Finite Group Databasebeta

← #13148020

Subgroups — #13148020

Order 1728.

1132 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6192C211yesmaximal, characteristic, complemented
2C2×#88658C211yesmaximal, characteristic, complemented
3#11721728C211yesmaximal, characteristic, complemented
4C3×#63444C211yesmaximal, characteristic, complemented
5C9×#877C211yesmaximal, characteristic, complemented
6#11721278C211yesmaximal, characteristic, complemented
7#11721276C211yesmaximal, characteristic, complemented
8#11721006C212yesmaximal, complemented
9#11721005C212yesmaximal, complemented
10#11720983C212yesmaximal, complemented
11#11720940C212yesmaximal, complemented
12#10594080—31nomaximal
13#10589627—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C23×C3×D9C2211yescharacteristic
16C2×#6233C2211yescharacteristic
17C2×C4⋊(C3⋊D9)C2212yescomplemented
18C2×C3×#611C2211yescharacteristic
19C22×C9×Dic3C2211yescharacteristic
20C2×C3×D36C2212yescomplemented
21C2×#6192—24no—
22C2×#6192C2212yes—
23C2×#6192C2212yes—
24C2×#6192C2211yescharacteristic, complemented
25C2×#6192C2212yescomplemented
26C2×#6192C2212yescomplemented
27C2×(C3⋊D9)⋊C4C2211yescharacteristic, complemented
28#88658—24no—
29C2×Dic3×D9C2211yescharacteristic, complemented
30C2×#6189C2211yescharacteristic, complemented
31C3×C9×C22⋊C4C2211yescharacteristic
32#88536C2212yescomplemented
33C9×#217C2211yescharacteristic
34C3×(C2×D9)⋊C4C2212yescomplemented
35C9×Dic3⋊C4C2212yescomplemented
36C4×C9×Dic3C2212yescomplemented
37#88445C2212yescomplemented
38(C2×D9)⋊Dic3C2212yescomplemented
39C22×#626—31no—
40C2×#3804—31no—
41#64091—31no—
42C3×#882—31no—
43C3×#877D311yescharacteristic
44#63981—31no—
45#63977—31no—
46#63911—32no—
47#63910—32no—
48#63886—32no—
49#63852—32no—
50C2×#3697—31no—
51C22×D36—31no—
52C9×#117—31no—
53#63457—32no—
54#63457—32no—
55#63444—31no—
56#63444—31no—
57#63444—31no—
58#63444—61no—
59#63444D311yescharacteristic, complemented
60#63437—32no—
61#63436—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9—22no—
64C22×C3⋊D9—42no—
65C22×C3⋊D9C2312yes—
66C22×C3⋊D9C2311yescharacteristic
67C22×C3×D9—22no—
68C22×C3×D9—42no—
69C22×C3×D9C2312yes—
70C22×C3×D9C2311yescharacteristic
71C2×C3×C4×C9C2312yes—
72#6233—22no—
73C2×C3⋊Dic9C2311yescharacteristic
74C4⋊(C3⋊D9)—42no—
75C3×#611—22no—
76C2×C9×Dic3D414yescomplemented
77C2×C9×Dic3C2312yes—
78C2×C9×Dic3C2311yescharacteristic
79C2×C9×Dic3C2311yescharacteristic
80C2×C3×Dic9C2311yescharacteristic
81C3×D36—42no—
82#6192—22no—
83#6192—24no—
84#6192—24no—
85#6192—42no—
86#6192—42no—
87(C3⋊D9)⋊C4—22no—
88Dic3×D9—22no—
89#6189—22no—
90#5592—91no—
91#5525—91no—
92C23×C3⋊D3—31no—
93C23×C3×D3—31no—
94C2×#638—31no—
95C2×C4⋊(C3⋊D3)—32no—
96C2×C3×#67—31no—
