D4—42no—
174C32×D4—42no—
175C32×D4—42no—
176C32×D4—62no—
177C32×D4C22×D314yescomplemented
178C32×D4C23×C312yescomplemented
179C2×C32×C4—21no—
180C2×C32×C4—22no—
181C2×C32×C4—41no—
182C2×C32×C4C22×D312yes—
183C2×C32×C4C23×C311yescharacteristic
184#638—24no—
185C2×C3⋊Dic3C3×D411yescharacteristic
186C4⋊(C3⋊D3)—21no—
187C4⋊(C3⋊D3)C23×C312yescomplemented
188C4×C3⋊D3C3×D412yes—
189C4×C3⋊D3C23×C312yes—
190C3×#67—68no—
191C3×#67—68no—
192C3×#67—68no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—61no—
195C3×D12—22no—
196C3×D12—22no—
197C3×D12—24no—
198C3×D12—34no—
199C3×D12—42no—
200C3×D12—62no—
201C3×D12—62no—
202C3×D12—62no—
203C3×D12C22×D314yescomplemented
204C3×D12C23×C314yescomplemented
205C3×C4×D3—34no—
206C3×C4×D3—34no—
207C3×C4×D3—62no—
208C3×C4×D3—62no—
209#625—24no—
210C32×C8—34no—
211C3×C3⋊C8—24no—
212C3×C3⋊C8C22×D314yescomplemented
213C3×C3⋊C8C23×C314yescomplemented
214C22×D8—91noSylow
215C2×C33—42no—
216C2×C33C2×D812yescomplemented
217C2×C33C22×D411yescharacteristic
218C3×C3⋊D3—42no—
219C3×C3⋊D3C2×D814yescomplemented
220C32×D3—44no—
221C24×C3—181no—
222C24×C3—181no—
223C23×D3—62no—
224C23×D3—62no—
225C23×D3—62no—
226C2×C3×D4—21no—
227C2×C3×D4—22no—
228C2×C3×D4—32no—
229C2×C3×D4—32no—
230C2×C3×D4—41no—
231C2×C3×D4—62no—
232C2×C3×D4—94no—
233C2×C3×D4—94no—
234C2×C3×D4—91no—
235C2×C3×D4—182no—
236C2×C3×D4—182no—
237C2×C3×D4—181no—
238C2×C3×D4—181no—
239C2×C3×D4D3211yescharacteristic, complemented
240C2×C3×D4C2×C3×D312yescomplemented
241C22×C3×C4—91no—
242C2×#67—62no—
243C2×#67—62no—
244C2×#67—62no—
245D3×D4—38no—
246D3×D4—38no—
247D3×D4—64no—
248D3×D4—64no—
249D3×D4—64no—
250D3×D4—64no—
251C2×D12—32no—
252C2×D12—61no—
253C2×D12C2×C3×D312yescomplemented
254C2×C4×D3—32no—
255C2×C4×D3—61no—
256C3×D8—38no—
257C3×D8—38no—
258C3×D8—68no—
259C3×D8—98no—
260C2×C3×C8—32no—
261C2×C3×C8—32no—
262C2×C3×C8—62no—
263C2×C3×C8—94no—
264#213—38no—
265#213C2×C3×D318yescomplemented
266C2×C3⋊C8C2×C3×D312yescomplemented
267D24—38no—
268D3×C8—38no—
269C22×C32—22no—
270C22×C32—24no—
271C22×C32—24no—
272C22×C32—21no—
273C22×C32—22no—
274C22×C32—41no—
275C22×C32—44no—
276C22×C32—44no—
277C22×C32—42no—
278C22×C32—44no—
279C22×C32—44no—
280C22×C32—64no—
281C22×C32—84no—
282C22×C32—84no—
283C22×C32—124no—
284C22×C32D3×D412yes—
285C22×C32C2×C3×D411yescharacteristic
286C2×C3⋊D3—22no—
287C2×C3⋊D3—42no—
288C2×C3⋊D3—48no—
289C2×C3⋊D3C3×D814yescomplemented
290C2×C3⋊D3C2×C3×D412yes—
291C2×C3×D3—24no—
292C2×C3×D3—24no—
293C2×C3×D3—38no—
294C2×C3×D3—34no—
295C2×C3×D3—44no—
