Finite Group Databasebeta

← C2×#11723717 #13178799

Subgroups — C2×#11723717 #13178799

Order 1728.

2546 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C212yesmaximal, complemented
2C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C22×#6295C211yesmaximal, characteristic, complemented
4C2×#88815C211yesmaximal, characteristic, complemented
5C2×#88814C212yesmaximal, complemented
6#11723717C218yesmaximal, direct product factor
7C2×#63980—31nomaximal
8C2×#63978—32nomaximal
9C24×C33C2211yescharacteristic
10C23×C3×C3⋊D3C2212yes—
11C2×C3×#638—28no—
12C2×C3×#638C2218yescomplemented
13C22×C3×C3⋊Dic3—24no—
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2212yes—
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2214yescomplemented
18C2×#6295—24no—
19C2×#6295C2214yescomplemented
20#88815C2214yescomplemented
21#88814C2218yescomplemented
22C22×#638D312yescomplemented
23C23×C3⋊Dic3D311yescharacteristic, complemented
24C22×C3×#67—32no—
25C22×C3×#67—32no—
26C22×C3×#67—62no—
27C23×C3×Dic3—32no—
28C23×C3×Dic3—61no—
29C22×#626—32no—
30C22×#625—31no—
31#63980—38no—
32#63978—316no—
33C2×#3724—31no—
34C2×Dic3⋊Dic3—32no—
35C2×#3722—32no—
36C2×(C2×D3)⋊Dic3—32no—
37C2×(C2×D3)⋊Dic3—32no—
38C23×C33—22no—
39C23×C33—22no—
40C23×C33D414yescomplemented
41C23×C33C2312yes—
42C23×C33C2311yescharacteristic
43C22×C3×C3⋊D3—24no—
44C22×C3×C3⋊D3—24no—
45C22×C3×C3⋊D3—24no—
46C22×C3×C3⋊D3C2314yes—
47C3×#638—216no—
48C2×C3×C3⋊Dic3—22no—
49C2×C3×C3⋊Dic3—24no—
50C2×C3×C3⋊Dic3—28no—
51C2×C3×C3⋊Dic3D414yescomplemented
52C2×C3×C3⋊Dic3C2312yes—
53C2×C3×C3⋊Dic3C2314yes—
54C2×C3×C3⋊Dic3C2312yes—
55#6295—28no—
56C2×#939—92no—
57C2×#930—91no—
58C24×C32—21no—
59C24×C32—22no—
60C24×C32—41no—
61C24×C32C2×D311yescharacteristic
62C24×C32C2×D312yes—
63C23×C3⋊D3C2×D312yes—
64C23×C3×D3—32no—
65C23×C3×D3—32no—
66C23×C3×D3—62no—
67C2×#638—28no—
68C2×#638C2×D318yescomplemented
69C22×C3⋊Dic3—24no—
70C22×C3⋊Dic3C2×D311yescharacteristic
71C22×C3⋊Dic3C2×D312yes—
72C22×C3⋊Dic3C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D314yescomplemented
74C2×C3×#67—38no—
75C2×C3×#67—38no—
76C2×C3×#67—68no—
77C2×C3×#67—616no—
78C2×C3×#67—68no—
79C2×C3×#67—68no—
80C22×C3×Dic3—32no—
81C22×C3×Dic3—32no—
82C22×C3×Dic3—38no—
83C22×C3×Dic3—32no—
84C22×C3×Dic3—32no—
85C22×C3×Dic3—68no—
86C22×C3×Dic3—61no—
87C22×C3×Dic3—61no—
88C22×C3×Dic3—64no—
89C22×C3×Dic3—62no—
90C22×C3×Dic3—68no—
91C2×#626—38no—
92C2×#626—68no—
93C2×#625—34no—
94C2×#625—64no—
95#3724—34no—
96Dic3⋊Dic3—38no—
97#3722—38no—
98(C2×D3)⋊Dic3—38no—
99(C2×D3)⋊Dic3—38no—
100C22×C33—28no—
101C22×C33—22no—
102C22×C33—22no—
103C22×C33C2×D412yes—
104C22×C33C2×D412yes—