97C22×C3×Dic3C2×D311yescharacteristic
98C2×C3×D12—32no—
99C2×#627—31no—
100C2×#626—31no—
101C2×#626—32no—
102C2×#626—32no—
103C2×#626—32no—
104C2×#626—32no—
105C2×#626—64no—
106C2×#624—31no—
107#3804—64no—
108C2×D3×Dic3—31no—
109C23×D9—31no—
110C23×D9—61no—
111C23×D9C2×D311yescharacteristic
112C32×C22⋊C4C2×D311yescharacteristic
113#3752—32no—
114C3×#217C2×D311yescharacteristic
115C3×(C2×D3)⋊C4—32no—
116C3×Dic3⋊C4C2×D312yes—
117C3×C4×Dic3C2×D312yes—
118#3722—32no—
119(C2×D3)⋊Dic3—32no—
120C22×C4×C9—31no—
121C2×#611—31no—
122C2×#611—61no—
123C2×#611C2×D311yescharacteristic
124#3697—64no—
125C2×D36—32no—
126C2×D36—32no—
127C2×D36—31no—
128C2×D36—32no—
129C2×D36—32no—
130C2×D36—32no—
131C2×D36—64no—
132C2×D36—62no—
133C2×D36C2×D312yescomplemented
134C2×C4×D9—31no—
135C2×C4×D9—31no—
136C2×Dic18—31no—
137C9×C4⋊C4—32no—
138C9×C22⋊C4—21no—
139C9×C22⋊C4—31no—
140C9×C22⋊C4C2×D311yescharacteristic
141C42×C9—32no—
142(C2×D9)⋊C4—32no—
143(C2×D9)⋊C4—32no—
144(C2×D9)⋊C4—32no—
145(C2×D9)⋊C4—62no—
146(C2×D9)⋊C4C2×D312yescomplemented
147C22×C3×C9—22no—
148C22×C3×C9C2×D412yes—
149C22×C3×C9C2411yescharacteristic
150C2×C3⋊D9—21no—
151C2×C3⋊D9—22no—
152C2×C3⋊D9—22no—
153C2×C3⋊D9—22no—
154C2×C3⋊D9—22no—
155C2×C3⋊D9—44no—
156C2×C3×D9—21no—
157C2×C3×D9—22no—
158C2×C3×D9—22no—
159C2×C3×D9—22no—
160C2×C3×D9—22no—
161C2×C3×D9—44no—
162C3×C4×C9—22no—
163C3⋊Dic9—21no—
164C9×Dic3—22no—
165C9×Dic3C2×D412yes—
166C9×Dic3C2×D412yes—
167C3×Dic9—21no—
168C22×#67—91no—
169C2×#237—91no—
170C2×#235—91no—
171C22×D12—91no—
172C3×#120—91no—
173C3×#117—31no—
174#953—91no—
175#940—92no—
176#939—92no—
177#937—91no—
178#895—92no—
179#895—92no—
180#885—92no—
181#884—92no—
182#882—31no—
183#882—91no—
184#882—91no—
185#882—91no—
186#882—91no—
187#882—181no—
188#877D911yescharacteristic, complemented
189#875—92no—
190#874—92no—
191C23×C32C22×D311yescharacteristic
192C22×C3⋊D3—32no—
193C22×C3⋊D3—31no—
194C22×C3⋊D3—62no—
195C22×C3⋊D3—122no—
196C22×C3×D3—32no—
197C22×C3×D3—31no—
198C22×C3×D3—62no—
199C22×C3×D3—122no—
200C2×C32×C4C22×D312yes—
201#638—62no—
202C2×C3⋊Dic3—31no—
203C4⋊(C3⋊D3)—122no—
204C3×#67—62no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3D1214yes—
207C2×C3×Dic3C22×D312yes—
208C2×C3×Dic3C22×D311yescharacteristic
209C2×C3×Dic3C22×D311yescharacteristic
210C3×D12—122no—
211#627—62no—
212#626—62no—
213#626—64no—
214#626—64no—
215#626—122no—
216#626—122no—
217#624—62no—
218D3×Dic3—62no—
219C23×C9—21no—
220C23×C9C22×D311yescharacteristic
221C22×D9—22no—
222C22×D9—32no—
223C22×D9—31no—
224C22×D9—42no—
225C22×D9—62no—
226C22×D9—61no—
227C22×D9—62no—
228C22×D9—122no—
229C22×D9—122no—
230C22×D9—124no—
231C22×D9C22×D312yes—
232C22×D9C22×D311yescharacteristic
233C2×C4×C9—22no—
234C2×C4×C9—34no—
235C2×C4×C9—31no—
