Finite Group Databasebeta

← C2×C3×#64308 #13181543

Subgroups — C2×C3×#64308 #13181543

Order 1728.

4210 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×C32×#235C212yesmaximal, complemented
3C2×C3×#3811C214yesmaximal, complemented
4C3×#64308C2116yesmaximal, direct product factor
5C2×C3×D3×D12C214yesmaximal, complemented
6C2×C3×#3799C214yesmaximal, complemented
7C2×C3×#1009—32nomaximal
8C2×#64308C311yesmaximal, characteristic, direct product factor
9C23×C3×C3⋊D3C2212yescomplemented
10C22×C3×D32C2214yescomplemented
11C22×C33×C4C2211yescharacteristic, complemented
12C2×C3×C4⋊(C3⋊D3)C2214yescomplemented
13C2×C3×C4⋊(C3⋊D3)C2218yescomplemented
14C2×C32×#67C2214yescomplemented
15C32×#235C22116yescomplemented
16C2×C32×D12C2212yescomplemented
17C2×C32×C4×D3C2214yescomplemented
18C2×C32×Dic6C2212yescomplemented
19C3×#3811C22132yescomplemented
20C2×C3×#626C2218yescomplemented
21C2×C3×#624C2214yescomplemented
22C3×D3×D12C22132yescomplemented
23C3×#3799C22132yescomplemented
24C2×C32×#41—32no—
25C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
26C3×#1009—332no—
27C2×C3×#239—34no—
28C2×C3×D3×D4—38no—
29C2×C3×D3×D4—34no—
30C2×C3×#235—22no—
31C2×C3×#235—34no—
32C2×C3×#235D312yescomplemented
33C2×C3×#235C2×C312yescomplemented
34C22×C3×D12—32no—
35C22×C3×D12—34no—
36C22×C3×D12—61no—
37C2×#3811C2×C314yescomplemented
38#64308C2×C3116yesdirect product factor
39C2×D3×D12C2×C314yescomplemented
40C2×#3799C2×C314yescomplemented
41C23×C33C2311yescharacteristic
42C22×C3×C3⋊D3—44no—
43C22×C3×C3⋊D3C2314yes—
44C22×C3×C3⋊D3C2318yescomplemented
45C22×C32×D3C2314yes—
46C2×C3×D32—48no—
47C2×C3×D32C23116yescomplemented
48C2×C33×C4C2312yes—
49C2×C33×C4C2314yescomplemented
50C3×C4⋊(C3⋊D3)C23116yescomplemented
51C32×#67C23116yescomplemented
52C2×C32×Dic3C2314yes—
53C32×D12C2318yescomplemented
54C32×C4×D3C23116yescomplemented
55C32×Dic6C2318yescomplemented
56C3×#626C23132yescomplemented
57C3×#624C23116yescomplemented
58C2×C3×H(5,𝔽2)—91no—
59C2×#1009—32no—
60C23×C3⋊D3C22×C312yescomplemented
61C23×C3×D3—34no—
62C23×C3×D3—38no—
63C23×C3×D3—62no—
64C22×D32C22×C314yescomplemented
65C32×#41—316no—
66C2×C32×Q8—32no—
67C2×C32×D4—34no—
68C2×C32×D4—32no—
69C22×C32×C4—21no—
70C22×C32×C4—22no—
71C22×C32×C4—34no—
72C22×C32×C4—41no—
73C22×C32×C4C2×D312yescomplemented
74C22×C32×C4C22×C311yescharacteristic, complemented
75C2×C4⋊(C3⋊D3)C22×C314yescomplemented
76C2×C4⋊(C3⋊D3)C22×C318yescomplemented
77C2×C3×#67—24no—
78C2×C3×#67—38no—
