Finite Group Databasebeta

← #13150362

Subgroups — #13150362

Order 1728.

1076 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C9×D12C211yesmaximal, characteristic, complemented
2C2×#88638C211yesmaximal, characteristic, complemented
3#11721813C211yesmaximal, characteristic, complemented
4C3×C4×#611C211yesmaximal, characteristic, complemented
5#11721286C212yesmaximal, complemented
6D3×#3676C212yesmaximal, complemented
7#11721036C211yesmaximal, characteristic, complemented
8D12×Dic9C211yesmaximal, characteristic, complemented
9#11721016C212yesmaximal, complemented
10#11721015C212yesmaximal, complemented
11#11720964C211yesmaximal, characteristic, complemented
12#10594886—31nomaximal
13#10589908—31nomaximal
14C23×D3×C9C2212yes—
15C22×C3×C4×C9C2211yescharacteristic
16C2×#6226C2211yescharacteristic, complemented
17C2×C3×#611C2211yescharacteristic
18C2×C9×D12—24no—
19C2×C9×D12C2211yescharacteristic
20C2×C9×D12C2212yes—
21C2×C9×D12C2211yescharacteristic
22C2×C3×C4×D9C2211yescharacteristic, complemented
23C2×#6194C2212yescomplemented
24C2×#6194C2212yescomplemented
25C2×D3×Dic9C2212yescomplemented
26C2×D3×Dic9C2212yescomplemented
27#88638—24no—
28#88551C2212yes—
29C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
30#88534C2212yescomplemented
31C3×#3676C2211yescharacteristic
32C3×(C2×D9)⋊C4C2211yescharacteristic, complemented
33C3×Dic9⋊C4C2211yescharacteristic, complemented
34C3×C4×Dic9C2211yescharacteristic, complemented
35(C2×D3)⋊Dic9C2212yescomplemented
36(C2×D3)⋊Dic9C2212yescomplemented
37(C2×D3)⋊Dic9C2212yescomplemented
38(C2×D3)⋊Dic9C2212yescomplemented
39C22×C3×D12D311yescharacteristic
40C2×#3795—31no—
41#64133—31no—
42C3×C4×#67—31no—
43#63976—32no—
44D3×#217—32no—
45#63908—31no—
46Dic3×D12—31no—
47#63885—32no—
48#63884—32no—
49#63860—31no—
50C22×D4×C9—31no—
51C2×#3699—31no—
52#63515—32no—
53C2×#3676—32no—
54#63501—32no—
55#63500—31no—
56#63498—31no—
57#63497—32no—
58D4×Dic9—31no—
59C4×#611D311yescharacteristic, complemented
60#63486—31no—
61#63486—61no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9—24no—
64C22×D3×C9—24no—
65C22×D3×C9—42no—
66C22×D3×C9C2312yes—
67C22×D3×C9C2312yes—
68C22×C3×D9C2311yescharacteristic
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3⋊Dic9C2312yes—
73C2×C3⋊Dic9C2312yes—
74#6226—22no—
75C3×#611—22no—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C9×D12—24no—
80C9×D12—22no—
81C9×D12D414yescomplemented
82C3×C4×D9—22no—
83#6194—24no—
84#6194—42no—
85D3×Dic9—24no—
86D3×Dic9—42no—
87#5806—91no—
88#5550—91no—
89C23×C3×D3C2×D312yes—
90C22×C32×C4C2×D311yescharacteristic
91C2×#634—31no—
92C2×C3×#67—31no—
93C2×C3×D12—24no—
94C2×C3×D12C2×D311yescharacteristic
95C2×C3×D12C2×D312yes—
96C2×C3×D12C2×D311yescharacteristic