236C2×C4×C9—31no—
237C2×C4×C9—32no—
238C2×C4×C9C22×D312yes—
239#611—22no—
240#611—62no—
241#611—122no—
242C2×Dic9—31no—
243C2×Dic9—61no—
244C2×Dic9C22×D311yescharacteristic
245D36—42no—
246D36—64no—
247D36—64no—
248D36—62no—
249D36—122no—
250D36—124no—
251D36—122no—
252D36—122no—
253C4×D9—62no—
254C4×D9—62no—
255Dic18—62no—
256#535—271noSylow
257C2×C3×C9—22no—
258C2×C3×C9C22×D411yescharacteristic
259C2×C3×C9H(5,𝔽2)12yes—
260C3⋊D9—42no—
261C3⋊D9—41no—
262C3×D9—41no—
263C3×D9—42no—
264C24×C3—91no—
265C23×D3—31no—
266C23×D3—91no—
267C23×D3—91no—
268C23×D3—181no—
269C2×C3×D4—91no—
270C2×C3×D4—92no—
271C22×C3×C4—31no—
272C2×#67—31no—
273C2×#67—91no—
274C2×#67—91no—
275C2×#67—92no—
276C2×#67—92no—
277C2×#67—91no—
278C2×#67—92no—
279C2×#67—92no—
280C2×#67—181no—
281C2×#67—184no—
282C22×Dic3—91no—
283C22×Dic3C2×D911yescharacteristic
284#237—184no—
285#235—184no—
286C2×D12—32no—
287C2×D12—91no—
288C2×D12—92no—
289C2×D12—92no—
290C2×D12—92no—
291C2×D12—92no—
292C2×D12—92no—
293C2×D12—92no—
294C2×D12—182no—
295C2×D12—184no—
296C2×C4×D3—91no—
297C2×C4×D3—91no—
298C2×C4×D3—91no—
299C2×Dic6—91no—
300C2×Dic6—91no—
301C3×C4⋊C4—32no—
302C3×C22⋊C4—21no—
303C3×C22⋊C4—31no—
304C3×C22⋊C4—92no—
305C3×C22⋊C4C2×D911yescharacteristic
306C3×C22⋊C4D3211yescharacteristic
307C3×C42—32no—
308#217—92no—
309#217C2×D911yescharacteristic
310(C2×D3)⋊C4—32no—
311(C2×D3)⋊C4—92no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—92no—
315(C2×D3)⋊C4—92no—
316(C2×D3)⋊C4—182no—
317Dic3⋊C4C2×D912yescomplemented
318C4×Dic3C2×D912yescomplemented
319C22×C32—22no—
320C22×C32C2×D1212yes—
321C22×C32C23×D311yescharacteristic
322C2×C3⋊D3—62no—
323C2×C3⋊D3—61no—
324C2×C3⋊D3—62no—
325C2×C3⋊D3—62no—
326C2×C3⋊D3—62no—
327C2×C3⋊D3—124no—
328C2×C3×D3—61no—
329C2×C3×D3—62no—
330C2×C3×D3—62no—
331C2×C3×D3—62no—
332C2×C3×D3—62no—
333C2×C3×D3—124no—
334C32×C4—22no—
335C3⋊Dic3—61no—
336C3×Dic3—22no—
337C3×Dic3—61no—
338C3×Dic3C2×D1212yes—
339C3×Dic3C2×D1212yes—
340C22×C9—22no—
341C22×C9—21no—
342C22×C9—22no—
343C22×C9—42no—
344C22×C9D3×D412yes—
345C22×C9C23×D311yescharacteristic
346C2×D9—21no—
347C2×D9—22no—
348C2×D9—22no—
349C2×D9—22no—
350C2×D9—22no—
351C2×D9—44no—
352C2×D9—64no—
353C2×D9—61no—
354C2×D9—62no—
355C2×D9—62no—
356C2×D9—62no—
357C2×D9—62no—
358C2×D9—122no—
359C2×D9—122no—
360C2×D9—122no—
361C2×D9—121no—
362C2×D9—124no—
363C2×D9—128no—
364C4×C9—22no—
365C4×C9—32no—
366C4×C9—32no—
367C4×C9—42no—
368C4×C9—62no—
369Dic9—21no—
370Dic9—61no—
371Dic9—121no—
372C2×#41—271no—
373C22×D4—271no—
374#127—272no—
375#124—272no—
376#121—272no—
377#121—272no—
378#120—91no—
379#120—271no—
380#120—271no—
381#120—271no—
382#117—31no—
383C3×C9#53511yescharacteristic, complemented, Sylow
384C23×C3—21no—
385C23×C3—92no—
