Finite Group Databasebeta

← C2×C3×#63982 #13175116

Subgroups — C2×C3×#63982 #13175116

Order 1728.

1922 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C22×C3×#627C211yesmaximal, characteristic, complemented
3C2×C32×#217C212yesmaximal
4C3×#63982C218yesmaximal, direct product factor
5C2×C3×#3724C211yesmaximal, characteristic, complemented
6C2×C3×Dic3⋊Dic3C212yesmaximal, complemented
7C2×C3×#878—32nomaximal
8C2×#63982C311yesmaximal, characteristic, direct product factor
9C24×C33C2211yescharacteristic
10C22×C3×C3⋊Dic3—24no—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C3×C3⋊Dic3C2214yes—
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C32×Dic3C2212yes—
15C22×C32×Dic3C2212yes—
16C2×C3×#627—24no—
17C2×C3×#627C2214yescomplemented
18C32×#217C2218yes—
19C3×#3724C2214yescomplemented
20C3×Dic3⋊Dic3C2218yescomplemented
21C23×C3⋊Dic3C2×C311yescharacteristic
22C23×C3×Dic3—32no—
23C23×C3×Dic3—61no—
24C22×C3×Dic6—32no—
25C22×#627C2×C311yescharacteristic, complemented
26C2×C32×C22⋊C4—32no—
27C2×C3×#217—22no—
28C2×C3×#217D312yes—
29C2×C3×#217C2×C312yes—
30C2×C3×C4⋊Dic3—32no—
31C2×C3×Dic3⋊C4—32no—
32C2×C3×Dic3⋊C4—32no—
33C3×#878—316no—
34#63982C2×C318yesdirect product factor
35C2×#3724C2×C311yescharacteristic, complemented
36C2×Dic3⋊Dic3C2×C312yescomplemented
37C23×C33—22no—
38C23×C33—22no—
39C23×C33Q814yescomplemented
40C23×C33C2312yes—
41C23×C33C2311yescharacteristic
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3—28no—
44C2×C3×C3⋊Dic3D414yes—
45C2×C3×C3⋊Dic3C2312yes—
46C2×C3×C3⋊Dic3C2312yes—
47C2×C32×Dic3—24no—
48C2×C32×Dic3—24no—
49C2×C32×Dic3C2314yes—
50C2×C32×Dic3C2314yes—
51C3×#627—28no—
52C2×C3×C22⋊Q8—91no—
53C2×#878—32no—
54C24×C32—21no—
55C24×C32—22no—
56C24×C32—41no—
57C24×C32C2×D312yes—
58C24×C32C22×C311yescharacteristic
59C22×C32×C4—32no—
60C22×C32×C4—32no—
61C22×C3⋊Dic3—24no—
62C22×C3⋊Dic3C22×C311yescharacteristic
63C22×C3⋊Dic3C22×C314yes—
64C22×C3⋊Dic3C22×C311yescharacteristic
65C22×C3×Dic3—22no—
66C22×C3×Dic3—22no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—38no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—68no—
71C22×C3×Dic3—61no—
72C22×C3×Dic3—64no—
73C22×C3×Dic3—61no—
74C22×C3×Dic3—68no—
75C22×C3×Dic3C2×D312yes—
76C22×C3×Dic3C2×D312yes—
77C22×C3×Dic3C22×C312yes—
78C22×C3×Dic3C22×C312yes—
79C2×C3×Dic6—38no—
80C2×C3×Dic6—68no—
81C2×#627—24no—
82C2×#627C22×C314yescomplemented
83C32×C22⋊C4—38no—
84C3×#217—28no—
85C3×#217C2×D318yes—
86C3×#217C22×C318yes—
87C3×C4⋊Dic3—38no—
88C3×Dic3⋊C4—38no—
89C3×Dic3⋊C4—38no—
90#3724C22×C314yescomplemented
91Dic3⋊Dic3C22×C318yescomplemented
92C22×C33—28no—
93C22×C33—22no—
94C22×C33—22no—
95C22×C33C2×D412yes—
96C22×C33C2×Q812yes—
97C22×C33C2×Q814yescomplemented
98C22×C33#4112yes—
99C22×C33C2411yescharacteristic
100C3×C3⋊Dic3—22no—
101C3×C3⋊Dic3C2×D414yes—
102C32×Dic3—24no—
103C32×Dic3—24no—
104C22×C3×Q8—91no—
105C23×C3×C4—91no—
106C23×Dic3—32no—
107C23×Dic3—61no—
108C22×Dic6—32no—
109C3×C22⋊Q8—98no—
110C2×C3×C4⋊C4—91no—
111C2×C3×C4⋊C4—92no—
