Finite Group Databasebeta

← Dic6×#611 #13150186

Subgroups — Dic6×#611 #13150186

Order 1728.

914 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C9×Dic6C211yesmaximal, characteristic, complemented
2C2×D9×Dic6C211yesmaximal, characteristic, complemented
3#11721813C211yesmaximal, characteristic, complemented
4C3×C4×#611C211yesmaximal, characteristic
5Dic3×#611C212yesmaximal
6#11721249C212yesmaximal, complemented
7#11720984C212yesmaximal, complemented
8#11720944C211yesmaximal, characteristic, complemented
9#11720924C211yesmaximal, characteristic, complemented
10Dic6×Dic9C211yesmaximal, characteristic, complemented
11#11720903C212yesmaximal, complemented
12Dic6×#67—31nomaximal
13Q8×#611—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×#611C2211yescharacteristic
17C22×C9×Dic3C2212yes—
18C2×C3×C4×D9C2211yescharacteristic
19C2×C9×Dic6—24no—
20C2×C9×Dic6C2212yes—
21C2×C9×Dic6C2211yescharacteristic
22C2×C9×Dic6C2211yescharacteristic
23C2×Dic3×D9C2212yes—
24C2×#6189C2212yes—
25D9×Dic6—24no—
26#88551C2212yes—
27C4⋊(C3⋊Dic9)C2211yescharacteristic
28#88534C2212yes—
29C3×#3676C2211yescharacteristic
30C3×(C2×D9)⋊C4C2211yescharacteristic
31C3×Dic9⋊C4C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic9⋊Dic3C2212yes—
37Dic9⋊Dic3C2212yes—
38Dic3×Dic9C2212yes—
39C22×C3×Dic6D311yescharacteristic
40C2×D3×Dic6—31no—
41#64133—31no—
42C3×C4×#67—31no—
43Dic3×#67—32no—
44#63964—32no—
45#63887—32no—
46#63866—31no—
47#63856—31no—
48Dic3×Dic6—31no—
49#63839—32no—
50C22×Q8×C9—31no—
51C2×Q8×D9—31no—
52#63508—31no—
53#63508—32no—
54Q8×Dic9—31no—
55#63506—32no—
56#63506—31no—
57C4×#611—32no—
58C4×#611D311yescharacteristic
59#63486—31no—
60#63486—32no—
61#63486—61no—
62C23×C3×C9C2311yescharacteristic
63C22×C3×D9C2311yescharacteristic
64C2×C3×C4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2312yes—
69#6226—22no—
70C3×#611Q814yescomplemented
71C2×C9×Dic3—24no—
72C2×C9×Dic3C2312yes—
73C2×C9×Dic3C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77C3×C4×D9—22no—
78C9×Dic6—22no—
79C9×Dic6—24no—
80C9×Dic6D414yescomplemented
81Dic3×D9—24no—
82#6189—24no—
83Q8×#67—91no—
84D4×Dic6—91no—
85C22×C32×C4C2×D311yescharacteristic
86C2×#634—31no—
87C2×C3×#67—31no—
88C22×C3×Dic3C2×D312yes—
89C2×C3×C4×D3—31no—
90C2×C3×Dic6—24no—
91C2×C3×Dic6C2×D312yes—
92C2×C3×Dic6C2×D311yescharacteristic
93C2×C3×Dic6C2×D311yescharacteristic
94C2×#627—32no—
95C2×D3×Dic3—32no—
96D3×Dic6—64no—
97#3757—32no—
98C4⋊(C3⋊Dic3)—31no—
99#3750—32no—
100C3×#217—31no—
101C3×(C2×D3)⋊C4—31no—
102C3×Dic3⋊C4—31no—
103C3×C4×Dic3—31no—
104#3724—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108(C2×D3)⋊Dic3—32no—
109Dic32—32no—
110C2×Q8×C9—31no—
111C2×Q8×C9—32no—
112C2×Q8×C9—31no—
113C2×Q8×C9—64no—
114C22×C4×C9—21no—
115C22×C4×C9—32no—
116C22×C4×C9C2×D311yescharacteristic
117C2×#611C2×D311yescharacteristic
118Q8×D9—64no—
119C2×C4×D9—32no—
120C2×C4×D9C2×D311yescharacteristic
121C2×Dic18—31no—
122C2×Dic18—32no—
123C2×Dic18—61no—
124#3676—32no—
