Finite Group Databasebeta

← C2×#11722083 #13162049

Subgroups — C2×#11722083 #13162049

Order 1728.

1474 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic
2C2×C3×#3757C212yesmaximal, complemented
3#11722083C214yesmaximal, direct product factor
4C2×#63581—31nomaximal
5C2×#63580—32nomaximal
6C24×C33C2211yescharacteristic
7C23×C3×C3⋊D3C411yescharacteristic, complemented
8C2×C3×#638—22no—
9C2×C3×#638—22no—
10C2×C3×#638C412yescomplemented
11C2×C3×#638C2212yes—
12C22×C3×C3⋊Dic3—22no—
13C22×C3×C3⋊Dic3C411yescharacteristic, complemented
14C3×#3757—22no—
15C3×#3757C2214yescomplemented
16C22×#638D311yescharacteristic
17C22×C3×#67—32no—
18C22×C3×#67—61no—
19C2×#3757D312yescomplemented
20C2×C3×#217—32no—
21C2×C3×#217—32no—
22C2×C3×#217—62no—
23#63581—34no—
24#63580—38no—
25C23×C33—42no—
26C23×C33C2×C411yescharacteristic
27C23×C33C2×C411yescharacteristic
28C23×C33D412yes—
29C23×C33D412yes—
30C23×C33C2311yescharacteristic
31C22×C3×C3⋊D3—21no—
32C22×C3×C3⋊D3—21no—
33C22×C3×C3⋊D3—42no—
34C22×C3×C3⋊D3C2×C412yescomplemented
35C3×#638—24no—
36C3×#638—24no—
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3—21no—
39C2×C3×C3⋊Dic3—24no—
40C2×C3×C3⋊Dic3—44no—
41C2×C3×C3⋊Dic3C2×C412yescomplemented
42C2×#834—91no—
43C2×#806—92no—
44C24×C32—22no—
45C24×C32—21no—
46C24×C32—41no—
47C24×C32C2×D312yes—
48C24×C32C2×D311yescharacteristic
49C23×C3⋊D3Dic311yescharacteristic, complemented
50C23×C3×D3—32no—
51C23×C3×D3—61no—
52C2×#638—22no—
53C2×#638—22no—
54C2×#638Dic312yescomplemented
55C2×#638C2×D312yes—
56C22×C3⋊Dic3—22no—
57C22×C3⋊Dic3Dic311yescharacteristic, complemented
58C2×C3×#67—34no—
59C2×C3×#67—34no—
60C2×C3×#67—62no—
61C2×C3×#67—62no—
62C2×C3×#67—64no—
63C2×C3×#67—64no—
64C2×C3×#67—62no—
65C2×C3×#67—62no—
66C2×C3×#67—62no—
67C2×C3×#67—62no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—61no—
70C22×C3×Dic3—64no—
71C22×C3×Dic3—62no—
72C22×C3×Dic3—62no—
73#3757—22no—
74#3757C2×D314yescomplemented
75C3×#217—34no—
76C3×#217—34no—
77C3×#217—64no—
78C3×#217—64no—
79C3×#217—62no—
80C3×#217—62no—
81C22×C33—24no—
82C22×C33—22no—
83C22×C33—41no—
84C22×C33—42no—
85C22×C33—41no—
86C22×C33C22⋊C411yescharacteristic
87C22×C33C22⋊C412yes—
88C22×C33C22⋊C411yescharacteristic
89C22×C33C22×C411yescharacteristic
90C22×C33C2×D412yes—
91C2×C3×C3⋊D3—22no—
92C2×C3×C3⋊D3—42no—
93C2×C3×C3⋊D3—42no—
94C2×C3×C3⋊D3—42no—
95C3×C3⋊Dic3—22no—
96C3×C3⋊Dic3—44no—
97C22×C3×D4—91no—
98C22×#67—32no—
99C22×#67—61no—
100C2×C3×C22⋊C4—92no—
101C2×#217—32no—
102C2×#217—32no—
103C2×#217—62no—
104#834—94no—
105#806—98no—
106C23×C32—22no—
107C23×C32—22no—
108C23×C32—21no—
109C23×C32—21no—
110C23×C32—22no—
111C23×C32—22no—
112C23×C32—22no—
113C23×C32—22no—
114C23×C32—22no—
115C23×C32—22no—
116C23×C32—22no—
117C23×C32—21no—
118C23×C32—41no—
119C23×C32—41no—
120C23×C32—42no—
121C23×C32—42no—
122C23×C32—41no—
123C23×C32—44no—
124C23×C32—42no—
125C23×C32—48no—
126C23×C32—44no—
127C23×C32—48no—
128C23×C32C4×D312yes—
129C23×C32C4×D312yes—
130C23×C32D1212yes—
131C23×C32D1212yes—
132C23×C32C2×Dic311yescharacteristic
133C23×C32C2×Dic311yescharacteristic