97C2×C3×C4×D3—31no—
98C2×#625—32no—
99C2×#625—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102#3795—64no—
103C24×C9—32no—
104#3757—32no—
105C4⋊(C3⋊Dic3)—31no—
106#3750—32no—
107C3×#217—31no—
108C3×(C2×D3)⋊C4—31no—
109C3×Dic3⋊C4—31no—
110C3×C4×Dic3—31no—
111(C2×D3)⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C2×D4×C9—31no—
116C2×D4×C9—32no—
117C2×D4×C9—31no—
118C2×D4×C9—64no—
119C22×C4×C9—21no—
120C22×C4×C9C2×D311yescharacteristic
121C2×#611—32no—
122C2×#611—32no—
123C2×#611C2×D311yescharacteristic
124C22×Dic9—32no—
125C22×Dic9—32no—
126#3699—64no—
127C2×C4×D9C2×D311yescharacteristic, complemented
128C2×Dic18—31no—
129C2×Dic18—61no—
130#3676—32no—
131#3676—32no—
132#3676—32no—
133#3676—32no—
134#3676—32no—
135#3676—62no—
136#3676C2×D311yescharacteristic
137(C2×D9)⋊C4C2×D311yescharacteristic, complemented
138C4⋊Dic9—31no—
139C4⋊Dic9—61no—
140Dic9⋊C4—32no—
141Dic9⋊C4—62no—
142Dic9⋊C4C2×D311yescharacteristic, complemented
143C4×Dic9C2×D311yescharacteristic, complemented
144C22×C3×C9—21no—
145C22×C3×C9—21no—
146C22×C3×C9#4112yescomplemented
147C22×C3×C9C2411yescharacteristic
148C2×D3×C9—22no—
149C2×D3×C9—22no—
150C2×D3×C9—22no—
151C2×D3×C9—22no—
152C2×D3×C9—22no—
153C2×D3×C9—48no—
154C2×D3×C9C2×D414yes—
155C2×C3×D9—21no—
156C2×C3×D9—21no—
157C2×C3×D9—21no—
158C3×C4×C9—21no—
159C3×C4×C9C2×D412yes—
160C3⋊Dic9—22no—
161C3⋊Dic9—22no—
162C3×Dic9—21no—
163C3×Dic9—21no—
164C3×Dic9—21no—
165C22×C3×D4—31no—
166C2×#237—91no—
167C2×#235—91no—
168C22×D12D911yescharacteristic, complemented
169C3×C4×D4—91no—
170#953—92no—
171C2×#217—92no—
172#939—92no—
173#938—91no—
174#937—92no—
175#936—91no—
176#935—92no—
177D4×Dic3—91no—
178#930—91no—
179C4×#67—31no—
180#924—91no—
181#924—91no—
182#924—181no—
183#894—92no—
184#884—92no—
185D3×C22⋊C4—92no—
186#875—91no—
187C4×D12—91no—
188C23×C32C22×D311yescharacteristic
189C22×C3×D3—24no—
190C22×C3×D3—24no—
191C22×C3×D3—31no—
192C22×C3×D3—42no—
193C22×C3×D3C22×D312yes—
194C22×C3×D3C22×D312yes—
195C2×C32×C4—22no—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C3⋊Dic3—32no—
199C2×C3⋊Dic3—32no—
200#634—62no—
201C3×#67—62no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C3×D12—24no—
206C3×D12—22no—
207C3×D12#6714yes—
208C3×C4×D3—62no—
209#625—64no—
210#625—122no—
211D3×Dic3—64no—
212D3×Dic3—122no—
213C23×C9—21no—
214C23×C9—32no—
215C23×C9—32no—
216C23×C9—64no—
217C23×C9—64no—
218C23×C9—122no—
219C23×C9C22×D311yescharacteristic
220C22×D9C22×D311yescharacteristic
221D4×C9—34no—
222D4×C9—64no—
223D4×C9—62no—
224C2×C4×C9—22no—
225C2×C4×C9—21no—
226C2×C4×C9—21no—
227C2×C4×C9—42no—
228C2×C4×C9C22×D311yescharacteristic
229C2×C4×C9C22×D311yescharacteristic
230#611—22no—
231#611—64no—
232#611—122no—
233C2×Dic9—32no—
234C2×Dic9—32no—