125#3676—62no—
126#3676C2×D311yescharacteristic
127(C2×D9)⋊C4—32no—
128(C2×D9)⋊C4C2×D311yescharacteristic
129C4⋊Dic9—31no—
130C4⋊Dic9—32no—
131C4⋊Dic9—61no—
132Dic9⋊C4—32no—
133Dic9⋊C4—32no—
134Dic9⋊C4—32no—
135Dic9⋊C4—32no—
136Dic9⋊C4—62no—
137Dic9⋊C4C2×D311yescharacteristic
138C4×Dic9—32no—
139C4×Dic9C2×D311yescharacteristic
140C22×C3×C9—21no—
141C22×C3×C9—21no—
142C22×C3×C9C2×Q812yescomplemented
143C22×C3×C9C2411yescharacteristic
144C2×C3×D9—21no—
145C2×C3×D9—21no—
146C2×C3×D9C2×Q812yescomplemented
147C3×C4×C9—21no—
148C3×C4×C9C2×D412yes—
149C3⋊Dic9—22no—
150C3⋊Dic9—22no—
151C9×Dic3—22no—
152C9×Dic3C2×D414yes—
153C3×Dic9—21no—
154C3×Dic9—21no—
155C3×Dic9C2×Q812yescomplemented
156C22×C3×Q8—31no—
157C2×D3×Q8—91no—
158C22×Dic6—91no—
159C22×Dic6D911yescharacteristic, complemented
160C3×C4×D4—91no—
161#946—92no—
162#946—91no—
163Q8×Dic3—91no—
164#944—91no—
165#944—92no—
166D4×Dic3—92no—
167C4×#67—31no—
168C4×#67—92no—
169#924—92no—
170#924—91no—
171#924—91no—
172#924—91no—
173#924—181no—
174#888—92no—
175#878—92no—
176#878—92no—
177#869—91no—
178C4×Dic6—91no—
179C23×C32C22×D311yescharacteristic
180C22×C3×D3—31no—
181C2×C32×C4—22no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—32no—
185C2×C3⋊Dic3—32no—
186#634—62no—
187C3×#67—34no—
188C2×C3×Dic3—24no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3C22×D312yes—
193C2×C3×Dic3C22×D312yes—
194C3×C4×D3—62no—
195C3×Dic6—22no—
196C3×Dic6—24no—
197C3×Dic6#6714yes—
198#627—64no—
199D3×Dic3—64no—
200C23×C9—21no—
201C23×C9C22×D311yescharacteristic
202C22×D9C22×D311yescharacteristic
203Q8×C9—34no—
204Q8×C9—64no—
205Q8×C9—62no—
206C2×C4×C9—22no—
207C2×C4×C9—21no—
208C2×C4×C9—21no—
209C2×C4×C9—32no—
210C2×C4×C9—32no—
211C2×C4×C9—42no—
212C2×C4×C9—64no—
213C2×C4×C9C22×D311yescharacteristic
214C2×C4×C9C22×D311yescharacteristic
215#611Dic614yesdirect product factor
216C2×Dic9—32no—
217C2×Dic9—32no—
218C2×Dic9—62no—
219C2×Dic9—62no—
220C2×Dic9C22×D311yescharacteristic
221C2×Dic9C22×D311yescharacteristic
222C2×Dic9C22×D311yescharacteristic
223C4×D9—22no—
224C4×D9—64no—
225Dic18—64no—
226Dic18—62no—
227Dic18—122no—
228D4×Q8—271noSylow
229C2×C3×C9—22no—
230C2×C3×C9C22×D411yescharacteristic
231C2×C3×C9C22×Q811yescharacteristic
232C2×C3×C92−1+411yescharacteristic
233C3×D9—22no—
234C2×C3×Q8—31no—
235C2×C3×Q8—32no—
236C2×C3×Q8—31no—
237C2×C3×Q8—64no—
238C2×C3×D4—91no—
239C22×C3×C4—21no—
240C22×C3×C4—32no—
241C22×C3×C4—91no—
242C22×C3×C4C2×D911yescharacteristic
243C22×C3×C4D3211yescharacteristic
244C2×#67—31no—
245C22×Dic3—92no—
246C22×Dic3C2×D912yes—
247D3×Q8—184no—
248C2×C4×D3—31no—
249C2×C4×D3—92no—
250C2×Dic6—24no—
251C2×Dic6—92no—
252C2×Dic6—91no—
253C2×Dic6—91no—
254C2×Dic6—92no—
255C2×Dic6—181no—
256C2×Dic6—184no—
257C2×Dic6C2×D911yescharacteristic
258C2×Dic6C2×D912yes—
259C2×Dic6C2×D911yescharacteristic
260C3×C4⋊C4—91no—
261C3×C22⋊C4—91no—
262C3×C22⋊C4—91no—
263C3×C42—91no—
264#217—31no—
265#217—92no—
266#217—92no—
267#217—92no—
268#217—182no—
