Finite Group Databasebeta

← #13148130

Subgroups — #13148130

Order 1728.

1146 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6192C211yesmaximal, characteristic
2C22⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C3×C22⋊C4×D9C211yesmaximal, characteristic, complemented
4C9×#877C211yesmaximal, characteristic, complemented
5#11721274C211yesmaximal, characteristic, complemented
6D9×#217C211yesmaximal, characteristic, complemented
7#11721244C211yesmaximal, characteristic, complemented
8#11721002C212yesmaximal, complemented
9C4×#6192C212yesmaximal, complemented
10#11720978C212yesmaximal, complemented
11#11720977C212yesmaximal, complemented
12#10594182—31nomaximal
13#10589623—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C23×C3×D9C2211yescharacteristic
16C2×C4×C3⋊D9C2212yescomplemented
17C22×C9×Dic3C2211yescharacteristic
18C2×C3×C4×D9C2212yescomplemented
19C2×#6192C418yescomplemented
20C2×#6192C2211yescharacteristic
21C2×#6192C2211yescharacteristic
22C2×#6192C2211yescharacteristic
23C2×#6192C2211yescharacteristic
24C2×(C3⋊D9)⋊C4C2212yescomplemented
25C2×Dic3×D9C2212yescomplemented
26C3×C9×C22⋊C4C2211yescharacteristic
27#88551C2211yescharacteristic
28#88536C2212yescomplemented
29C9×#217C2211yescharacteristic
30C3×#3676C2211yescharacteristic
31C3×(C2×D9)⋊C4C2212yescomplemented
32C9×Dic3⋊C4C2212yescomplemented
33C4×C9×Dic3C2212yescomplemented
34#88445C2212yescomplemented
35(C2×D9)⋊Dic3C2212yescomplemented
36Dic3⋊Dic9C2212yescomplemented
37Dic3×Dic9C2212yescomplemented
38C22×#626—31no—
39C22⋊C4×C3⋊D3—31no—
40C3×D3×C22⋊C4—31no—
41C3×#877D311yescharacteristic
42#63974—31no—
43D3×#217—31no—
44#63955—31no—
45#63904—32no—
46C4×#626—32no—
47#63881—32no—
48#63880—32no—
49C22×D36—31no—
50C9×#117—31no—
51#63488—31no—
52#63455—32no—
53#63455—32no—
54C22⋊C4×D9—31no—
55C22⋊C4×D9—31no—
56C22⋊C4×D9—31no—
57C22⋊C4×D9—61no—
58C22⋊C4×D9D311yescharacteristic, complemented
59C4×D36—32no—
60C4×D36—32no—
61C23×C3×C9C2311yescharacteristic
62C22×C3⋊D9—42no—
63C22×C3⋊D9C2×C414yescomplemented
64C22×C3⋊D9C2311yescharacteristic
65C22×C3⋊D9C2311yescharacteristic
66C22×C3×D9—42no—
67C22×C3×D9C2×C414yescomplemented
68C22×C3×D9C2311yescharacteristic
69C22×C3×D9C2311yescharacteristic
70C2×C3×C4×C9C2312yes—
71C2×C3⋊Dic9C2312yes—
72C4×C3⋊D9—42no—
73C2×C9×Dic3C2×C414yescomplemented
74C2×C9×Dic3C2312yes—
75C2×C9×Dic3C2311yescharacteristic
76C2×C9×Dic3C2311yescharacteristic
77C2×C3×Dic9C2312yes—
78C3×C4×D9—42no—
79#6192C2×C414yescomplemented
80#6192C2×C414yescomplemented
81#6192C2×C414yescomplemented
82#6192C2×C414yescomplemented
83(C3⋊D9)⋊C4—42no—
84Dic3×D9—42no—
85#5521—91no—
86#5486—91no—
87C23×C3⋊D3—31no—
88C23×C3×D3—31no—
89C2×C4×C3⋊D3—32no—
90C22×C3×Dic3C2×D311yescharacteristic
91C2×C3×C4×D3—32no—
92C2×#626—38no—
93C2×#626—31no—
94C2×#626—31no—