79C2×C3×#67C2×D314yescomplemented
80C2×C3×#67C22×C314yescomplemented
81C3×#239—332no—
82C3×D3×D4—364no—
83C3×D3×D4—332no—
84C3×#235—216no—
85C3×#235—332no—
86C3×#235C2×D3116yescomplemented
87C3×#235C22×C3116yescomplemented
88C2×C3×D12—22no—
89C2×C3×D12—38no—
90C2×C3×D12—34no—
91C2×C3×D12—34no—
92C2×C3×D12—316no—
93C2×C3×D12—332no—
94C2×C3×D12—64no—
95C2×C3×D12—68no—
96C2×C3×D12C2×D312yescomplemented
97C2×C3×D12C22×C312yescomplemented
98C2×C3×C4×D3—24no—
99C2×C3×C4×D3—38no—
100C2×C3×C4×D3C2×D314yescomplemented
101C2×C3×C4×D3C22×C314yescomplemented
102C2×C3×Dic6—22no—
103C2×C3×Dic6C2×D312yescomplemented
104C2×C3×Dic6C22×C312yescomplemented
105#3811C22×C3132yescomplemented
106C2×#626C22×C318yescomplemented
107C2×#624C22×C314yescomplemented
108D3×D12C22×C3132yescomplemented
109#3799C22×C3132yescomplemented
110C22×C33—22no—
111C22×C33C2412yes—
112C22×C33C2411yescharacteristic
113C2×C3×C3⋊D3—28no—
114C2×C3×C3⋊D3—48no—
115C2×C3×C3⋊D3C2418yes—
116C2×C32×D3—28no—
117C2×C32×D3C2418yes—
118C3×D32—416no—
119C33×C4C2414yes—
120C32×Dic3C2418yes—
121C3×H(5,𝔽2)—916no—
122C2×C3×#41—32no—
123C2×C3×#41—32no—
124C2×C3×#41—62no—
125C2×C3×#41—94no—
126C22×C3×D4—94no—
127C22×C3×D4—94no—
128C22×C3×D4—91no—
129#1009—332no—
130C2×#239—34no—
131C2×D3×D4—38no—
132C2×D3×D4—34no—
133C2×#235—34no—
134C2×#235C3×D312yescomplemented
135C22×D12—32no—
136C22×D12—34no—
137C22×D12—61no—
138C23×C32—21no—
139C23×C32—22no—
140C23×C32—34no—
141C23×C32—41no—
142C23×C32C22×D312yes—
143C23×C32C23×C311yescharacteristic
144C22×C3⋊D3—44no—
145C22×C3⋊D3C23×C314yes—
146C22×C3⋊D3C23×C318yescomplemented
147C22×C3×D3—24no—
148C22×C3×D3—38no—
149C22×C3×D3—316no—
150C22×C3×D3—332no—
151C22×C3×D3—64no—
152C22×C3×D3—68no—
153C22×C3×D3—1216no—
154C22×C3×D3—128no—
155C22×C3×D3—128no—
156C22×C3×D3C22×D314yes—
157C22×C3×D3C23×C314yes—
158C2×D32—48no—
159C2×D32C23×C3116yescomplemented
160C32×Q8—38no—
161C32×D4—316no—
162C32×D4—38no—
163C2×C32×C4—22no—
164C2×C32×C4—24no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4—28no—
168C2×C32×C4—34no—
169C2×C32×C4—316no—
170C2×C32×C4—42no—
171C2×C32×C4—44no—
172C2×C32×C4C22×D312yes—
173C2×C32×C4C22×D312yes—
174C2×C32×C4C22×D318yescomplemented
175C2×C32×C4C23×C312yes—
176C2×C32×C4C23×C314yescomplemented
177C4⋊(C3⋊D3)C23×C3116yescomplemented
178C3×#67—216no—
179C3×#67—332no—
180C3×#67C22×D3116yescomplemented
181C3×#67C23×C3116yescomplemented
182C2×C3×Dic3—24no—
183C2×C3×Dic3C22×D314yes—
184C2×C3×Dic3C23×C314yes—