269(C2×D3)⋊C4—31no—
270(C2×D3)⋊C4—92no—
271C4⋊Dic3—92no—
272C4⋊Dic3—91no—
273C4⋊Dic3—91no—
274C4⋊Dic3—92no—
275C4⋊Dic3—92no—
276C4⋊Dic3—181no—
277Dic3⋊C4—31no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—182no—
286C4×Dic3—31no—
287C4×Dic3—92no—
288C4×Dic3—92no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32D3×Q812yes—
292C22×C32C23×D311yescharacteristic
293C2×C3×D3—32no—
294C2×C3×D3—61no—
295C2×C3×D3—61no—
296C32×C4—21no—
297C32×C4C2×#6712yes—
298C3⋊Dic3—62no—
299C3⋊Dic3—62no—
300C3×Dic3—22no—
301C3×Dic3—32no—
302C3×Dic3—61no—
303C3×Dic3—61no—
304C3×Dic3C2×#6714yes—
305C22×C9—21no—
306C22×C9—21no—
307C22×C9—21no—
308C22×C9—22no—
309C22×C9—41no—
310C22×C9—41no—
311C22×C9C2×Dic612yescomplemented
312C22×C9C23×D311yescharacteristic
313C2×D9—21no—
314C2×D9—21no—
315C2×D9C2×Dic612yescomplemented
316C4×C9—21no—
317C4×C9—22no—
318C4×C9—34no—
319C4×C9—41no—
320C4×C9—62no—
321C4×C9D3×D412yes—
322Dic9—21no—
323Dic9—21no—
324Dic9—62no—
325Dic9—62no—
326Dic9—122no—
327Dic9—122no—
328Dic9C2×Dic612yescomplemented
329C22×Q8—31no—
330C22×Q8—271no—
331C4⋊Q8—271no—
332C4⋊Q8—272no—
333C22⋊Q8—272no—
334C22⋊Q8—272no—
335C22⋊Q8—271no—
336C22⋊Q8—271no—
337C4×Q8—271no—
338C4×D4—91no—
339C4×D4—272no—
340C3×C9D4×Q811yescharacteristic, complemented, Sylow
341C23×C3—21no—
342C23×C3—91no—
343C23×C3C22×D911yescharacteristic
344C23×C3C2×D3211yescharacteristic
345C22×D3—31no—
346C3×Q8—34no—
347C3×Q8—64no—
348C3×Q8—62no—
349C3×D4—94no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—32no—
355C2×C3×C4—32no—
356C2×C3×C4—42no—
357C2×C3×C4—64no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—182no—
362C2×C3×C4C22×D911yescharacteristic
363C2×C3×C4C22×D911yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365C2×C3×C4C2×D3211yescharacteristic
366#67—34no—
367C2×Dic3—24no—
368C2×Dic3—31no—
369C2×Dic3—31no—
370C2×Dic3—31no—
371C2×Dic3—92no—
372C2×Dic3—92no—
373C2×Dic3—92no—
374C2×Dic3—92no—
375C2×Dic3—182no—
376C2×Dic3—182no—
377C2×Dic3—184no—
378C2×Dic3C22×D912yes—
379C2×Dic3C22×D912yes—
380C4×D3—62no—
381C4×D3—184no—
382Dic6—22no—
383Dic6—24no—
384Dic6—184no—
385Dic6—182no—
386Dic6—182no—
387Dic6—184no—
388Dic6—362no—
389Dic6#61114yesdirect product factor
390C2×C32—22no—
391C2×C32C2×D3×Q811yescharacteristic
392C2×C32#100711yescharacteristic
393C2×C32C22×#6711yescharacteristic
394C3×D3—62no—
395C2×C9—22no—
396C2×C9—21no—
397C2×C9—21no—
398C2×C9—21no—
399C2×C9—42no—
400C2×C9C22×Dic611yescharacteristic
401C2×C9C2×D3×D411yescharacteristic
402C2×C9#101011yescharacteristic
403D9—22no—
404C2×Q8—31no—
405C2×Q8—32no—
406C2×Q8—31no—
407C2×Q8—64no—
408C2×Q8—272no—
409C2×Q8—271no—
410C2×Q8—544no—
411C2×D4—91no—
412C22×C4—32no—
413C22×C4—91no—
414C22×C4—272no—
415C22×C4D3×D911yescharacteristic
416C4⋊C4—91no—
417C4⋊C4—272no—
418C4⋊C4—272no—
419C4⋊C4—272no—
420C4⋊C4—272no—
421C4⋊C4—271no—
422C4⋊C4—272no—
423C22⋊C4—91no—
424C22⋊C4—91no—
425C22⋊C4—272no—
426C22⋊C4—272no—
427C42—91no—