235C2×Dic9—62no—
236C2×Dic9—62no—
237C2×Dic9—64no—
238C2×Dic9—122no—
239C2×Dic9C22×D311yescharacteristic
240C2×Dic9C22×D311yescharacteristic
241C2×Dic9C22×D311yescharacteristic
242C4×D9—22no—
243Dic18—62no—
244Dic18—122no—
245#546—271noSylow
246C2×C3×C9—22no—
247C2×C3×C9C22×D411yescharacteristic
248C2×C3×C9C2×#4111yescharacteristic
249C2×C3×C9H(5,𝔽2)11yescharacteristic
250D3×C9—24no—
251D3×C9—42no—
252C3×D9—41no—
253C24×C3—32no—
254C23×D3C2×D912yes—
255C2×C3×D4—31no—
256C2×C3×D4—32no—
257C2×C3×D4—31no—
258C2×C3×D4—64no—
259C2×C3×D4—91no—
260C22×C3×C4—21no—
261C22×C3×C4—91no—
262C22×C3×C4C2×D911yescharacteristic
263C22×C3×C4D3211yescharacteristic
264C2×#67—31no—
265C2×#67—92no—
266C2×#67—92no—
267C2×#67—92no—
268C2×#67—92no—
269C22×Dic3—92no—
270C22×Dic3—92no—
271#237—184no—
272#235—184no—
273C2×D12—24no—
274C2×D12C2×D911yescharacteristic
275C2×D12C2×D912yes—
276C2×D12C2×D911yescharacteristic
277C2×C4×D3—31no—
278C2×C4×D3—92no—
279C2×C4×D3—92no—
280C2×Dic6—91no—
281C2×Dic6—91no—
282C2×Dic6—181no—
283C3×C4⋊C4—91no—
284C3×C22⋊C4—91no—
285C3×C22⋊C4—91no—
286C3×C42—91no—
287#217—31no—
288#217—92no—
289#217—92no—
290#217—92no—
291#217—92no—
292#217—92no—
293#217—92no—
294#217—182no—
295(C2×D3)⋊C4—31no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300C4⋊Dic3—91no—
301C4⋊Dic3—91no—
302C4⋊Dic3—181no—
303Dic3⋊C4—31no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—182no—
307C4×Dic3—31no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32#23712yes—
311C22×C32C23×D311yescharacteristic
312C2×C3×D3—22no—
313C2×C3×D3—22no—
314C2×C3×D3—22no—
315C2×C3×D3—22no—
316C2×C3×D3—22no—
317C2×C3×D3—48no—
318C2×C3×D3—61no—
319C2×C3×D3—61no—
320C2×C3×D3—61no—
321C2×C3×D3C2×#6714yes—
322C32×C4—21no—
323C32×C4C2×#6712yes—
324C3⋊Dic3—62no—
325C3⋊Dic3—62no—
326C3×Dic3—61no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C22×C9—21no—
330C22×C9—21no—
331C22×C9—21no—
332C22×C9—22no—
333C22×C9—34no—
334C22×C9—41no—
335C22×C9—41no—
336C22×C9—62no—
337C22×C9—62no—
338C22×C9—62no—
339C22×C9—62no—
340C22×C9—62no—
341C22×C9—128no—
342C22×C9#23512yescomplemented
343C22×C9C23×D311yescharacteristic
344C2×D9—21no—
345C2×D9—21no—
346C2×D9—21no—
347C4×C9—21no—
348C4×C9—22no—
349C4×C9—41no—
350C4×C9D3×D412yes—
351Dic9—21no—
352Dic9—21no—
353Dic9—21no—
354Dic9—62no—
355Dic9—62no—
356Dic9—122no—
357Dic9—122no—
358C2×#41—271no—
359C22×D4—31no—
360#124—271no—
361#123—272no—
362C22⋊Q8—271no—
363#121—272no—
364#121—271no—
365#120—272no—
366C4×D4—91no—
367C4×D4—271no—
368C2×C22⋊C4—272no—
369C3×C9#54611yescharacteristic, complemented, Sylow
370C23×C3—21no—
371C23×C3—32no—
372C23×C3—32no—
373C23×C3—64no—
374C23×C3—64no—
375C23×C3—91no—
376C23×C3—122no—
377C23×C3C22×D911yescharacteristic