185C3×D12—28no—
186C3×D12—332no—
187C3×D12—316no—
188C3×D12—316no—
189C3×D12—616no—
190C3×D12C22×D318yescomplemented
191C3×D12C23×C318yescomplemented
192C3×C4×D3—216no—
193C3×C4×D3—332no—
194C3×C4×D3C22×D3116yescomplemented
195C3×C4×D3C23×C3116yescomplemented
196C3×Dic6—28no—
197C3×Dic6C22×D318yescomplemented
198C3×Dic6C23×C318yescomplemented
199#626C23×C3132yescomplemented
200#624C23×C3116yescomplemented
201C2×H(5,𝔽2)—91noSylow
202C2×C33—22no—
203C2×C33H(5,𝔽2)12yescomplemented
204C2×C33C2511yescharacteristic
205C3×C3⋊D3—28no—
206C32×D3—28no—
207C24×C3—94no—
208C24×C3—92no—
209C23×D3—34no—
210C23×D3—38no—
211C23×D3—62no—
212C3×#41—316no—
213C3×#41—316no—
214C3×#41—616no—
215C3×#41—932no—
216C2×C3×Q8—32no—
217C2×C3×Q8—32no—
218C2×C3×Q8—62no—
219C2×C3×D4—34no—
220C2×C3×D4—34no—
221C2×C3×D4—32no—
222C2×C3×D4—32no—
223C2×C3×D4—62no—
224C2×C3×D4—64no—
225C2×C3×D4—932no—
226C2×C3×D4—98no—
227C2×C3×D4—932no—
228C2×C3×D4—94no—
229C2×C3×D4—98no—
230C22×C3×C4—21no—
231C22×C3×C4—22no—
232C22×C3×C4—34no—
233C22×C3×C4—34no—
234C22×C3×C4—41no—
235C22×C3×C4—64no—
236C22×C3×C4—94no—
237C22×C3×C4D3211yescharacteristic, complemented
238C22×C3×C4C2×C3×D312yescomplemented
239C2×#67—38no—
240C2×#67C2×C3×D314yescomplemented
241#239—332no—
242D3×D4—364no—
243D3×D4—332no—
244#235—332no—
245#235C2×C3×D3116yescomplemented
246C2×D12—38no—
247C2×D12—34no—
248C2×D12—34no—
249C2×D12—316no—
250C2×D12—332no—
251C2×D12—64no—
252C2×D12—68no—
253C2×D12C2×C3×D312yescomplemented
254C2×C4×D3—38no—
255C2×C4×D3C2×C3×D314yescomplemented
256C2×Dic6C2×C3×D312yescomplemented
257C22×C32—22no—
258C22×C32—24no—
259C22×C32—24no—
260C22×C32—22no—
261C22×C32—21no—
262C22×C32—22no—
263C22×C32—24no—
264C22×C32—38no—
265C22×C32—41no—
266C22×C32—42no—
267C22×C32—44no—
268C22×C32—44no—
269C22×C32—68no—
270C22×C32C23×D314yes—
271C22×C32C23×D312yes—
272C22×C32C24×C312yes—
273C22×C32C24×C311yescharacteristic
274C2×C3⋊D3—28no—
275C2×C3⋊D3—48no—
276C2×C3⋊D3C24×C318yes—
277C2×C3×D3—28no—
278C2×C3×D3—28no—
279C2×C3×D3—28no—
280C2×C3×D3—316no—
281C2×C3×D3—48no—
282C2×C3×D3—68no—
283C2×C3×D3—616no—
284C2×C3×D3—128no—
285C2×C3×D3—1232no—
286C2×C3×D3—1216no—
287C2×C3×D3—1216no—
288C2×C3×D3C23×D318yes—
289C2×C3×D3C24×C318yes—
290D32—416no—
291C32×C4—24no—
292C32×C4—24no—
293C32×C4—24no—
294C32×C4—38no—
295C32×C4—44no—
296C32×C4C23×D314yes—
297C32×C4C23×D314yes—
298C32×C4C24×C314yes—
299C3×Dic3—28no—
300C3×Dic3C23×D318yes—
301C3×Dic3C24×C318yes—