134C23×C32#6712yes—
135C23×C32#6712yes—
136C23×C32#6712yes—
137C23×C32#6712yes—
138C23×C32C22×D312yes—
139C23×C32C22×D311yescharacteristic
140C22×C3⋊D3—21no—
141C22×C3⋊D3—21no—
142C22×C3⋊D3—42no—
143C22×C3⋊D3C2×Dic312yescomplemented
144C22×C3×D3—34no—
145C22×C3×D3—62no—
146C22×C3×D3—62no—
147C22×C3×D3—62no—
148C22×C3×D3—62no—
149C22×C3×D3—62no—
150C22×C3×D3—124no—
151C22×C3×D3—124no—
152#638—24no—
153#638—24no—
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—24no—
157C2×C3⋊Dic3—44no—
158C2×C3⋊Dic3C2×Dic312yescomplemented
159C3×#67—68no—
160C3×#67—68no—
161C3×#67—64no—
162C3×#67—64no—
163C3×#67—64no—
164C3×#67—64no—
165C2×C3×Dic3—34no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—68no—
172C2×C3×Dic3—64no—
173C2×C3×Dic3—64no—
174C2×C3×Dic3—128no—
175C2×C3×Dic3—124no—
176C2×C3×Dic3—124no—
177C2×#99—271noSylow
178C2×C33—22no—
179C2×C33—21no—
180C2×C33—21no—
181C2×C33—22no—
182C2×C33#9912yescomplemented
183C2×C33C2×C22⋊C411yescharacteristic
184C3×C3⋊D3—42no—
185C24×C3—22no—
186C24×C3—21no—
187C24×C3—41no—
188C24×C3—91no—
189C24×C3D3212yes—
190C24×C3D3211yescharacteristic
191C23×D3—32no—
192C23×D3—61no—
193C2×C3×D4—92no—
194C2×C3×D4—92no—
195C2×C3×D4—182no—
196C2×C3×D4—182no—
197C22×C3×C4—91no—
198C22×C3×C4—182no—
199C2×#67—34no—
200C2×#67—34no—
201C2×#67—62no—
202C2×#67—62no—
203C2×#67—64no—
204C2×#67—64no—
205C2×#67—62no—
206C2×#67—62no—
207C2×#67—62no—
208C2×#67—62no—
209C22×Dic3—32no—
210C22×Dic3—61no—
211C22×Dic3—64no—
212C22×Dic3—62no—
213C22×Dic3—62no—
214C3×C22⋊C4—94no—
215C3×C22⋊C4—182no—
216#217—34no—
217#217—34no—
218#217—64no—
219#217—64no—
220#217—62no—
221#217—62no—
222C22×C32—24no—
223C22×C32—22no—
224C22×C32—28no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—24no—
228C22×C32—24no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—21no—
234C22×C32—21no—
235C22×C32—22no—
236C22×C32—22no—
237C22×C32—21no—
238C22×C32—22no—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—41no—
242C22×C32—42no—
243C22×C32—41no—
244C22×C32—41no—
245C22×C32—42no—
246C22×C32—44no—
247C22×C32—44no—
248C22×C32—44no—
249C22×C32—48no—
250C22×C32—44no—
251C22×C32—42no—
252C22×C32—42no—
253C22×C32—41no—
254C22×C32—42no—
255C22×C32—42no—
256C22×C32—42no—
257C22×C32—41no—
258C22×C32—44no—
259C22×C32—42no—
260C22×C32—44no—
261C22×C32—44no—
262C22×C32—44no—
263C22×C32—44no—
264C22×C32—42no—
265C22×C32—44no—
266C22×C32—48no—
267C22×C32—44no—
268C22×C32(C2×D3)⋊C412yes—
269C22×C32(C2×D3)⋊C412yes—
270C22×C32(C2×D3)⋊C412yes—
271C22×C32(C2×D3)⋊C412yes—
272C22×C32#21711yescharacteristic
273C22×C32#21712yes—
274C22×C32#21711yescharacteristic
275C22×C32C2×C4×D312yes—
276C22×C32C2×D1212yes—
277C22×C32C22×Dic311yescharacteristic
278C22×C32C2×#6712yes—
279C22×C32C2×#6712yes—
280C2×C3⋊D3—22no—
281C2×C3⋊D3—42no—
282C2×C3⋊D3—42no—
283C2×C3⋊D3—42no—
284C2×C3×D3—64no—
285C2×C3×D3—64no—
286C2×C3×D3—124no—
287C2×C3×D3—124no—
288C2×C3×D3—124no—
289C2×C3×D3—124no—
290C2×C3×D3—124no—
291C2×C3×D3—124no—
292C3⋊Dic3—22no—
293C3⋊Dic3—44no—
294C3×Dic3—64no—
295C3×Dic3—64no—
296C3×Dic3—128no—