105C22×C33C2×D414yescomplemented
106C22×C33#4112yes—
107C22×C33C2411yescharacteristic
108C2×C3×C3⋊D3—24no—
109C2×C3×C3⋊D3—24no—
110C2×C3×C3⋊D3—24no—
111C2×C3×C3⋊D3—48no—
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3—24no—
114C3×C3⋊Dic3C2×D414yescomplemented
115C22×C3×D4—92no—
116C23×C3×C4—91no—
117C22×#67—32no—
118C22×#67—32no—
119C22×#67—62no—
120C22×#67—92no—
121C23×Dic3—32no—
122C23×Dic3—61no—
123C22×D12—91no—
124C2×#217—92no—
125#939—916no—
126#930—98no—
127C2×(C2×D3)⋊C4—92no—
128C2×(C2×D3)⋊C4—92no—
129C2×C4⋊Dic3—91no—
130C2×Dic3⋊C4—92no—
131C23×C32—22no—
132C23×C32—22no—
133C23×C32—24no—
134C23×C32—24no—
135C23×C32—24no—
136C23×C32—24no—
137C23×C32—24no—
138C23×C32—28no—
139C23×C32—22no—
140C23×C32—22no—
141C23×C32—21no—
142C23×C32—24no—
143C23×C32—44no—
144C23×C32—42no—
145C23×C32—41no—
146C23×C32—44no—
147C23×C32—44no—
148C23×C32—44no—
149C23×C32—44no—
150C23×C32D1214yescomplemented
151C23×C32#6718yescomplemented
152C23×C32C22×D312yes—
153C23×C32C22×D314yes—
154C23×C32C22×D311yescharacteristic
155C23×C32C22×D312yes—
156C22×C3⋊D3—24no—
157C22×C3⋊D3—24no—
158C22×C3⋊D3—24no—
159C22×C3⋊D3C22×D314yes—
160C22×C3×D3—34no—
161C22×C3×D3—34no—
162C22×C3×D3—64no—
163C22×C3×D3—68no—
164C22×C3×D3—64no—
165C22×C3×D3—64no—
166C22×C3×D3—64no—
167C22×C3×D3—64no—
168C22×C3×D3—64no—
169C22×C3×D3—64no—
170C22×C3×D3—124no—
171C22×C3×D3—124no—
172#638—216no—
173C2×C3⋊Dic3—22no—
174C2×C3⋊Dic3—24no—
175C2×C3⋊Dic3—28no—
176C2×C3⋊Dic3#6714yescomplemented
177C2×C3⋊Dic3C22×D312yes—
178C2×C3⋊Dic3C22×D314yes—
179C2×C3⋊Dic3C22×D312yes—
180C3×#67—632no—
181C3×#67—616no—
182C3×#67—616no—
183C2×C3×Dic3—38no—
184C2×C3×Dic3—34no—
185C2×C3×Dic3—34no—
186C2×C3×Dic3—34no—
187C2×C3×Dic3—34no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—64no—
192C2×C3×Dic3—616no—
193C2×C3×Dic3—616no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3—68no—
196C2×C3×Dic3—64no—
197C2×C3×Dic3—64no—
198C2×C3×Dic3—64no—
199#626—616no—
200#625—68no—
201C2×#121—271noSylow
202C2×C33—24no—
203C2×C33#12114yescomplemented
204C2×C33C22×D411yescharacteristic
205C2×C33C22×D411yescharacteristic
206C2×C33C2×#4111yescharacteristic
207C3×C3⋊D3—44no—
208C24×C3—21no—
209C24×C3—22no—
210C24×C3—41no—
211C24×C3—92no—
212C24×C3D3211yescharacteristic
213C24×C3D3212yes—
214C23×D3—32no—
215C23×D3—32no—
216C23×D3—62no—
217C2×C3×D4—98no—
218C2×C3×D4—188no—
219C22×C3×C4—91no—
220C22×C3×C4—92no—
221C22×C3×C4—91no—
222C22×C3×C4—94no—
223C22×C3×C4—184no—
224C2×#67—38no—
225C2×#67—38no—
226C2×#67—68no—
227C2×#67—616no—
228C2×#67—68no—
229C2×#67—68no—
230C2×#67—98no—
231C2×#67—188no—
232C22×Dic3—32no—
233C22×Dic3—32no—
234C22×Dic3—32no—
235C22×Dic3—32no—
