Finite Group Databasebeta

← C2×#11722946 #13172283

Subgroups — C2×#11722946 #13172283

Order 1728.

3202 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D32C211yesmaximal, characteristic, complemented
2C22×#6290C211yesmaximal, characteristic, complemented
3C22×C3×#625C212yesmaximal, complemented
4C2×#88814C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊(C3⋊Dic3)C212yesmaximal, complemented
6#11722946C218yesmaximal, direct product factor
7C2×#63983—31nomaximal
8C2×#63976—32nomaximal
9C23×C3×C3⋊D3C2211yescharacteristic
10C23×C32×D3C2212yes—
11C23×C32×D3C2211yescharacteristic, complemented
12C22×C3×D32—24no—
13C22×C3×D32—28no—
14C22×C3×D32C2214yescomplemented
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
16C22×C3×C3⋊Dic3C2212yescomplemented
17C2×#6290—24no—
18C2×#6290C2214yescomplemented
19C2×C3×#625—28no—
20C2×C3×#625C2218yescomplemented
21#88814C2214yescomplemented
22(C2×D3)⋊(C3⋊Dic3)C2218yescomplemented
23C24×C3×D3—32no—
24C23×D32D311yescharacteristic, complemented
25C22×#638—31no—
26C22×C3×#67—32no—
27C22×C3×#67—32no—
28C22×#625—32no—
29C22×#625—61no—
30C22×#625D312yescomplemented
31C2×#3757—32no—
32#63983—38no—
33#63976—316no—
34C2×#3722—31no—
35C2×(C2×D3)⋊Dic3—32no—
36C2×(C2×D3)⋊Dic3—32no—
37C2×(C2×D3)⋊Dic3—32no—
38C2×(C2×D3)⋊Dic3—62no—
39C23×C33C2311yescharacteristic
40C22×C3×C3⋊D3—24no—
41C22×C3×C3⋊D3D414yescomplemented
42C22×C3×C3⋊D3C2312yes—
43C22×C32×D3—24no—
44C22×C32×D3—24no—
45C22×C32×D3—22no—
46C22×C32×D3—24no—
47C22×C32×D3D418yescomplemented
48C22×C32×D3C2314yes—
49C22×C32×D3C2312yescomplemented
50C2×C3×D32—232no—
51C2×C3×D32—28no—
52C2×C3×D32—216no—
53C2×C3⋊(C3⋊Dic3)—22no—
54C2×C3⋊(C3⋊Dic3)C2312yescomplemented
55C2×C3×C3⋊Dic3—24no—
56C2×C3×C3⋊Dic3C2314yescomplemented
57#6290—28no—
58C3×#625—216no—
59C2×#953—91no—
60C2×#937—92no—
61C24×C32—32no—
62C24×C32—31no—
63C23×C3⋊D3C2×D311yescharacteristic
64C23×C3×D3—22no—
65C23×C3×D3—21no—
66C23×C3×D3—32no—
67C23×C3×D3—38no—
68C23×C3×D3—68no—
69C23×C3×D3—68no—
70C23×C3×D3—68no—
71C23×C3×D3—61no—
72C23×C3×D3C2×D312yes—
73C23×C3×D3C2×D312yes—
74C23×C3×D3C2×D312yescomplemented
75C22×D32—24no—
76C22×D32—28no—
77C22×D32C2×D314yescomplemented
78C2×#638—34no—
79C2×#638—64no—
80C22×C3⋊Dic3—32no—
81C22×C3⋊Dic3—31no—
82C22×C3⋊Dic3—61no—
83C22×C3⋊Dic3—62no—
84C22×C3⋊Dic3—121no—
85C22×C3⋊Dic3C2×D312yescomplemented
86C2×C3×#67—38no—
87C2×C3×#67—38no—
88C2×C3×#67—68no—
89C2×C3×#67—68no—
90C22×C3×Dic3—32no—
91C22×C3×Dic3—32no—
92C22×C3×Dic3—62no—
93C2×#625—28no—
94C2×#625—38no—
95C2×#625—68no—
96C2×#625—64no—
97C2×#625—68no—
98C2×#625C2×D318yescomplemented
99#3757—38no—
100#3722—34no—
101(C2×D3)⋊Dic3—38no—
102(C2×D3)⋊Dic3—38no—
103(C2×D3)⋊Dic3—38no—