297C3×Dic3—124no—
298C3×Dic3—124no—
299C22×D4—91no—
300C2×C22⋊C4—92no—
301#99—274no—
302C33C2×#9911yescharacteristic, complemented, Sylow
303C23×C3—21no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—21no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—44no—
316C23×C3—42no—
317C23×C3—48no—
318C23×C3—44no—
319C23×C3—48no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—41no—
325C23×C3—92no—
326C23×C3—181no—
327C23×C3—181no—
328C23×C3—362no—
329C23×C3D3×Dic312yes—
330C23×C3D3×Dic312yes—
331C23×C3#62411yescharacteristic
332C23×C3#62411yescharacteristic
333C23×C3#62512yes—
334C23×C3#62512yes—
335C23×C3#62612yes—
336C23×C3#62612yes—
337C23×C3#62612yes—
338C23×C3#62612yes—
339C23×C3C2×D3212yes—
340C23×C3C2×D3211yescharacteristic
341C22×D3—34no—
342C22×D3—62no—
343C22×D3—62no—
344C22×D3—62no—
345C22×D3—62no—
346C22×D3—62no—
347C22×D3—124no—
348C22×D3—124no—
349C3×D4—184no—
350C3×D4—184no—
351C2×C3×C4—92no—
352C2×C3×C4—181no—
353C2×C3×C4—181no—
354C2×C3×C4—184no—
355C2×C3×C4—364no—
356#67—68no—
357#67—68no—
358#67—64no—
359#67—64no—
360#67—64no—
361#67—64no—
362C2×Dic3—34no—
363C2×Dic3—62no—
364C2×Dic3—62no—
365C2×Dic3—62no—
366C2×Dic3—62no—
367C2×Dic3—62no—
368C2×Dic3—68no—
369C2×Dic3—64no—
370C2×Dic3—64no—
371C2×Dic3—128no—
372C2×Dic3—124no—
373C2×Dic3—124no—
374C2×C32—22no—
375C2×C32—22no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—24no—
379C2×C32—22no—
380C2×C32—22no—
381C2×C32—24no—
382C2×C32—22no—
383C2×C32—22no—
384C2×C32—22no—
385C2×C32—22no—
386C2×C32—22no—
387C2×C32—22no—
388C2×C32—24no—
389C2×C32—22no—
390C2×C32—22no—
391C2×C32—21no—
392C2×C32—42no—
393C2×C32—41no—
394C2×C32—42no—
395C2×C32—42no—
396C2×C32—42no—
397C2×C32—42no—
398C2×C32—42no—
399C2×C32—42no—
400C2×C32—44no—
401C2×C32—42no—
402C2×C32—42no—
403C2×C32—44no—
404C2×C32#80614yescomplemented
405C2×C32#83412yescomplemented
406C2×C32C2×(C2×D3)⋊C412yes—
407C2×C32C2×#21711yescharacteristic
408C3⋊D3—42no—
409C3×D3—124no—
410C3×D3—124no—
411C24—91no—
412C24#111311yescharacteristic
413C2×D4—92no—
414C2×D4—92no—
415C2×D4—182no—
416C2×D4—182no—
417C22×C4—91no—
418C22×C4—182no—
419C22⋊C4—94no—
420C22⋊C4—182no—
421C22×C3—28no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3—24no—
427C22×C3—22no—
428C22×C3—21no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—21no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—21no—
439C22×C3—22no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3—44no—
443C22×C3—48no—
444C22×C3—44no—
445C22×C3—42no—
446C22×C3—41no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—41no—
452C22×C3—42no—
453C22×C3—44no—
454C22×C3—44no—
455C22×C3—44no—
456C22×C3—44no—
457C22×C3—44no—
458C22×C3—42no—
459C22×C3—44no—
460C22×C3—48no—
461C22×C3—44no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—41no—
465C22×C3—42no—
466C22×C3—41no—
467C22×C3—182no—
468C22×C3—362no—
469C22×C3—362no—
470C22×C3—362no—
471C22×C3(C2×D3)⋊Dic312yes—
472C22×C3(C2×D3)⋊Dic312yes—
473C22×C3(C2×D3)⋊Dic312yes—
474C22×C3(C2×D3)⋊Dic312yes—
475C22×C3#372211yescharacteristic
476C22×C3#372212yes—