386C23×C3—91no—
387C23×C3—182no—
388C23×C3—362no—
389C23×C3C22×D911yescharacteristic
390C23×C3C2×D3211yescharacteristic
391C22×D3—32no—
392C22×D3—31no—
393C22×D3—62no—
394C22×D3—92no—
395C22×D3—91no—
396C22×D3—92no—
397C22×D3—91no—
398C22×D3—122no—
399C22×D3—182no—
400C22×D3—181no—
401C22×D3—182no—
402C22×D3—182no—
403C22×D3—362no—
404C22×D3—362no—
405C22×D3—362no—
406C22×D3—364no—
407C3×D4—182no—
408C3×D4—362no—
409C2×C3×C4—22no—
410C2×C3×C4—31no—
411C2×C3×C4—31no—
412C2×C3×C4—34no—
413C2×C3×C4—32no—
414C2×C3×C4—91no—
415C2×C3×C4C22×D912yes—
416C2×C3×C4C2×D3212yes—
417#67—62no—
418#67—182no—
419#67—182no—
420#67—182no—
421#67—184no—
422#67—184no—
423#67—362no—
424#67—362no—
425#67—362no—
426C2×Dic3—31no—
427C2×Dic3—91no—
428C2×Dic3—91no—
429C2×Dic3—181no—
430C2×Dic3—182no—
431C2×Dic3D3614yescomplemented
432C2×Dic3C22×D912yes—
433C2×Dic3C22×D911yescharacteristic
434C2×Dic3C22×D911yescharacteristic
435D12—122no—
436D12—184no—
437D12—184no—
438D12—182no—
439D12—362no—
440D12—362no—
441D12—364no—
442D12—362no—
443D12—362no—
444C4×D3—182no—
445C4×D3—182no—
446C4×D3—182no—
447Dic6—182no—
448Dic6—182no—
449C2×C32—22no—
450C2×C32C22×D1211yescharacteristic
451C2×C32#100912yes—
452C3⋊D3—122no—
453C3⋊D3—121no—
454C3×D3—121no—
455C3×D3—122no—
456C2×C9—22no—
457C2×C9—21no—
458C2×C9—22no—
459C2×C9—42no—
460C2×C9C2×D3×D411yescharacteristic
461C2×C9#100412yes—
462D9—41no—
463D9—42no—
464D9—124no—
465D9—121no—
466D9—122no—
467D9—241no—
468C24—91no—
469C24—271no—
470#41—544no—
471C2×Q8—271no—
472C2×D4—91no—
473C2×D4—92no—
474C2×D4—272no—
475C2×D4—272no—
476C2×D4—271no—
477C2×D4—272no—
478C2×D4—272no—
479C2×D4—271no—
480C2×D4—272no—
481C2×D4—544no—
482C22×C4—31no—
483C22×C4—271no—
484C22×C4—271no—
485C4⋊C4—32no—
486C22⋊C4—31no—
487C22⋊C4—92no—
488C22⋊C4—272no—
489C22⋊C4—272no—
490C22⋊C4—272no—
491C22⋊C4D3×D911yescharacteristic
492C42—32no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—21no—
496C22×C3—22no—
497C22×C3—42no—
498C22×C3—181no—
499C22×C3—182no—
500C22×C3—182no—
501C22×C3—182no—
502C22×C3—182no—
503C22×C3—364no—
504C22×C3C2×D3612yes—
505C22×C3C23×D911yescharacteristic
506C22×C3D3×D1212yes—
507C22×C3C22×D3211yescharacteristic
508C2×D3—61no—
509C2×D3—62no—
510C2×D3—62no—
511C2×D3—62no—
512C2×D3—62no—
513C2×D3—124no—
514C2×D3—184no—
515C2×D3—181no—
516C2×D3—182no—
517C2×D3—182no—
518C2×D3—182no—
519C2×D3—182no—
520C2×D3—181no—
521C2×D3—182no—
522C2×D3—182no—
523C2×D3—182no—
524C2×D3—182no—
525C2×D3—362no—
526C2×D3—362no—
527C2×D3—362no—
528C2×D3—361no—
529C2×D3—364no—
530C2×D3—364no—
531C2×D3—368no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—32no—
535C3×C4—32no—
536C3×C4—42no—
537C3×C4—62no—
538C3×C4—181no—
539Dic3—22no—
540Dic3—61no—
541Dic3—181no—