302H(5,𝔽2)—916no—
303C2×#41—32no—
304C2×#41—94no—
305C22×D4—94no—
306C22×D4—94no—
307C22×D4—91no—
308C33C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
309C23×C3—21no—
310C23×C3—22no—
311C23×C3—34no—
312C23×C3—34no—
313C23×C3—41no—
314C23×C3—64no—
315C23×C3—916no—
316C23×C3—94no—
317C23×C3—98no—
318C23×C3—364no—
319C23×C3—368no—
320C23×C3C2×D3211yescharacteristic
321C23×C3C22×C3×D312yes—
322C22×D3—38no—
323C22×D3—316no—
324C22×D3—332no—
325C22×D3—64no—
326C22×D3—68no—
327C22×D3—1216no—
328C22×D3—128no—
329C22×D3—128no—
330C22×D3C22×C3×D314yes—
331C3×Q8—38no—
332C3×Q8—38no—
333C3×Q8—68no—
334C3×D4—316no—
335C3×D4—316no—
336C3×D4—38no—
337C3×D4—38no—
338C3×D4—68no—
339C3×D4—616no—
340C3×D4—932no—
341C3×D4—916no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—28no—
346C2×C3×C4—24no—
347C2×C3×C4—34no—
348C2×C3×C4—34no—
349C2×C3×C4—316no—
350C2×C3×C4—316no—
351C2×C3×C4—42no—
352C2×C3×C4—44no—
353C2×C3×C4—616no—
354C2×C3×C4—64no—
355C2×C3×C4—916no—
356C2×C3×C4C2×D3212yes—
357C2×C3×C4C2×D3214yescomplemented
358C2×C3×C4C22×C3×D312yes—
359C2×C3×C4C22×C3×D312yes—
360C2×C3×C4C22×C3×D318yescomplemented
361#67—332no—
362#67C22×C3×D3116yescomplemented
363C2×Dic3C22×C3×D314yes—
364D12—332no—
365D12—316no—
366D12—316no—
367D12—616no—
368D12C22×C3×D318yescomplemented
369C4×D3—332no—
370C4×D3C22×C3×D3116yescomplemented
371Dic6C22×C3×D318yescomplemented
372C2×C32—22no—
373C2×C32—24no—
374C2×C32—24no—
375C2×C32—22no—
376C2×C32—24no—
377C2×C32—21no—
378C2×C32—22no—
379C2×C32—42no—
380C2×C32—41no—
381C2×C32—44no—
382C2×C32—44no—
383C2×C32—68no—
384C2×C32#100914yescomplemented
385C2×C32C3×H(5,𝔽2)12yescomplemented
386C2×C32C24×D312yes—
387C2×C32C25×C311yescharacteristic
388C3⋊D3—28no—
389C3×D3—28no—
390C3×D3—28no—
391C3×D3—48no—
392C3×D3—616no—
393C3×D3—128no—
394C24—94no—
395C24—92no—
396#41—316no—
397#41—932no—
398C2×Q8—32no—
399C2×D4—34no—
400C2×D4—32no—
401C2×D4—932no—
402C2×D4—98no—
403C2×D4—932no—
404C2×D4—94no—
405C2×D4—98no—
406C22×C4—34no—
407C22×C4—94no—
408C22×C4C3×D3211yescharacteristic, complemented
409C22×C3—24no—
410C22×C3—22no—
411C22×C3—24no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—21no—
415C22×C3—24no—
416C22×C3—38no—
417C22×C3—38no—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—42no—
421C22×C3—41no—
422C22×C3—68no—
423C22×C3—68no—
424C22×C3—68no—
425C22×C3—98no—
426C22×C3—128no—
427C22×C3—188no—
428C22×C3—368no—
429C22×C3—3616no—
430C22×C3C22×D3212yes—
431C22×C3C22×D3211yescharacteristic