477C22×C3#372211yescharacteristic
478C22×C3C2×D3×Dic312yes—
479C22×C3C2×#62411yescharacteristic
480C22×C3C2×#62512yes—
481C22×C3C2×#62612yes—
482C22×C3C2×#62612yes—
483C2×D3—64no—
484C2×D3—64no—
485C2×D3—124no—
486C2×D3—124no—
487C2×D3—124no—
488C2×D3—124no—
489C2×D3—124no—
490C2×D3—124no—
491C3×C4—182no—
492C3×C4—364no—
493Dic3—64no—
494Dic3—64no—
495Dic3—128no—
496Dic3—124no—
497Dic3—124no—
498C32—22no—
499C32—21no—
500C32—41no—
501C32C2×#80612yescomplemented
502C32C2×#83411yescharacteristic, complemented
503C23—42no—
504C23—92no—
505C23—181no—
506C23—181no—
507C23—362no—
508C23#629411yescharacteristic
509C23#629411yescharacteristic
510C23#629512yes—
511C23#629512yes—
512C23C2×#111311yescharacteristic
513D4—184no—
514D4—184no—
515C2×C4—92no—
516C2×C4—181no—
517C2×C4—181no—
518C2×C4—184no—
519C2×C4—364no—
520C2×C3—24no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—24no—
524C2×C3—22no—
525C2×C3—22no—
526C2×C3—22no—
527C2×C3—22no—
528C2×C3—22no—
529C2×C3—22no—
530C2×C3—22no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—24no—
536C2×C3—21no—
537C2×C3—22no—
538C2×C3—42no—
539C2×C3—42no—
540C2×C3—42no—
541C2×C3—42no—
542C2×C3—42no—
543C2×C3—42no—
544C2×C3—44no—
545C2×C3—42no—
546C2×C3—42no—
547C2×C3—44no—
548C2×C3—42no—
549C2×C3—41no—
550C2×C3—362no—
551C2×C3#6358014yescomplemented
552C2×C3#6358112yescomplemented
553C2×C3C2×(C2×D3)⋊Dic312yes—
554C2×C3C2×#372211yescharacteristic
555D3—124no—
556D3—124no—
557C22—24no—
558C22—22no—
559C22—41no—
560C22—42no—
561C22—41no—
562C22—182no—
563C22—362no—
564C22—362no—
565C22—362no—
566C22#8881411yescharacteristic
567C22#8881412yes—
568C22#8881411yescharacteristic
569C22C2×#629411yescharacteristic
570C22C2×#629512yes—
571C4—182no—
572C4—364no—
573C3—21no—
574C3—22no—
575C3—41no—
576C3C2×#6358012yescomplemented
577C3C2×#6358111yescharacteristic, complemented
578C2—22no—
579C2—21no—
580C2—21no—
581C2—22no—
582C2—362no—
583C2#1172208312yesdirect product factor
584C2C2×#8881411yescharacteristic
subgroupG/H
centerC22#88814
commutatorC22×C33C22×C4
FrattiniC23C2×#1113
FittingC24×C33C22
socleC22×C33C22⋊C4
prime radicalC22×C3×#638C2
cosocleC23×C33C23
hypercenterC24#1113
hypocenterC33C2×#99
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3757C2semidirect
3. #11722083C2direct (N × H)
7. C23×C3×C3⋊D3C4semidirect
10. C2×C3×#638C4semidirect
13. C22×C3×C3⋊Dic3C4semidirect
15. C3×#3757C22semidirect
19. C2×#3757D3semidirect
34. C22×C3×C3⋊D3C2×C4semidirect
41. C2×C3×C3⋊Dic3C2×C4semidirect
49. C23×C3⋊D3Dic3semidirect
54. C2×#638Dic3semidirect
57. C22×C3⋊Dic3Dic3semidirect
74. #3757C2×D3semidirect
143. C22×C3⋊D3C2×Dic3semidirect
158. C2×C3⋊Dic3C2×Dic3semidirect
182. C2×C33#99semidirect
302. C33C2×#99semidirect
404. C2×C32#806semidirect
405. C2×C32#834semidirect
501. C32C2×#806semidirect
502. C32C2×#834semidirect
551. C2×C3#63580semidirect
552. C2×C3#63581semidirect
576. C3C2×#63580semidirect
577. C3C2×#63581semidirect
583. C2#11722083direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C2×C3×#37572
#117220832
C2×#635813
C2×#635803