236C22×Dic3—38no—
237C22×Dic3—61no—
238C22×Dic3—61no—
239C22×Dic3—64no—
240C22×Dic3—62no—
241C22×Dic3—68no—
242C22×Dic3—68no—
243C2×D12—94no—
244C2×D12—184no—
245#217—98no—
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—98no—
248C4⋊Dic3—94no—
249Dic3⋊C4—98no—
250C22×C32—28no—
251C22×C32—22no—
252C22×C32—22no—
253C22×C32—24no—
254C22×C32—24no—
255C22×C32—216no—
256C22×C32—216no—
257C22×C32—24no—
258C22×C32—24no—
259C22×C32—21no—
260C22×C32—22no—
261C22×C32—24no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—24no—
265C22×C32—24no—
266C22×C32—28no—
267C22×C32—24no—
268C22×C32—22no—
269C22×C32—42no—
270C22×C32—42no—
271C22×C32—44no—
272C22×C32—42no—
273C22×C32—41no—
274C22×C32—44no—
275C22×C32—44no—
276C22×C32—416no—
277C22×C32—44no—
278C22×C32—416no—
279C22×C32—44no—
280C22×C32C2×D1212yes—
281C22×C32C2×D1214yescomplemented
282C22×C32#23512yes—
283C22×C32D3×D414yes—
284C22×C32#23714yes—
285C22×C32C2×#6714yes—
286C22×C32C2×#6712yes—
287C22×C32C2×#6718yescomplemented
288C22×C32C23×D311yescharacteristic
289C22×C32C23×D312yes—
290C2×C3⋊D3—24no—
291C2×C3⋊D3—24no—
292C2×C3⋊D3—24no—
293C2×C3⋊D3—48no—
294C2×C3×D3—68no—
295C2×C3×D3—64no—
296C2×C3×D3—64no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C2×C3×D3—64no—
300C2×C3×D3—64no—
301C2×C3×D3—124no—
302C2×C3×D3—124no—
303C2×C3×D3—1216no—
304C2×C3×D3—128no—
305C2×C3×D3—128no—
306C3⋊Dic3—22no—
307C3⋊Dic3—24no—
308C3⋊Dic3C2×#6714yescomplemented
309C3×Dic3—38no—
310C3×Dic3—64no—
311C3×Dic3—64no—
312C3×Dic3—68no—
313C3×Dic3—64no—
314C3×Dic3—64no—
315C3×Dic3—64no—
316C22×D4—92no—
317C22×D4—271no—
318C23×C4—91no—
319#121—278no—
320C2×C4⋊C4—271no—
321C2×C22⋊C4—272no—
322C33C2×#12111yescharacteristic, complemented, Sylow
323C23×C3—24no—
324C23×C3—24no—
325C23×C3—24no—
326C23×C3—22no—
327C23×C3—24no—
328C23×C3—22no—
329C23×C3—24no—
330C23×C3—28no—
331C23×C3—22no—
332C23×C3—24no—
333C23×C3—21no—
334C23×C3—22no—
335C23×C3—44no—
336C23×C3—44no—
337C23×C3—44no—
338C23×C3—44no—
339C23×C3—44no—
340C23×C3—42no—
341C23×C3—41no—
342C23×C3—94no—
343C23×C3—184no—
344C23×C3—184no—
345C23×C3—184no—
346C23×C3#62514yescomplemented
347C23×C3#62618yescomplemented
348C23×C3C2×D3212yes—
349C23×C3C2×D3214yes—
350C23×C3C2×D3212yes—
351C23×C3C2×D3211yescharacteristic
352C22×D3—34no—
353C22×D3—34no—
354C22×D3—64no—
355C22×D3—68no—
356C22×D3—64no—
357C22×D3—64no—
358C22×D3—64no—
359C22×D3—64no—
360C22×D3—64no—
361C22×D3—64no—
362C22×D3—124no—
363C22×D3—124no—
364C3×D4—1816no—
365C2×C3×C4—92no—
366C2×C3×C4—94no—
367C2×C3×C4—94no—
368C2×C3×C4—92no—
369C2×C3×C4—188no—
370C2×C3×C4—184no—
371C2×C3×C4—182no—
372#67—632no—
373#67—616no—
374#67—616no—
375#67—1816no—