542Dic3—181no—
543Dic3—361no—
544Dic3C2×D3612yes—
545Dic3C2×D3612yes—
546C32#552511yescharacteristic
547C9—21no—
548C9#559211yescharacteristic, complemented
549C23—92no—
550C23—91no—
551C23—182no—
552C23—272no—
553C23—271no—
554C23—362no—
555C23—542no—
556C23—1082no—
557C23C2×D3×D911yescharacteristic
558Q8—542no—
559D4—182no—
560D4—362no—
561D4—544no—
562D4—544no—
563D4—542no—
564D4—542no—
565D4—1082no—
566D4—1082no—
567D4—1082no—
568C2×C4—31no—
569C2×C4—31no—
570C2×C4—34no—
571C2×C4—32no—
572C2×C4—91no—
573C2×C4—271no—
574C2×C4—542no—
575C2×C4—542no—
576C2×C4C2×D3×D912yes—
577C2×C3—22no—
578C2×C3—22no—
579C2×C3—22no—
580C2×C3—21no—
581C2×C3—42no—
582C2×C3—362no—
583C2×C3—361no—
584C2×C3C22×D3611yescharacteristic
585C2×C3#6371512yes—
586C2×C3C2×D3×D1211yescharacteristic
587C2×C3#6431412yes—
588D3—122no—
589D3—121no—
590D3—364no—
591D3—362no—
592D3—361no—
593D3—362no—
594D3—361no—
595D3—721no—
596C22—22no—
597C22—182no—
598C22—182no—
599C22—182no—
600C22—181no—
601C22—182no—
602C22—364no—
603C22—542no—
604C22—542no—
605C22—542no—
606C22—541no—
607C22—542no—
608C22—1084no—
609C22D3×D3612yes—
610C22C22×D3×D911yescharacteristic
611C4—22no—
612C4—32no—
613C4—32no—
614C4—62no—
615C4—181no—
616C4—541no—
617C3—21no—
618C3#1058962711yescharacteristic, complemented
619C3#1059408011yescharacteristic
620C2—22no—
621C2—362no—
622C2—361no—
623C2—1082no—
624C2—1081no—
625C2C2×D3×D3611yescharacteristic
626C2#1172250112yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#535
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×#6192C2semidirect
2. C2×#88658C2semidirect
3. #11721728C2semidirect
4. C3×#63444C2semidirect
5. C9×#877C2semidirect
6. #11721278C2semidirect
7. #11721276C2semidirect
8. #11721006C2semidirect
9. #11721005C2semidirect
10. #11720983C2semidirect
11. #11720940C2semidirect
17. C2×C4⋊(C3⋊D9)C22semidirect
20. C2×C3×D36C22semidirect
24. C2×#6192C22semidirect
25. C2×#6192C22semidirect
26. C2×#6192C22semidirect
27. C2×(C3⋊D9)⋊C4C22semidirect
29. C2×Dic3×D9C22semidirect
30. C2×#6189C22semidirect
32. #88536C22semidirect
34. C3×(C2×D9)⋊C4C22semidirect
35. C9×Dic3⋊C4C22semidirect
36. C4×C9×Dic3C22semidirect
37. #88445C22semidirect
38. (C2×D9)⋊Dic3C22semidirect
59. #63444D3semidirect
76. C2×C9×Dic3D4semidirect
133. C2×D36C2×D3semidirect
146. (C2×D9)⋊C4C2×D3semidirect
188. #877D9semidirect
317. Dic3⋊C4C2×D9semidirect
318. C4×Dic3C2×D9semidirect
383. C3×C9#535semidirect
431. C2×Dic3D36semidirect
548. C9#5592semidirect
618. C3#10589627semidirect

Maximal subgroups

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

subgroupindex
C22×#61922
C2×#886582
#117217282
C3×#634442
C9×#8772
#117212782
#117212762
#117210062
#117210052
#117209832
#117209402
#105940803
#105896273