296C2×C3×D3—44no—
297C2×C3×D3—44no—
298C2×C3×D3—64no—
299C2×C3×D3—62no—
300C2×C3×D3—64no—
301C2×C3×D3—64no—
302C2×C3×D3—84no—
303C2×C3×D3—1216no—
304C2×C3×D3—1216no—
305C2×C3×D3—1216no—
306C2×C3×D3—124no—
307C2×C3×D3—124no—
308D32—48no—
309C32×C4—21no—
310C32×C4—22no—
311C32×C4—21no—
312C32×C4—22no—
313C32×C4—41no—
314C32×C4—41no—
315C32×C4D3×D412yes—
316C32×C4C2×C3×D411yescharacteristic
317C32×C4C23×D312yes—
318C32×C4C24×C311yescharacteristic
319C3⋊Dic3C2×C3×D412yes—
320C3×Dic3—34no—
321C3×Dic3—62no—
322C22×D4—91no—
323C22×D4—91no—
324C2×D8—32no—
325C2×D8—92no—
326C2×D8—98no—
327C22×C8—91no—
328C33C22×D811yescharacteristic, complemented, Sylow
329C23×C3—22no—
330C23×C3—21no—
331C23×C3—22no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—62no—
335C23×C3—62no—
336C23×C3—91no—
337C23×C3—122no—
338C23×C3—184no—
339C23×C3—184no—
340C23×C3—181no—
341C23×C3—364no—
342C23×C3—364no—
343C22×D3—22no—
344C22×D3—32no—
345C22×D3—61no—
346C22×D3—68no—
347C22×D3—68no—
348C22×D3—62no—
349C22×D3—68no—
350C22×D3—62no—
351C22×D3—128no—
352C22×D3—128no—
353C22×D3—128no—
354C3×D4—22no—
355C3×D4—21no—
356C3×D4—22no—
357C3×D4—22no—
358C3×D4—24no—
359C3×D4—34no—
360C3×D4—34no—
361C3×D4—42no—
362C3×D4—42no—
363C3×D4—42no—
364C3×D4—64no—
365C3×D4—62no—
366C3×D4—62no—
367C3×D4—92no—
368C3×D4—122no—
369C3×D4—184no—
370C3×D4—184no—
371C3×D4—181no—
372C3×D4C2×D3212yescomplemented
373C3×D4C22×C3×D314yescomplemented
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—41no—
377C2×C3×C4—91no—
378C2×C3×C4—92no—
379C2×C3×C4—92no—
380C2×C3×C4C2×D3211yescharacteristic
381C2×C3×C4C22×C3×D312yes—
382#67—68no—
383#67—68no—
384#67—68no—
385C2×Dic3—32no—
386C2×Dic3—61no—
387D12—22no—
388D12—34no—
389D12—62no—
390D12—62no—
391D12—62no—
392D12C22×C3×D314yescomplemented
393C4×D3—34no—
394C4×D3—34no—
395C4×D3—62no—
396C4×D3—62no—
397C3×C8—34no—
398C3×C8—34no—
399C3×C8—64no—
400C3⋊C8C22×C3×D314yescomplemented
401C2×C32—24no—
402C2×C32—21no—
403C2×C32—22no—
404C2×C32—22no—
405C2×C32—42no—
406C2×C32—41no—
407C2×C32—42no—
408C2×C32—44no—
409C2×C32—44no—
410C2×C32—84no—
411C2×C32—84no—
412C2×C32—124no—
413C2×C32D3×D814yescomplemented
414C2×C32C2×C3×D812yescomplemented
415C2×C32C2×D3×D412yes—
416C2×C32C22×C3×D411yescharacteristic
417C3⋊D3—42no—
418C3⋊D3C2×C3×D814yescomplemented
419C3×D3—38no—
420C3×D3—44no—
421C3×D3—44no—
422C3×D3—64no—
423C3×D3—84no—
424C3×D3—124no—
425C3×D3—124no—
426C24—181no—
427C24—181no—
428C2×D4—32no—
429C2×D4—94no—
430C2×D4—94no—
431C2×D4—91no—