376C2×Dic3—34no—
377C2×Dic3—38no—
378C2×Dic3—34no—
379C2×Dic3—34no—
380C2×Dic3—34no—
381C2×Dic3—64no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—64no—
385C2×Dic3—616no—
386C2×Dic3—616no—
387C2×Dic3—64no—
388C2×Dic3—68no—
389C2×Dic3—64no—
390C2×Dic3—64no—
391C2×Dic3—64no—
392D12—188no—
393C2×C32—24no—
394C2×C32—28no—
395C2×C32—28no—
396C2×C32—22no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—22no—
400C2×C32—22no—
401C2×C32—21no—
402C2×C32—28no—
403C2×C32—24no—
404C2×C32—44no—
405C2×C32—41no—
406C2×C32—41no—
407C2×C32—41no—
408C2×C32—48no—
409C2×C32—48no—
410C2×C32#93014yescomplemented
411C2×C32#93918yescomplemented
412C2×C32C22×D1211yescharacteristic
413C2×C32C2×#23511yescharacteristic
414C2×C32C2×D3×D412yes—
415C2×C32C2×#23712yes—
416C2×C32C22×#6712yes—
417C2×C32C22×#6711yescharacteristic
418C3⋊D3—44no—
419C3×D3—128no—
420C3×D3—124no—
421C3×D3—124no—
422C24—92no—
423C24#111311yescharacteristic
424C2×D4—98no—
425C2×D4—188no—
426C2×D4—274no—
427C2×D4—544no—
428C22×C4—91no—
429C22×C4—94no—
430C22×C4—91no—
431C22×C4—92no—
432C22×C4—184no—
433C4⋊C4—274no—
434C22⋊C4—278no—
435C22×C3—24no—
436C22×C3—24no—
437C22×C3—216no—
438C22×C3—24no—
439C22×C3—22no—
440C22×C3—24no—
441C22×C3—22no—
442C22×C3—216no—
443C22×C3—28no—
444C22×C3—21no—
445C22×C3—22no—
446C22×C3—24no—
447C22×C3—22no—
448C22×C3—24no—
449C22×C3—22no—
450C22×C3—24no—
451C22×C3—28no—
452C22×C3—22no—
453C22×C3—24no—
454C22×C3—44no—
455C22×C3—44no—
456C22×C3—416no—
457C22×C3—44no—
458C22×C3—44no—
459C22×C3—416no—
460C22×C3—41no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—44no—
464C22×C3—42no—
465C22×C3—184no—
466C22×C3—184no—
467C22×C3—184no—
468C22×C3—368no—
469C22×C3#380414yes—
470C22×C3#380512yes—
471C22×C3C2×#62512yes—
472C22×C3C2×#62514yescomplemented
473C22×C3C2×#62614yes—
474C22×C3C2×#62618yescomplemented
475C22×C3D3×#6714yes—
476C22×C3#381112yes—
477C22×C3C22×D3212yes—
478C22×C3C22×D3211yescharacteristic
479C2×D3—68no—
480C2×D3—64no—
481C2×D3—64no—
482C2×D3—64no—
483C2×D3—64no—
484C2×D3—64no—
485C2×D3—64no—
486C2×D3—124no—
487C2×D3—124no—
488C2×D3—1216no—
489C2×D3—128no—
490C2×D3—128no—
491C3×C4—94no—
492C3×C4—184no—
493C3×C4—182no—
494Dic3—38no—
495Dic3—64no—
496Dic3—64no—
497Dic3—68no—
498Dic3—64no—
499Dic3—64no—
500Dic3—64no—
501C32—22no—
502C32—21no—
503C32—41no—
504C32C2×#93011yescharacteristic, complemented
505C32C2×#93912yescomplemented
506C23—22no—
507C23—22no—
508C23—94no—
509C23—184no—
510C23—184no—
511C23—184no—
512C23#629514yescomplemented
513C23C2×#111312yes—
514C23C2×#111311yescharacteristic
515D4—1816no—
516D4—548no—
517C2×C4—92no—
518C2×C4—94no—
519C2×C4—92no—
520C2×C4—94no—
521C2×C4—188no—