112C2×C3×C22⋊C4—32no—
113C2×C3×C22⋊C4—32no—
114C2×C3×C22⋊C4—62no—
115C2×#217C3×D312yes—
116C2×C4⋊Dic3—32no—
117C2×Dic3⋊C4—32no—
118C2×Dic3⋊C4—32no—
119#878—316no—
120C23×C32—22no—
121C23×C32—22no—
122C23×C32—24no—
123C23×C32—24no—
124C23×C32—24no—
125C23×C32—24no—
126C23×C32—24no—
127C23×C32—28no—
128C23×C32—22no—
129C23×C32—21no—
130C23×C32—22no—
131C23×C32—24no—
132C23×C32—44no—
133C23×C32—41no—
134C23×C32—42no—
135C23×C32—44no—
136C23×C32—44no—
137C23×C32—44no—
138C23×C32—44no—
139C23×C32Dic618yescomplemented
140C23×C32C3×Q814yescomplemented
141C23×C32C22×D314yes—
142C23×C32C22×D312yes—
143C23×C32C23×C312yes—
144C23×C32C23×C311yescharacteristic
145C2×C32×C4—34no—
146C2×C32×C4—34no—
147C2×C32×C4—64no—
148C2×C32×C4—64no—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—28no—
151C2×C3⋊Dic3C3×D414yes—
152C2×C3⋊Dic3C23×C312yes—
153C2×C3⋊Dic3C23×C312yes—
154C2×C3×Dic3—24no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3—38no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3—34no—
163C2×C3×Dic3—44no—
164C2×C3×Dic3—44no—
165C2×C3×Dic3—64no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—616no—
169C2×C3×Dic3—616no—
170C2×C3×Dic3—64no—
171C2×C3×Dic3—64no—
172C2×C3×Dic3C22×D314yes—
173C2×C3×Dic3C22×D314yes—
174C2×C3×Dic3C23×C314yes—
175C2×C3×Dic3C23×C314yes—
176C3×Dic6—616no—
177#627—28no—
178C2×C22⋊Q8—91noSylow
179C2×C33—24no—
180C2×C33C22⋊Q814yescomplemented
181C2×C33C22×D411yescharacteristic
182C2×C33C22×Q811yescharacteristic
183C2×C33C2×#4111yescharacteristic
184C24×C3—21no—
185C24×C3—22no—
186C24×C3—41no—
187C24×C3D3211yescharacteristic
188C24×C3C2×C3×D312yes—
189C2×C3×Q8—94no—
190C2×C3×Q8—184no—
191C22×C3×C4—32no—
192C22×C3×C4—32no—
193C22×C3×C4—32no—
194C22×C3×C4—32no—
195C22×C3×C4—62no—
196C22×C3×C4—62no—
197C22×C3×C4—91no—
198C22×C3×C4—94no—
199C22×C3×C4—91no—
200C22×C3×C4—184no—
201C22×Dic3—32no—
202C22×Dic3—38no—
203C22×Dic3—32no—
204C22×Dic3—61no—
205C22×Dic3—61no—
206C22×Dic3—64no—
207C22×Dic3—68no—
208C22×Dic3—68no—
209C22×Dic3C2×C3×D312yes—
210C22×Dic3C2×C3×D312yes—
211C2×Dic6—38no—
212C2×Dic6—68no—
213C3×C4⋊C4—94no—
214C3×C4⋊C4—98no—
215C3×C22⋊C4—38no—
216C3×C22⋊C4—38no—
217C3×C22⋊C4—68no—
218#217C2×C3×D318yes—
219C4⋊Dic3—38no—
220Dic3⋊C4—38no—
221Dic3⋊C4—38no—
222C22×C32—28no—
223C22×C32—22no—
224C22×C32—22no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—216no—
228C22×C32—216no—
229C22×C32—24no—
230C22×C32—24no—
231C22×C32—21no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—22no—
235C22×C32—24no—
236C22×C32—22no—
237C22×C32—24no—
238C22×C32—28no—
239C22×C32—24no—
240C22×C32—24no—
241C22×C32—42no—
242C22×C32—42no—
243C22×C32—44no—
244C22×C32—42no—
245C22×C32—41no—
246C22×C32—416no—
247C22×C32—44no—
248C22×C32—44no—
249C22×C32—44no—
250C22×C32—416no—
251C22×C32—44no—
252C22×C32C2×Dic614yes—
253C22×C32C2×Dic618yescomplemented
254C22×C32D3×D414yes—
255C22×C32#23714yes—
256C22×C32C2×C3×D412yes—
257C22×C32C2×C3×Q812yes—
258C22×C32C2×C3×Q814yescomplemented
259C22×C32C3×#4112yes—