104(C2×D3)⋊Dic3—68no—
105C22×C33C2×D412yes—
106C22×C33C2×D414yes—
107C22×C33C2411yescharacteristic
108C2×C3×C3⋊D3—28no—
109C2×C3×C3⋊D3—24no—
110C2×C3×C3⋊D3C2×D414yescomplemented
111C2×C32×D3—216no—
112C2×C32×D3—24no—
113C2×C32×D3—24no—
114C2×C32×D3—22no—
115C2×C32×D3—24no—
116C2×C32×D3—44no—
117C2×C32×D3C2×D418yescomplemented
118C3×D32—232no—
119C3⋊(C3⋊Dic3)—22no—
120C3×C3⋊Dic3—24no—
121C25×C3—91no—
122C24×D3—32no—
123C22×C3×D4—92no—
124C22×#67—32no—
125C22×#67—32no—
126C22×#67—92no—
127C22×#67—91no—
128C22×#67—181no—
129#953—98no—
130C2×#217—91no—
131C2×#217—92no—
132C2×#217—92no—
133C2×#217—182no—
134#937—916no—
135C2×(C2×D3)⋊C4—92no—
136C2×(C2×D3)⋊C4—92no—
137C23×C32—21no—
138C23×C32—22no—
139C23×C32—38no—
140C23×C32—34no—
141C23×C32—32no—
142C23×C32—41no—
143C23×C32—62no—
144C23×C32—64no—
145C23×C32—64no—
146C23×C32—64no—
147C23×C32C22×D312yes—
148C23×C32C22×D311yescharacteristic
149C22×C3⋊D3—24no—
150C22×C3⋊D3#6714yescomplemented
151C22×C3⋊D3C22×D312yes—
152C22×C3×D3—24no—
153C22×C3×D3—24no—
154C22×C3×D3—24no—
155C22×C3×D3—24no—
156C22×C3×D3—24no—
157C22×C3×D3—24no—
158C22×C3×D3—24no—
159C22×C3×D3—24no—
160C22×C3×D3—28no—
161C22×C3×D3—24no—
162C22×C3×D3—22no—
163C22×C3×D3—38no—
164C22×C3×D3—34no—
165C22×C3×D3—44no—
166C22×C3×D3—44no—
167C22×C3×D3—44no—
168C22×C3×D3—64no—
169C22×C3×D3—62no—
170C22×C3×D3—616no—
171C22×C3×D3—616no—
172C22×C3×D3—664no—
173C22×C3×D3—64no—
174C22×C3×D3—64no—
175C22×C3×D3—616no—
176C22×C3×D3—124no—
177C22×C3×D3#6718yescomplemented
178C22×C3×D3#6718yescomplemented
179C22×C3×D3C22×D314yes—
180C22×C3×D3C22×D314yes—
181C22×C3×D3C22×D314yescomplemented
182C2×D32—232no—
183C2×D32—28no—
184C2×D32—216no—
185#638—68no—
186C2×C3⋊Dic3—24no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—34no—
189C2×C3⋊Dic3—62no—
190C2×C3⋊Dic3—64no—
191C2×C3⋊Dic3—62no—
192C2×C3⋊Dic3—64no—
193C2×C3⋊Dic3—64no—
194C2×C3⋊Dic3—122no—
195C2×C3⋊Dic3—124no—
196C2×C3⋊Dic3—124no—
197C2×C3⋊Dic3C22×D314yescomplemented
198C3×#67—616no—
199C3×#67—616no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—64no—
203C2×C3×Dic3—64no—
204C2×C3×Dic3—64no—
205C2×C3×Dic3—124no—
206#625—216no—
207#625—616no—
208#625—616no—
209C2×#120—271noSylow
210C2×C33#12014yescomplemented
211C2×C33C22×D411yescharacteristic
212C2×C33C22×D412yes—
213C3×C3⋊D3—24no—
214C32×D3—28no—
215C32×D3—42no—
216C24×C3—32no—
217C24×C3—32no—
218C24×C3—32no—
219C24×C3—62no—
220C24×C3—61no—
221C24×C3—94no—
222C24×C3—91no—
223C24×C3—188no—
224C24×C3—184no—
225C23×D3—32no—
226C23×D3—38no—
227C23×D3—61no—
228C23×D3—68no—
229C23×D3—68no—
230C23×D3—68no—
231C23×D3D3212yes—
232C23×D3D3211yescharacteristic, complemented
233C2×C3×D4—98no—
234C2×C3×D4—188no—
235C22×C3×C4—92no—