432C22×C3C23×C3×D314yes—
433C22×C3C23×C3×D312yes—
434C2×D3—28no—
435C2×D3—316no—
436C2×D3—68no—
437C2×D3—616no—
438C2×D3—128no—
439C2×D3—1232no—
440C2×D3—1216no—
441C2×D3—1216no—
442C2×D3C23×C3×D318yes—
443C3×C4—24no—
444C3×C4—24no—
445C3×C4—24no—
446C3×C4—38no—
447C3×C4—38no—
448C3×C4—44no—
449C3×C4—68no—
450C3×C4C22×D3214yes—
451C3×C4C23×C3×D314yes—
452C3×C4C23×C3×D314yes—
453Dic3C23×C3×D318yes—
454C32—21no—
455C32—22no—
456C32—41no—
457C32C2×#100912yescomplemented
458C32C2×C3×H(5,𝔽2)11yescharacteristic, complemented
459C23—34no—
460C23—916no—
461C23—94no—
462C23—98no—
463C23—364no—
464C23—368no—
465C23C2×C3×D3211yescharacteristic
466Q8—38no—
467D4—316no—
468D4—38no—
469D4—932no—
470D4—916no—
471C2×C4—34no—
472C2×C4—316no—
473C2×C4—916no—
474C2×C4C2×C3×D3212yes—
475C2×C4C2×C3×D3214yescomplemented
476C2×C3—24no—
477C2×C3—22no—
478C2×C3—24no—
479C2×C3—22no—
480C2×C3—21no—
481C2×C3—22no—
482C2×C3—24no—
483C2×C3—44no—
484C2×C3—44no—
485C2×C3—41no—
486C2×C3—42no—
487C2×C3—68no—
488C2×C3—68no—
489C2×C3—128no—
490C2×C3—188no—
491C2×C3#6430812yesdirect product factor
492C2×C3C3×#100914yescomplemented
493C2×C3C23×D3211yescharacteristic
494C2×C3C24×C3×D312yes—
495D3—28no—
496D3—616no—
497D3—128no—
498C22—22no—
499C22—38no—
500C22—68no—
501C22—98no—
502C22—188no—
503C22—368no—
504C22—3616no—
505C22C22×C3×D3212yes—
506C22C22×C3×D3211yescharacteristic
507C4—38no—
508C4C22×C3×D3214yes—
509C3—21no—
510C3—22no—
511C3—41no—
512C3C2×#6430811yescharacteristic, direct product factor
513C3C2×C3×#100912yescomplemented
514C2—22no—
515C2—68no—
516C2—188no—
517C2C3×#6430812yesdirect product factor
518C2C23×C3×D3211yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC2×C32C25×C3
FrattiniC2C23×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊(C3⋊D3)C2semidirect
2. C2×C32×#235C2semidirect
3. C2×C3×#3811C2semidirect
4. C3×#64308C2direct (N × H)
5. C2×C3×D3×D12C2semidirect
6. C2×C3×#3799C2semidirect
8. C2×#64308C3direct (N × H)
9. C23×C3×C3⋊D3C22semidirect
10. C22×C3×D32C22semidirect
11. C22×C33×C4C22semidirect
12. C2×C3×C4⋊(C3⋊D3)C22semidirect
13. C2×C3×C4⋊(C3⋊D3)C22semidirect
14. C2×C32×#67C22semidirect
15. C32×#235C22semidirect
16. C2×C32×D12C22semidirect
17. C2×C32×C4×D3C22semidirect
18. C2×C32×Dic6C22semidirect
19. C3×#3811C22semidirect
20. C2×C3×#626C22semidirect
21. C2×C3×#624C22semidirect
22. C3×D3×D12C22semidirect
23. C3×#3799C22semidirect
25. C22×C4⋊(C3⋊D3)C2×C3semidirect