522C2×C4—184no—
523C2×C4—182no—
524C2×C3—28no—
525C2×C3—28no—
526C2×C3—24no—
527C2×C3—22no—
528C2×C3—21no—
529C2×C3—22no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—22no—
533C2×C3—28no—
534C2×C3—24no—
535C2×C3—48no—
536C2×C3—48no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—41no—
540C2×C3—44no—
541C2×C3—364no—
542C2×C3#6397818yescomplemented
543C2×C3#6398014yescomplemented
544C2×C3C2×#380412yes—
545C2×C3C2×#380511yescharacteristic
546C2×C3C22×#62511yescharacteristic
547C2×C3C22×#62612yes—
548C2×C3C2×D3×#6712yes—
549C2×C3C2×#381111yescharacteristic
550D3—128no—
551D3—124no—
552D3—124no—
553C22—22no—
554C22—22no—
555C22—28no—
556C22—184no—
557C22—184no—
558C22—184no—
559C22—368no—
560C22#8904612yes—
561C22C2×#629512yes—
562C22C2×#629514yescomplemented
563C22#8904912yes—
564C22C22×#111311yescharacteristic
565C4—94no—
566C4—184no—
567C4—182no—
568C3—22no—
569C3—21no—
570C3—41no—
571C3C2×#6397812yescomplemented
572C3C2×#6398011yescharacteristic, complemented
573C2—24no—
574C2—364no—
575C2#1172371714yesdirect product factor
576C2C2×#8904611yescharacteristic
577C2C22×#629511yescharacteristic
578C2C2×#8904911yescharacteristic
subgroupG/H
centerC23C2×#1113
commutator, cosocleC22×C33C24
FrattiniC22C22×#1113
FittingC24×C33C22
socleC23×C33C23
hypercenterC24#1113
hypocenterC33C2×#121
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C3×#638C2semidirect
2. C23×C3×C3⋊Dic3C2semidirect
3. C22×#6295C2semidirect
4. C2×#88815C2semidirect
5. C2×#88814C2semidirect
6. #11723717C2direct (N × H)
12. C2×C3×#638C22semidirect
17. C22×C3×C3⋊Dic3C22semidirect
19. C2×#6295C22semidirect
20. #88815C22semidirect
21. #88814C22semidirect
22. C22×#638D3semidirect
23. C23×C3⋊Dic3D3semidirect
40. C23×C33D4semidirect
51. C2×C3×C3⋊Dic3D4semidirect
68. C2×#638C2×D3semidirect
73. C22×C3⋊Dic3C2×D3semidirect
105. C22×C33C2×D4semidirect
114. C3×C3⋊Dic3C2×D4semidirect
150. C23×C32D12semidirect
151. C23×C32#67semidirect
176. C2×C3⋊Dic3#67semidirect
203. C2×C33#121semidirect
281. C22×C32C2×D12semidirect
287. C22×C32C2×#67semidirect
308. C3⋊Dic3C2×#67semidirect
322. C33C2×#121semidirect
346. C23×C3#625semidirect
347. C23×C3#626semidirect
410. C2×C32#930semidirect
411. C2×C32#939semidirect
472. C22×C3C2×#625semidirect
474. C22×C3C2×#626semidirect
504. C32C2×#930semidirect
505. C32C2×#939semidirect
512. C23#6295semidirect
542. C2×C3#63978semidirect
543. C2×C3#63980semidirect
562. C22C2×#6295semidirect
571. C3C2×#63978semidirect
572. C3C2×#63980semidirect
575. C2#11723717direct (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
C23×C3×C3⋊Dic32
C22×#62952
C2×#888152
C2×#888142
#117237172
C2×#639803
C2×#639783