236C2×#67—38no—
237C2×#67—38no—
238C2×#67—68no—
239C2×#67—68no—
240C2×#67—98no—
241C2×#67—94no—
242C2×#67—184no—
243C2×#67—184no—
244C2×#67—188no—
245C2×#67—188no—
246C22×Dic3—32no—
247C22×Dic3—32no—
248C22×Dic3—62no—
249C22×Dic3—92no—
250C22×Dic3—91no—
251C22×Dic3—181no—
252C22×Dic3—182no—
253C22×Dic3—361no—
254#217—94no—
255#217—98no—
256#217—98no—
257#217—188no—
258(C2×D3)⋊C4—98no—
259(C2×D3)⋊C4—98no—
260C22×C32—22no—
261C22×C32—21no—
262C22×C32—24no—
263C22×C32—24no—
264C22×C32—24no—
265C22×C32—24no—
266C22×C32—22no—
267C22×C32—38no—
268C22×C32—42no—
269C22×C32—44no—
270C22×C32—41no—
271C22×C32—62no—
272C22×C32—616no—
273C22×C32—64no—
274C22×C32—64no—
275C22×C32—64no—
276C22×C32—124no—
277C22×C32D3×D414yes—
278C22×C32D3×D414yes—
279C22×C32C2×#6714yes—
280C22×C32C2×#6714yes—
281C22×C32C2×#6712yes—
282C22×C32C23×D312yes—
283C22×C32C23×D311yescharacteristic
284C2×C3⋊D3—28no—
285C2×C3⋊D3—24no—
286C2×C3⋊D3C2×#6714yescomplemented
287C2×C3×D3—216no—
288C2×C3×D3—216no—
289C2×C3×D3—24no—
290C2×C3×D3—24no—
291C2×C3×D3—24no—
292C2×C3×D3—24no—
293C2×C3×D3—24no—
294C2×C3×D3—24no—
295C2×C3×D3—24no—
296C2×C3×D3—24no—
297C2×C3×D3—28no—
298C2×C3×D3—38no—
299C2×C3×D3—44no—
300C2×C3×D3—44no—
301C2×C3×D3—416no—
302C2×C3×D3—44no—
303C2×C3×D3—48no—
304C2×C3×D3—48no—
305C2×C3×D3—64no—
306C2×C3×D3—664no—
307C2×C3×D3—616no—
308C2×C3×D3—64no—
309C2×C3×D3—64no—
310C2×C3×D3—124no—
311C2×C3×D3—128no—
312C2×C3×D3C2×#6718yescomplemented
313C2×C3×D3C2×#6718yescomplemented
314D32—232no—
315C3⋊Dic3—24no—
316C3⋊Dic3—62no—
317C3⋊Dic3—64no—
318C3⋊Dic3—64no—
319C3⋊Dic3—124no—
320C3⋊Dic3—124no—
321C3×Dic3—64no—
322C3×Dic3—64no—
323C3×Dic3—124no—
324C25—91no—
325C22×D4—92no—
326C22×D4—271no—
327#120—278no—
328C2×C22⋊C4—271no—
329C2×C22⋊C4—272no—
330C33C2×#12011yescharacteristic, complemented, Sylow
331C23×C3—22no—
332C23×C3—21no—
333C23×C3—34no—
334C23×C3—34no—
335C23×C3—38no—
336C23×C3—38no—
337C23×C3—34no—
338C23×C3—41no—
339C23×C3—64no—
340C23×C3—62no—
341C23×C3—68no—
342C23×C3—64no—
343C23×C3—64no—
344C23×C3—64no—
345C23×C3—64no—
346C23×C3—64no—
347C23×C3—64no—
348C23×C3—64no—
349C23×C3—64no—
350C23×C3—92no—
351C23×C3—94no—
352C23×C3—124no—
353C23×C3—124no—
354C23×C3—124no—
355C23×C3—1816no—
356C23×C3—188no—
357C23×C3—1832no—
358C23×C3—184no—
359C23×C3C2×D3212yes—
360C23×C3C2×D3211yescharacteristic
361C22×D3—24no—
362C22×D3—24no—
363C22×D3—24no—
364C22×D3—22no—
365C22×D3—34no—
366C22×D3—38no—
367C22×D3—64no—
368C22×D3—62no—
369C22×D3—616no—
370C22×D3—616no—
371C22×D3—664no—
372C22×D3—64no—
373C22×D3—64no—
374C22×D3—616no—
375C22×D3—124no—
376C22×D3#62518yescomplemented
377C22×D3C2×D3214yes—
378C22×D3C2×D3212yescomplemented
379C3×D4—1816no—
380C2×C3×C4—94no—