32. C2×C3×#235D3semidirect
33. C2×C3×#235C2×C3semidirect
37. C2×#3811C2×C3semidirect
38. #64308C2×C3direct (N × H)
39. C2×D3×D12C2×C3semidirect
40. C2×#3799C2×C3semidirect
44. C22×C3×C3⋊D3C23semidirect
47. C2×C3×D32C23semidirect
49. C2×C33×C4C23semidirect
50. C3×C4⋊(C3⋊D3)C23semidirect
51. C32×#67C23semidirect
53. C32×D12C23semidirect
54. C32×C4×D3C23semidirect
55. C32×Dic6C23semidirect
56. C3×#626C23semidirect
57. C3×#624C23semidirect
60. C23×C3⋊D3C22×C3semidirect
64. C22×D32C22×C3semidirect
73. C22×C32×C4C2×D3semidirect
74. C22×C32×C4C22×C3semidirect
75. C2×C4⋊(C3⋊D3)C22×C3semidirect
76. C2×C4⋊(C3⋊D3)C22×C3semidirect
79. C2×C3×#67C2×D3semidirect
80. C2×C3×#67C22×C3semidirect
86. C3×#235C2×D3semidirect
87. C3×#235C22×C3semidirect
96. C2×C3×D12C2×D3semidirect
97. C2×C3×D12C22×C3semidirect
100. C2×C3×C4×D3C2×D3semidirect
101. C2×C3×C4×D3C22×C3semidirect
103. C2×C3×Dic6C2×D3semidirect
104. C2×C3×Dic6C22×C3semidirect
105. #3811C22×C3semidirect
106. C2×#626C22×C3semidirect
107. C2×#624C22×C3semidirect
108. D3×D12C22×C3semidirect
109. #3799C22×C3semidirect
134. C2×#235C3×D3semidirect
146. C22×C3⋊D3C23×C3semidirect
159. C2×D32C23×C3semidirect
174. C2×C32×C4C22×D3semidirect
176. C2×C32×C4C23×C3semidirect
177. C4⋊(C3⋊D3)C23×C3semidirect
180. C3×#67C22×D3semidirect
181. C3×#67C23×C3semidirect
190. C3×D12C22×D3semidirect
191. C3×D12C23×C3semidirect
194. C3×C4×D3C22×D3semidirect
195. C3×C4×D3C23×C3semidirect
197. C3×Dic6C22×D3semidirect
198. C3×Dic6C23×C3semidirect
199. #626C23×C3semidirect
200. #624C23×C3semidirect
203. C2×C33H(5,𝔽2)semidirect
237. C22×C3×C4D32semidirect
238. C22×C3×C4C2×C3×D3semidirect
240. C2×#67C2×C3×D3semidirect
245. #235C2×C3×D3semidirect
253. C2×D12C2×C3×D3semidirect
255. C2×C4×D3C2×C3×D3semidirect
256. C2×Dic6C2×C3×D3semidirect
308. C33C2×H(5,𝔽2)semidirect
357. C2×C3×C4C2×D32semidirect
360. C2×C3×C4C22×C3×D3semidirect
362. #67C22×C3×D3semidirect
368. D12C22×C3×D3semidirect
370. C4×D3C22×C3×D3semidirect
371. Dic6C22×C3×D3semidirect
384. C2×C32#1009semidirect
385. C2×C32C3×H(5,𝔽2)semidirect
408. C22×C4C3×D32semidirect
457. C32C2×#1009semidirect
458. C32C2×C3×H(5,𝔽2)semidirect
475. C2×C4C2×C3×D32semidirect
491. C2×C3#64308direct (N × H)
492. C2×C3C3×#1009semidirect
512. C3C2×#64308direct (N × H)
513. C3C2×C3×#1009semidirect
517. C2C3×#64308direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊(C3⋊D3)2
C2×C32×#2352
C2×C3×#38112
C3×#643082
C2×C3×D3×D122
C2×C3×#37992
C2×C3×#10093
C2×#643083