260C22×C32C23×D312yes—
261C22×C32C24×C311yescharacteristic
262C32×C4—64no—
263C32×C4—64no—
264C3⋊Dic3—22no—
265C3⋊Dic3C2×C3×D414yes—
266C3×Dic3—24no—
267C3×Dic3—24no—
268C3×Dic3—24no—
269C3×Dic3—24no—
270C3×Dic3—38no—
271C3×Dic3—44no—
272C3×Dic3—44no—
273C3×Dic3—64no—
274C3×Dic3—64no—
275C3×Dic3—64no—
276C22×Q8—91no—
277C23×C4—91no—
278C22⋊Q8—98no—
279C2×C4⋊C4—91no—
280C2×C4⋊C4—92no—
281C2×C22⋊C4—32no—
282C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—24no—
288C23×C3—24no—
289C23×C3—24no—
290C23×C3—28no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—21no—
294C23×C3—24no—
295C23×C3—44no—
296C23×C3—44no—
297C23×C3—44no—
298C23×C3—44no—
299C23×C3—44no—
300C23×C3—42no—
301C23×C3—41no—
302C23×C3#62714yescomplemented
303C23×C3C3×Dic618yescomplemented
304C23×C3C2×D3212yes—
305C23×C3C2×D3211yescharacteristic
306C23×C3C22×C3×D314yes—
307C23×C3C22×C3×D312yes—
308C3×Q8—188no—
309C2×C3×C4—34no—
310C2×C3×C4—34no—
311C2×C3×C4—34no—
312C2×C3×C4—34no—
313C2×C3×C4—64no—
314C2×C3×C4—64no—
315C2×C3×C4—64no—
316C2×C3×C4—64no—
317C2×C3×C4—64no—
318C2×C3×C4—64no—
319C2×C3×C4—92no—
320C2×C3×C4—94no—
321C2×C3×C4—92no—
322C2×C3×C4—124no—
323C2×C3×C4—124no—
324C2×C3×C4—188no—
325C2×C3×C4—182no—
326C2×Dic3—24no—
327C2×Dic3—24no—
328C2×Dic3—34no—
329C2×Dic3—38no—
330C2×Dic3—34no—
331C2×Dic3—64no—
332C2×Dic3—62no—
333C2×Dic3—62no—
334C2×Dic3—616no—
335C2×Dic3—616no—
336C2×Dic3—64no—
337C2×Dic3—64no—
338C2×Dic3C22×C3×D314yes—
339C2×Dic3C22×C3×D314yes—
340Dic6—616no—
341C2×C32—24no—
342C2×C32—28no—
343C2×C32—28no—
344C2×C32—28no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—22no—
351C2×C32—24no—
352C2×C32—44no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32—41no—
356C2×C32—48no—
357C2×C32—48no—
358C2×C32#87818yescomplemented
359C2×C32C3×C22⋊Q814yescomplemented
360C2×C32C22×Dic612yes—
361C2×C32C2×D3×D412yes—
362C2×C32C2×#23712yes—
363C2×C32C22×C3×D411yescharacteristic
364C2×C32C22×C3×Q811yescharacteristic
365C2×C32C2×C3×#4111yescharacteristic
366C24C3×D3211yescharacteristic
367C2×Q8—94no—
368C2×Q8—184no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—94no—
374C22×C4—184no—
375C4⋊C4—94no—
376C4⋊C4—98no—
377C22⋊C4—38no—
378C22×C3—24no—
379C22×C3—24no—
380C22×C3—216no—
381C22×C3—216no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—24no—
385C22×C3—24no—
386C22×C3—28no—
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—24no—
390C22×C3—22no—
391C22×C3—24no—
392C22×C3—22no—
393C22×C3—24no—
394C22×C3—28no—
395C22×C3—22no—
396C22×C3—24no—
397C22×C3—44no—
398C22×C3—44no—
399C22×C3—416no—
400C22×C3—44no—
401C22×C3—44no—
402C22×C3—416no—
403C22×C3—41no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—44no—
407C22×C3—42no—
408C22×C3#380512yes—
409C22×C3C2×#62712yes—
410C22×C3C2×#62714yescomplemented
411C22×C3#381112yes—
412C22×C3C2×C3×Dic614yes—
413C22×C3C2×C3×Dic618yescomplemented
414C22×C3C3×D3×D414yes—
415C22×C3C3×#23714yes—
416C22×C3C22×D3211yescharacteristic
417C22×C3C23×C3×D312yes—
418C3×C4—64no—
419C3×C4—64no—