381C2×C3×C4—184no—
382#67—616no—
383#67—616no—
384#67—188no—
385#67—1816no—
386#67—1816no—
387C2×Dic3—34no—
388C2×Dic3—34no—
389C2×Dic3—64no—
390C2×Dic3—64no—
391C2×Dic3—64no—
392C2×Dic3—94no—
393C2×Dic3—92no—
394C2×Dic3—124no—
395C2×Dic3—182no—
396C2×Dic3—184no—
397C2×Dic3—184no—
398C2×Dic3—184no—
399C2×Dic3—182no—
400C2×Dic3—362no—
401C2×Dic3—364no—
402C2×Dic3—364no—
403C2×C32—22no—
404C2×C32—22no—
405C2×C32—22no—
406C2×C32—22no—
407C2×C32—21no—
408C2×C32—28no—
409C2×C32—24no—
410C2×C32—44no—
411C2×C32—41no—
412C2×C32—42no—
413C2×C32—68no—
414C2×C32—122no—
415C2×C32#93718yescomplemented
416C2×C32#95314yescomplemented
417C2×C32C2×D3×D412yes—
418C2×C32C2×D3×D412yes—
419C2×C32C22×#6712yes—
420C2×C32C22×#6712yes—
421C2×C32C22×#6711yescharacteristic
422C3⋊D3—24no—
423C3×D3—28no—
424C3×D3—28no—
425C3×D3—48no—
426C3×D3—44no—
427C3×D3—44no—
428C3×D3—68no—
429C3×D3—124no—
430C24—32no—
431C24—31no—
432C24—94no—
433C24—91no—
434C24—188no—
435C24—184no—
436C2×D4—98no—
437C2×D4—188no—
438C2×D4—274no—
439C2×D4—544no—
440C22×C4—92no—
441C22×C4—271no—
442C22⋊C4—274no—
443C22⋊C4—278no—
444C22×C3—22no—
445C22×C3—21no—
446C22×C3—24no—
447C22×C3—24no—
448C22×C3—24no—
449C22×C3—22no—
450C22×C3—24no—
451C22×C3—38no—
452C22×C3—38no—
453C22×C3—41no—
454C22×C3—44no—
455C22×C3—42no—
456C22×C3—68no—
457C22×C3—64no—
458C22×C3—64no—
459C22×C3—64no—
460C22×C3—64no—
461C22×C3—64no—
462C22×C3—616no—
463C22×C3—616no—
464C22×C3—64no—
465C22×C3—64no—
466C22×C3—64no—
467C22×C3—94no—
468C22×C3—124no—
469C22×C3—124no—
470C22×C3—1216no—
471C22×C3—124no—
472C22×C3—128no—
473C22×C3—128no—
474C22×C3—1832no—
475C22×C3—184no—
476C22×C3—188no—
477C22×C3C2×#62514yes—
478C22×C3D3×#6714yes—
479C22×C3D3×#6714yes—
480C22×C3D3×#6714yes—
481C22×C3#381112yes—
482C22×C3C22×D3212yes—
483C22×C3C22×D3211yescharacteristic
484C2×D3—24no—
485C2×D3—24no—
486C2×D3—216no—
487C2×D3—24no—
488C2×D3—22no—
489C2×D3—38no—
490C2×D3—44no—
491C2×D3—64no—
492C2×D3—664no—
493C2×D3—64no—
494C2×D3—64no—
495C2×D3—616no—
496C2×D3—124no—
497C2×D3—128no—
498C2×D3C2×#62518yescomplemented
499C3×C4—184no—
500Dic3—64no—
501Dic3—64no—
502Dic3—124no—
503Dic3—184no—
504Dic3—184no—
505Dic3—182no—
506Dic3—364no—
507Dic3—364no—
508C32—22no—
509C32—21no—
510C32—41no—
511C32C2×#93712yescomplemented
512C32C2×#95311yescharacteristic, complemented
513C23—34no—
514C23—38no—
515C23—32no—
516C23—64no—
517C23—64no—
518C23—64no—
519C23—62no—
520C23—92no—
521C23—94no—
522C23—1816no—
523C23—188no—
524C23—1832no—
525C23—184no—
526C23D3311yescharacteristic
527D4—1816no—
528D4—548no—
529C2×C4—94no—
530C2×C4—184no—
531C2×C4—272no—
532C2×C4—542no—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—22no—
536C2×C3—21no—