378C23×C3C2×D3211yescharacteristic
379C22×D3—24no—
380C22×D3—24no—
381C22×D3—31no—
382C22×D3—42no—
383C22×D3C22×D912yes—
384C22×D3C22×D912yes—
385C3×D4—34no—
386C3×D4—64no—
387C3×D4—62no—
388C3×D4—182no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—42no—
394C2×C3×C4—91no—
395C2×C3×C4—91no—
396C2×C3×C4—91no—
397C2×C3×C4—182no—
398C2×C3×C4C22×D911yescharacteristic
399C2×C3×C4C22×D911yescharacteristic
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C2×D3211yescharacteristic
402#67—62no—
403#67—184no—
404#67—184no—
405#67—362no—
406#67—362no—
407C2×Dic3—31no—
408C2×Dic3—31no—
409C2×Dic3—31no—
410C2×Dic3—92no—
411C2×Dic3—92no—
412C2×Dic3—92no—
413C2×Dic3—92no—
414C2×Dic3—182no—
415C2×Dic3—182no—
416C2×Dic3—184no—
417C2×Dic3—362no—
418D12—24no—
419D12—22no—
420D12#61114yescomplemented
421C4×D3—62no—
422C4×D3—184no—
423C4×D3—362no—
424Dic6—182no—
425Dic6—182no—
426Dic6—362no—
427C2×C32—22no—
428C2×C32C2×#23711yescharacteristic
429C2×C32#100411yescharacteristic
430C2×C32C22×#6711yescharacteristic
431C3×D3—24no—
432C3×D3—42no—
433C3×D3—121no—
434C2×C9—22no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—21no—
438C2×C9—42no—
439C2×C9—64no—
440C2×C9—122no—
441C2×C9C2×#23511yescharacteristic
442C2×C9C2×D3×D411yescharacteristic
443C2×C9#100911yescharacteristic
444D9—41no—
445C24—32no—
446#41—544no—
447C2×Q8—271no—
448C2×D4—31no—
449C2×D4—32no—
450C2×D4—31no—
451C2×D4—64no—
452C2×D4—91no—
453C2×D4—272no—
454C2×D4—272no—
455C22×C4—91no—
456C22×C4—272no—
457C22×C4—272no—
458C22×C4D3×D911yescharacteristic
459C4⋊C4—91no—
460C4⋊C4—271no—
461C4⋊C4—272no—
462C22⋊C4—91no—
463C22⋊C4—91no—
464C22⋊C4—272no—
465C22⋊C4—272no—
466C22⋊C4—272no—
467C22⋊C4—272no—
468C22⋊C4—272no—
469C42—91no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—22no—
475C22×C3—21no—
476C22×C3—34no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—62no—
480C22×C3—62no—
481C22×C3—62no—
482C22×C3—62no—
483C22×C3—62no—
484C22×C3—128no—
485C22×C3—181no—
486C22×C3—181no—
487C22×C3—181no—
488C22×C3#369912yescomplemented
489C22×C3C23×D911yescharacteristic
490C22×C3#379512yes—
491C22×C3C22×D3211yescharacteristic
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—22no—
495C2×D3—22no—
496C2×D3—22no—
497C2×D3—48no—
498C2×D3—61no—
499C2×D3—61no—
500C2×D3—61no—
501C2×D3C2×#61114yes—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—22no—
505C3×C4—41no—
506C3×C4—181no—
507C3×C4—181no—
508C3×C4—181no—
509C3×C4C2×#61112yes—
510C3×C4D3×#6712yes—
511Dic3—61no—
512Dic3—61no—
513Dic3—61no—
514Dic3—182no—
515Dic3—182no—
516Dic3—182no—
517Dic3—182no—
518Dic3—362no—
519Dic3—362no—
520C32#580611yescharacteristic
521C9—21no—
522C9#555011yescharacteristic, complemented
523C23—32no—
524C23—32no—
525C23—64no—
526C23—64no—