95C2×#626—31no—
96C2×#626—31no—
97C2×#624—32no—
98C2×D3×Dic3—32no—
99C23×D9—31no—
100C23×D9—61no—
101C23×D9C2×D311yescharacteristic
102C32×C22⋊C4C2×D311yescharacteristic
103#3757—31no—
104#3752—32no—
105C3×#217—31no—
106C3×#217C2×D311yescharacteristic
107C3×(C2×D3)⋊C4—32no—
108C3×Dic3⋊C4C2×D312yes—
109C3×C4×Dic3C2×D312yes—
110Dic3⋊Dic3—32no—
111#3722—32no—
112(C2×D3)⋊Dic3—32no—
113Dic32—32no—
114C22×C4×C9—31no—
115C2×D36—38no—
116C2×D36—31no—
117C2×D36—31no—
118C2×D36—31no—
119C2×D36—31no—
120C2×C4×D9—32no—
121C2×C4×D9—32no—
122C2×C4×D9—32no—
123C2×C4×D9—62no—
124C2×C4×D9C2×D312yescomplemented
125C9×C4⋊C4—32no—
126C9×C22⋊C4—21no—
127C9×C22⋊C4—31no—
128C9×C22⋊C4C2×D311yescharacteristic
129C42×C9—32no—
130#3676—31no—
131#3676—61no—
132#3676C2×D311yescharacteristic
133(C2×D9)⋊C4—32no—
134(C2×D9)⋊C4—32no—
135(C2×D9)⋊C4—32no—
136(C2×D9)⋊C4—62no—
137(C2×D9)⋊C4C2×D312yescomplemented
138C4⋊Dic9—32no—
139C4×Dic9—32no—
140C22×C3×C9—22no—
141C22×C3×C9C22×C412yes—
142C22×C3×C9C2411yescharacteristic
143C2×C3⋊D9—22no—
144C2×C3⋊D9—22no—
145C2×C3⋊D9—44no—
146C2×C3⋊D9C22×C412yes—
147C2×C3⋊D9C22×C412yes—
148C2×C3×D9—22no—
149C2×C3×D9—22no—
150C2×C3×D9—44no—
151C2×C3×D9C22×C412yes—
152C2×C3×D9C22×C412yes—
153C3×C4×C9—22no—
154C3⋊Dic9—22no—
155C9×Dic3—22no—
156C9×Dic3C22×C412yes—
157C9×Dic3C22×C412yes—
158C3×Dic9—22no—
159C22×#67—91no—
160C22×D12—91no—
161C3×#117—31no—
162C2×C3×C22⋊C4—91no—
163C2×#217—91no—
164C4×#67—92no—
165C4×#67—92no—
166#926—91no—
167#893—92no—
168#893—92no—
169#67⋊C4—92no—
170#67⋊C4—92no—
171D3×C22⋊C4—31no—
172D3×C22⋊C4—91no—
173D3×C22⋊C4—91no—
174D3×C22⋊C4—91no—
175D3×C22⋊C4—91no—
176D3×C22⋊C4—91no—
177D3×C22⋊C4—181no—
178#877—91no—
179#877D911yescharacteristic, complemented
180C4×D12—92no—
181C4×D12—92no—
182C23×C32C22×D311yescharacteristic
183C22×C3⋊D3—31no—
184C22×C3⋊D3—31no—
185C22×C3⋊D3—34no—
186C22×C3⋊D3—122no—
187C22×C3×D3—31no—
188C22×C3×D3—31no—
189C22×C3×D3—34no—
190C22×C3×D3—122no—
191C2×C32×C4C22×D312yes—
192C2×C3⋊Dic3—32no—
193C4×C3⋊D3—122no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3C4×D314yes—
196C2×C3×Dic3C22×D312yes—
197C2×C3×Dic3C22×D311yescharacteristic
198C2×C3×Dic3C22×D311yescharacteristic
199C3×C4×D3—122no—
200#626—34no—
201#626—34no—
202#626—34no—
203#626—34no—
204#624—122no—
205D3×Dic3—122no—
206C23×C9—21no—
207C23×C9C22×D311yescharacteristic
208C22×D9—31no—
209C22×D9—31no—
210C22×D9—34no—
211C22×D9—42no—
212C22×D9—61no—
213C22×D9—61no—
214C22×D9—64no—
215C22×D9—122no—
216C22×D9—124no—
217C22×D9C4×D314yescomplemented
218C22×D9C22×D311yescharacteristic
219C22×D9C22×D311yescharacteristic
220C2×C4×C9—22no—
221C2×C4×C9—34no—
222C2×C4×C9—31no—
223C2×C4×C9—31no—
224C2×C4×C9—32no—
225C2×C4×C9C22×D312yes—
226C2×Dic9—32no—
227C2×Dic9—62no—
228C2×Dic9C22×D312yes—
229D36—34no—
230D36—34no—