432C2×D4—182no—
433C2×D4—182no—
434C2×D4—181no—
435C2×D4—181no—
436C2×D4C3×D3211yescharacteristic, complemented
437C22×C4—91no—
438D8—38no—
439D8—98no—
440C2×C8—32no—
441C2×C8—94no—
442C22×C3—24no—
443C22×C3—22no—
444C22×C3—24no—
445C22×C3—21no—
446C22×C3—22no—
447C22×C3—44no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—42no—
452C22×C3—41no—
453C22×C3—64no—
454C22×C3—64no—
455C22×C3—84no—
456C22×C3—84no—
457C22×C3—92no—
458C22×C3—94no—
459C22×C3—124no—
460C22×C3—124no—
461C22×C3—124no—
462C22×C3—182no—
463C22×C3—244no—
464C22×C3—368no—
465C22×C3—368no—
466C22×C3—362no—
467C22×C3#381111yescharacteristic
468C22×C3C3×D3×D412yes—
469C2×D3—24no—
470C2×D3—38no—
471C2×D3—34no—
472C2×D3—44no—
473C2×D3—62no—
474C2×D3—64no—
475C2×D3—64no—
476C2×D3—64no—
477C2×D3—1216no—
478C2×D3—1216no—
479C2×D3—1216no—
480C2×D3—124no—
481C2×D3—124no—
482C3×C4—22no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—22no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—92no—
489C3×C4#381111yescharacteristic
490C3×C4C3×D3×D412yes—
491C3×C4C22×D3211yescharacteristic
492C3×C4C23×C3×D312yes—
493Dic3—34no—
494Dic3—62no—
495C32—22no—
496C32—21no—
497C32—41no—
498C32C2×D3×D812yescomplemented
499C32C22×C3×D811yescharacteristic, complemented
500C23—21no—
501C23—62no—
502C23—91no—
503C23—184no—
504C23—184no—
505C23—181no—
506C23—364no—
507C23—364no—
508D4—21no—
509D4—34no—
510D4—62no—
511D4—92no—
512D4—184no—
513D4—184no—
514D4—181no—
515D4C2×C3×D3212yescomplemented
516C2×C4—91no—
517C2×C4—92no—
518C2×C4—92no—
519C2×C4C2×C3×D3211yescharacteristic
520C8—34no—
521C2×C3—21no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—24no—
525C2×C3—44no—
526C2×C3—44no—
527C2×C3—42no—
528C2×C3—41no—
529C2×C3—42no—
530C2×C3—84no—
531C2×C3—84no—
532C2×C3—94no—
533C2×C3—124no—
534C2×C3—124no—
535C2×C3—244no—
536C2×C3—362no—
537C2×C3#6392612yesdirect product factor
538C2×C3C3×D3×D814yescomplemented
539C2×C3C2×#381111yescharacteristic
540C2×C3C2×C3×D3×D412yes—
541D3—38no—
542D3—44no—
543D3—64no—
544D3—124no—
545D3—124no—
546C22—22no—
547C22—42no—
548C22—64no—
549C22—92no—
550C22—94no—
551C22—124no—
552C22—182no—
553C22—368no—
554C22—368no—
555C22—362no—
556C22C3×#381111yescharacteristic
557C4—92no—
558C4C3×#381111yescharacteristic
559C4C22×C3×D3211yescharacteristic
560C3—21no—
561C3—22no—
562C3—41no—
563C3C2×#6392611yescharacteristic, direct product factor
564C3C2×C3×D3×D812yescomplemented
565C2—42no—
566C2—94no—
567C2—124no—
568C2—362no—
569C2C3×#6392612yesdirect product factor