527C23—91no—
528C23—122no—
529C23C2×D3×D911yescharacteristic
530Q8—542no—
531D4—34no—
532D4—64no—
533D4—62no—
534D4—182no—
535D4—544no—
536D4—1082no—
537C2×C4—22no—
538C2×C4—91no—
539C2×C4—91no—
540C2×C4—91no—
541C2×C4—182no—
542C2×C4—272no—
543C2×C4—272no—
544C2×C4—544no—
545C2×C4—1082no—
546C2×C4C2×D3×D911yescharacteristic
547C2×C4C2×D3×D911yescharacteristic
548C2×C3—22no—
549C2×C3—22no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—42no—
554C2×C3—64no—
555C2×C3—122no—
556C2×C3—361no—
557C2×C3C2×#369911yescharacteristic
558C2×C3#6371011yescharacteristic
559C2×C3C22×#61111yescharacteristic
560C2×C3C2×#379511yescharacteristic
561C2×C3#6430711yescharacteristic
562C2×C3C2×D3×#6711yescharacteristic
563D3—24no—
564D3—42no—
565D3—121no—
566C22—21no—
567C22—21no—
568C22—34no—
569C22—62no—
570C22—62no—
571C22—62no—
572C22—62no—
573C22—62no—
574C22—128no—
575C22—181no—
576C22—181no—
577C22—181no—
578C22#8863812yescomplemented
579C22C22×D3×D911yescharacteristic
580C4—21no—
581C4—181no—
582C4—181no—
583C4—181no—
584C4—542no—
585C4—542no—
586C4D3×#61112yes—
587C3—21no—
588C3#1058990811yescharacteristic, complemented
589C3#1059488611yescharacteristic
590C2—22no—
591C2—64no—
592C2—122no—
593C2—361no—
594C2C2×#8863811yescharacteristic
595C2#1172247611yescharacteristic
596C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#546
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×D12C2semidirect
2. C2×#88638C2semidirect
3. #11721813C2semidirect
4. C3×C4×#611C2semidirect
5. #11721286C2semidirect
6. D3×#3676C2semidirect
7. #11721036C2semidirect
8. D12×Dic9C2semidirect
9. #11721016C2semidirect
10. #11721015C2semidirect
11. #11720964C2semidirect
16. C2×#6226C22semidirect
22. C2×C3×C4×D9C22semidirect
23. C2×#6194C22semidirect
24. C2×#6194C22semidirect
25. C2×D3×Dic9C22semidirect
26. C2×D3×Dic9C22semidirect
29. C4⋊(C3⋊Dic9)C22semidirect
30. #88534C22semidirect
32. C3×(C2×D9)⋊C4C22semidirect
33. C3×Dic9⋊C4C22semidirect
34. C3×C4×Dic9C22semidirect
35. (C2×D3)⋊Dic9C22semidirect
36. (C2×D3)⋊Dic9C22semidirect
37. (C2×D3)⋊Dic9C22semidirect
38. (C2×D3)⋊Dic9C22semidirect
59. C4×#611D3semidirect
81. C9×D12D4semidirect
127. C2×C4×D9C2×D3semidirect
137. (C2×D9)⋊C4C2×D3semidirect
142. Dic9⋊C4C2×D3semidirect
143. C4×Dic9C2×D3semidirect
146. C22×C3×C9#41semidirect
168. C22×D12D9semidirect
342. C22×C9#235semidirect
369. C3×C9#546semidirect
420. D12#611semidirect
488. C22×C3#3699semidirect
522. C9#5550semidirect
578. C22#88638semidirect
588. C3#10589908semidirect

Maximal subgroups

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

subgroupindex
C22×C9×D122
C2×#886382
#117218132
C3×C4×#6112
#117212862
D3×#36762
#117210362
D12×Dic92
#117210162
#117210152
#117209642
#105948863
#105899083