428C42—272no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—22no—
434C22×C3—21no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—92no—
438C22×C3—181no—
439C22×C3—181no—
440C22×C3Q8×D912yescomplemented
441C22×C3C23×D911yescharacteristic
442C22×C3D3×Dic612yes—
443C22×C3C22×D3211yescharacteristic
444C2×D3—32no—
445C2×D3—61no—
446C2×D3—61no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—22no—
450C3×C4—34no—
451C3×C4—41no—
452C3×C4—62no—
453C3×C4—92no—
454C3×C4—181no—
455C3×C4—181no—
456C3×C4C2×#61112yes—
457C3×C4D3×#6712yes—
458Dic3—22no—
459Dic3—32no—
460Dic3—61no—
461Dic3—61no—
462Dic3—182no—
463Dic3—182no—
464Dic3—182no—
465Dic3—182no—
466Dic3—362no—
467Dic3—362no—
468Dic3C2×#61114yes—
469C32Q8×#6711yescharacteristic
470C9—21no—
471C9D4×Dic611yescharacteristic, complemented
472C23—91no—
473C23C2×D3×D911yescharacteristic
474Q8—34no—
475Q8—64no—
476Q8—62no—
477Q8—544no—
478Q8—542no—
479D4—94no—
480C2×C4—22no—
481C2×C4—32no—
482C2×C4—32no—
483C2×C4—64no—
484C2×C4—91no—
485C2×C4—91no—
486C2×C4—91no—
487C2×C4—182no—
488C2×C4—272no—
489C2×C4—272no—
490C2×C4—544no—
491C2×C4C2×D3×D911yescharacteristic
492C2×C4C2×D3×D911yescharacteristic
493C2×C3—22no—
494C2×C3—22no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—42no—
499C2×C3—182no—
500C2×C3C2×Q8×D911yescharacteristic
501C2×C3#6371311yescharacteristic
502C2×C3C22×#61111yescharacteristic
503C2×C3C2×D3×Dic611yescharacteristic
504C2×C3#6429711yescharacteristic
505C2×C3C2×D3×#6711yescharacteristic
506D3—62no—
507C22—21no—
508C22—21no—
509C22—92no—
510C22—181no—
511C22—181no—
512C22D9×Dic612yescomplemented
513C22C22×D3×D911yescharacteristic
514C4—21no—
515C4—34no—
516C4—62no—
517C4—92no—
518C4—181no—
519C4—181no—
520C4—542no—
521C4—542no—
522C4D3×#61112yes—
523C3—21no—
524C3Q8×#61111yescharacteristic, complemented
525C3Dic6×#6711yescharacteristic
526C2—22no—
527C2—182no—
528C2C2×D9×Dic611yescharacteristic
529C2#1172245811yescharacteristic
530C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×#611C22
hypercenterC22×C4D3×D9
hypocenterC3×C9D4×Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C9×Dic6C2semidirect
2. C2×D9×Dic6C2semidirect
3. #11721813C2semidirect
6. #11721249C2semidirect
7. #11720984C2semidirect
8. #11720944C2semidirect
9. #11720924C2semidirect
10. Dic6×Dic9C2semidirect
11. #11720903C2semidirect
70. C3×#611Q8semidirect
80. C9×Dic6D4semidirect
142. C22×C3×C9C2×Q8semidirect
146. C2×C3×D9C2×Q8semidirect
155. C3×Dic9C2×Q8semidirect
159. C22×Dic6D9semidirect
215. #611Dic6direct (N × H)
311. C22×C9C2×Dic6semidirect
315. C2×D9C2×Dic6semidirect
328. Dic9C2×Dic6semidirect
340. C3×C9D4×Q8semidirect
389. Dic6#611direct (N × H)
440. C22×C3Q8×D9semidirect
471. C9D4×Dic6semidirect
512. C22D9×Dic6semidirect
524. C3Q8×#611semidirect

Maximal subgroups

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

subgroupindex
C22×C9×Dic62
C2×D9×Dic62
#117218132
C3×C4×#6112
Dic3×#6112
#117212492
#117209842
#117209442
#117209242
Dic6×Dic92
#117209032
Dic6×#673
Q8×#6113