570C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC32×C4C24×C3
FrattiniC4C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C22×C3×D8
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ 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×#3802C2semidirect
3. C2×C32×#213C2semidirect
4. C3×#63926C2direct (N × H)
5. C2×C3×#3714C2semidirect
6. C2×C3×#3710C2semidirect
8. C2×#63926C3direct (N × H)
11. C2×C33×D4C22semidirect
14. C3×D4×C3⋊D3C22semidirect
15. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C2×C32×D12C22semidirect
20. C3×#3802C22semidirect
21. C32×#213C22semidirect
22. C2×C32×C3⋊C8C22semidirect
23. C3×#3714C22semidirect
24. C3×#3710C22semidirect
25. C2×D4×C3⋊D3C2×C3semidirect
29. C2×#3802C2×C3semidirect
33. C2×C3×#213D3semidirect
34. C2×C3×#213C2×C3semidirect
38. #63926C2×C3direct (N × H)
39. C2×#3714C2×C3semidirect
40. C2×#3710C2×C3semidirect
50. C33×D4C23semidirect
55. C3×C4⋊(C3⋊D3)C23semidirect
59. C32×D12C23semidirect
61. C32×C3⋊C8C23semidirect
73. C2×C32×D4C2×D3semidirect
74. C2×C32×D4C22×C3semidirect
77. D4×C3⋊D3C22×C3semidirect
78. C2×C4⋊(C3⋊D3)C22×C3semidirect
92. C2×C3×D12C2×D3semidirect
93. C2×C3×D12C22×C3semidirect
98. #3802C22×C3semidirect
103. C3×#213C2×D3semidirect
104. C3×#213C22×C3semidirect
106. C2×C3×C3⋊C8C2×D3semidirect
107. C2×C3×C3⋊C8C22×C3semidirect
110. #3714C22×C3semidirect
111. #3710C22×C3semidirect
118. C2×C3×C3⋊D3D8semidirect
138. C2×#213C3×D3semidirect
177. C32×D4C22×D3semidirect
178. C32×D4C23×C3semidirect
187. C4⋊(C3⋊D3)C23×C3semidirect
203. C3×D12C22×D3semidirect
204. C3×D12C23×C3semidirect
212. C3×C3⋊C8C22×D3semidirect
213. C3×C3⋊C8C23×C3semidirect
216. C2×C33C2×D8semidirect
219. C3×C3⋊D3C2×D8semidirect
239. C2×C3×D4D32semidirect
240. C2×C3×D4C2×C3×D3semidirect
253. C2×D12C2×C3×D3semidirect
265. #213C2×C3×D3semidirect
266. C2×C3⋊C8C2×C3×D3semidirect
289. C2×C3⋊D3C3×D8semidirect
328. C33C22×D8semidirect
372. C3×D4C2×D32semidirect
373. C3×D4C22×C3×D3semidirect
392. D12C22×C3×D3semidirect
400. C3⋊C8C22×C3×D3semidirect
413. C2×C32D3×D8semidirect
414. C2×C32C2×C3×D8semidirect
418. C3⋊D3C2×C3×D8semidirect
436. C2×D4C3×D32semidirect
498. C32C2×D3×D8semidirect
499. C32C22×C3×D8semidirect
515. D4C2×C3×D32semidirect
537. C2×C3#63926direct (N × H)
538. C2×C3C3×D3×D8semidirect
563. C3C2×#63926direct (N × H)
564. C3C2×C3×D3×D8semidirect
569. C2C3×#63926direct (N × H)

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×#38022
C2×C32×#2132
C3×#639262
C2×C3×#37142
C2×C3×#37102
C2×C3×D3×D83
C2×#639263