537C2×C3—22no—
538C2×C3—24no—
539C2×C3—28no—
540C2×C3—42no—
541C2×C3—41no—
542C2×C3—44no—
543C2×C3—68no—
544C2×C3—68no—
545C2×C3—128no—
546C2×C3—124no—
547C2×C3—124no—
548C2×C3—184no—
549C2×C3#6397618yescomplemented
550C2×C3#6398314yescomplemented
551C2×C3C22×#62512yes—
552C2×C3C2×D3×#6712yes—
553C2×C3C2×D3×#6712yes—
554C2×C3C2×D3×#6712yes—
555C2×C3C2×#381111yescharacteristic
556D3—28no—
557D3—42no—
558D3—68no—
559D3—124no—
560C22—38no—
561C22—64no—
562C22—64no—
563C22—616no—
564C22—64no—
565C22—62no—
566C22—94no—
567C22—124no—
568C22—1832no—
569C22—184no—
570C22—188no—
571C22D3×#62514yes—
572C22#8895212yes—
573C22C2×D3311yescharacteristic
574C4—184no—
575C4—542no—
576C3—21no—
577C3—22no—
578C3—41no—
579C3C2×#6397612yescomplemented
580C3C2×#6398311yescharacteristic, complemented
581C2—68no—
582C2—122no—
583C2—184no—
584C2#1172294614yesdirect product factor
585C2C2×D3×#62512yes—
586C2C2×#8895211yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutator, cosocleC22×C33C24
FrattiniC22C2×D33
Fitting, socleC23×C33C23
hypocenterC33C2×#120
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3×D32C2semidirect
2. C22×#6290C2semidirect
3. C22×C3×#625C2semidirect
4. C2×#88814C2semidirect
5. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
6. #11722946C2direct (N × H)
11. C23×C32×D3C22semidirect
14. C22×C3×D32C22semidirect
15. C22×C3⋊(C3⋊Dic3)C22semidirect
16. C22×C3×C3⋊Dic3C22semidirect
18. C2×#6290C22semidirect
20. C2×C3×#625C22semidirect
21. #88814C22semidirect
22. (C2×D3)⋊(C3⋊Dic3)C22semidirect
24. C23×D32D3semidirect
30. C22×#625D3semidirect
41. C22×C3×C3⋊D3D4semidirect
47. C22×C32×D3D4semidirect
49. C22×C32×D3C23semidirect
54. C2×C3⋊(C3⋊Dic3)C23semidirect
56. C2×C3×C3⋊Dic3C23semidirect
74. C23×C3×D3C2×D3semidirect
77. C22×D32C2×D3semidirect
85. C22×C3⋊Dic3C2×D3semidirect
98. C2×#625C2×D3semidirect
110. C2×C3×C3⋊D3C2×D4semidirect
117. C2×C32×D3C2×D4semidirect
150. C22×C3⋊D3#67semidirect
177. C22×C3×D3#67semidirect
178. C22×C3×D3#67semidirect
181. C22×C3×D3C22×D3semidirect
197. C2×C3⋊Dic3C22×D3semidirect
210. C2×C33#120semidirect
232. C23×D3D32semidirect
286. C2×C3⋊D3C2×#67semidirect
312. C2×C3×D3C2×#67semidirect
313. C2×C3×D3C2×#67semidirect
330. C33C2×#120semidirect
376. C22×D3#625semidirect
378. C22×D3C2×D32semidirect
415. C2×C32#937semidirect
416. C2×C32#953semidirect
498. C2×D3C2×#625semidirect
511. C32C2×#937semidirect
512. C32C2×#953semidirect
549. C2×C3#63976semidirect
550. C2×C3#63983semidirect
579. C3C2×#63976semidirect
580. C3C2×#63983semidirect
584. C2#11722946direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D322
C22×#62902
C22×C3×#6252
C2×#888142
C2×(C2×D3)⋊(C3⋊Dic3)2
#117229462
C2×#639833
C2×#639763