420C3×C4—64no—
421C3×C4—64no—
422C3×C4—94no—
423C3×C4—124no—
424C3×C4—124no—
425C3×C4—182no—
426Dic3—24no—
427Dic3—24no—
428Dic3—38no—
429Dic3—64no—
430Dic3—64no—
431Dic3—64no—
432C32—21no—
433C32—22no—
434C32—41no—
435C32C2×#87812yescomplemented
436C32C2×C3×C22⋊Q811yescharacteristic, complemented
437C23—22no—
438C23—22no—
439C23C3×#62714yescomplemented
440C23C2×C3×D3212yes—
441C23C2×C3×D3211yescharacteristic
442Q8—188no—
443C2×C4—34no—
444C2×C4—34no—
445C2×C4—64no—
446C2×C4—64no—
447C2×C4—92no—
448C2×C4—94no—
449C2×C4—92no—
450C2×C4—188no—
451C2×C4—182no—
452C2×C3—28no—
453C2×C3—24no—
454C2×C3—28no—
455C2×C3—22no—
456C2×C3—21no—
457C2×C3—22no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—22no—
461C2×C3—24no—
462C2×C3—28no—
463C2×C3—48no—
464C2×C3—48no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—44no—
469C2×C3#6398214yesdirect product factor
470C2×C3C3×#87818yescomplemented
471C2×C3C2×#380511yescharacteristic
472C2×C3C22×#62711yescharacteristic
473C2×C3C2×#381111yescharacteristic
474C2×C3C22×C3×Dic612yes—
475C2×C3C2×C3×D3×D412yes—
476C2×C3C2×C3×#23712yes—
477C22—22no—
478C22—22no—
479C22—28no—
480C22C3×#380512yes—
481C22C2×C3×#62712yes—
482C22C2×C3×#62714yescomplemented
483C22C3×#381112yes—
484C22C22×C3×D3211yescharacteristic
485C4—64no—
486C4—64no—
487C4—94no—
488C4—182no—
489C3—21no—
490C3—22no—
491C3—41no—
492C3C2×#6398211yescharacteristic, direct product factor
493C3C2×C3×#87812yescomplemented
494C2—24no—
495C2C3×#6398214yesdirect product factor
496C2C2×C3×#380511yescharacteristic
497C2C22×C3×#62711yescharacteristic
498C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
Fitting, prime radicalC24×C33C22
socleC23×C33C23
hypercenterC24×C3D32
hypocenterC32C2×C3×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C22×C3×#627C2semidirect
4. C3×#63982C2direct (N × H)
5. C2×C3×#3724C2semidirect
6. C2×C3×Dic3⋊Dic3C2semidirect
8. C2×#63982C3direct (N × H)
17. C2×C3×#627C22semidirect
19. C3×#3724C22semidirect
20. C3×Dic3⋊Dic3C22semidirect
25. C22×#627C2×C3semidirect
34. #63982C2×C3direct (N × H)
35. C2×#3724C2×C3semidirect
36. C2×Dic3⋊Dic3C2×C3semidirect
39. C23×C33Q8semidirect
82. C2×#627C22×C3semidirect
90. #3724C22×C3semidirect
91. Dic3⋊Dic3C22×C3semidirect
97. C22×C33C2×Q8semidirect
139. C23×C32Dic6semidirect
140. C23×C32C3×Q8semidirect
180. C2×C33C22⋊Q8semidirect
253. C22×C32C2×Dic6semidirect
258. C22×C32C2×C3×Q8semidirect
282. C33C2×C22⋊Q8semidirect
302. C23×C3#627semidirect
303. C23×C3C3×Dic6semidirect
358. C2×C32#878semidirect
359. C2×C32C3×C22⋊Q8semidirect
410. C22×C3C2×#627semidirect
413. C22×C3C2×C3×Dic6semidirect
435. C32C2×#878semidirect
436. C32C2×C3×C22⋊Q8semidirect
439. C23C3×#627semidirect
469. C2×C3#63982direct (N × H)
470. C2×C3C3×#878semidirect
482. C22C2×C3×#627semidirect
492. C3C2×#63982direct (N × H)
493. C3C2×C3×#878semidirect
495. C2C3×#63982direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C22×C3×#6272
C2×C32×#2172
C3×#639822
C2×C3×#37242
C2×C3×Dic3⋊Dic32
C2×C3×#8783
C2×#639823