231D36—34no—
232D36—34no—
233C4×D9—42no—
234C4×D9—122no—
235C4×D9—122no—
236C4×D9—122no—
237C4×D9—124no—
238#518—271noSylow
239C2×C3×C9—22no—
240C2×C3×C9C23×C411yescharacteristic
241C2×C3×C9H(5,𝔽2)12yes—
242C3⋊D9—22no—
243C3⋊D9—22no—
244C3×D9—22no—
245C3×D9—22no—
246C24×C3—91no—
247C23×D3—31no—
248C23×D3—91no—
249C23×D3—91no—
250C23×D3—181no—
251C22×C3×C4—31no—
252C22×C3×C4—92no—
253C2×#67—98no—
254C2×#67—91no—
255C2×#67—91no—
256C2×#67—91no—
257C2×#67—91no—
258C22×Dic3—92no—
259C22×Dic3C2×D911yescharacteristic
260C2×D12—91no—
261C2×D12—91no—
262C2×D12—91no—
263C2×D12—91no—
264C2×D12—98no—
265C2×C4×D3—32no—
266C2×C4×D3—92no—
267C2×C4×D3—92no—
268C2×C4×D3—92no—
269C2×C4×D3—92no—
270C2×C4×D3—92no—
271C2×C4×D3—182no—
272C3×C4⋊C4—32no—
273C3×C22⋊C4—21no—
274C3×C22⋊C4—31no—
275C3×C22⋊C4—91no—
276C3×C22⋊C4—92no—
277C3×C22⋊C4C2×D911yescharacteristic
278C3×C22⋊C4D3211yescharacteristic
279C3×C42—32no—
280#217—31no—
281#217—91no—
282#217—91no—
283#217—92no—
284#217—181no—
285#217C2×D911yescharacteristic
286(C2×D3)⋊C4—32no—
287(C2×D3)⋊C4—92no—
288(C2×D3)⋊C4—92no—
289(C2×D3)⋊C4—92no—
290(C2×D3)⋊C4—92no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—182no—
293C4⋊Dic3—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4C2×D912yescomplemented
296C4×Dic3—92no—
297C4×Dic3—92no—
298C4×Dic3C2×D912yescomplemented
299C22×C32—22no—
300C22×C32C2×C4×D312yes—
301C22×C32C23×D311yescharacteristic
302C2×C3⋊D3—32no—
303C2×C3⋊D3—32no—
304C2×C3⋊D3—62no—
305C2×C3⋊D3—62no—
306C2×C3⋊D3—124no—
307C2×C3×D3—32no—
308C2×C3×D3—32no—
309C2×C3×D3—62no—
310C2×C3×D3—62no—
311C2×C3×D3—124no—
312C32×C4—22no—
313C3⋊Dic3—62no—
314C3×Dic3—22no—
315C3×Dic3—62no—
316C3×Dic3C2×C4×D312yes—
317C3×Dic3C2×C4×D312yes—
318C22×C9—22no—
319C22×C9—21no—
320C22×C9—22no—
321C22×C9—42no—
322C22×C9C2×C4×D312yes—
323C22×C9C23×D311yescharacteristic
324C2×D9—22no—
325C2×D9—22no—
326C2×D9—32no—
327C2×D9—32no—
328C2×D9—44no—
329C2×D9—62no—
330C2×D9—62no—
331C2×D9—64no—
332C2×D9—62no—
333C2×D9—62no—
334C2×D9—122no—
335C2×D9—124no—
336C2×D9—128no—
337C2×D9C2×C4×D312yes—
338C2×D9C2×C4×D312yes—
339C4×C9—22no—
340C4×C9—32no—
341C4×C9—32no—
342C4×C9—42no—
343C4×C9—62no—
344Dic9—22no—
345Dic9—62no—
346Dic9—122no—
347C22×D4—271no—
348C4×D4—272no—
349C4×D4—272no—
350C4×D4—272no—
351C4×D4—272no—
352#117—31no—
353#117—271no—
354C2×C22⋊C4—91no—
355C2×C22⋊C4—271no—
356C2×C22⋊C4—271no—
357C2×C22⋊C4—271no—
358C3×C9#51811yescharacteristic, complemented, Sylow
359C23×C3—21no—
360C23×C3—91no—
361C23×C3—91no—
362C23×C3—94no—
363C23×C3—362no—
364C23×C3C22×D911yescharacteristic
365C23×C3C2×D3211yescharacteristic
366C22×D3—31no—
367C22×D3—31no—
368C22×D3—34no—
369C22×D3—91no—
370C22×D3—91no—
371C22×D3—91no—
372C22×D3—91no—
373C22×D3—94no—
374C22×D3—94no—
375C22×D3—122no—
376C22×D3—181no—
377C22×D3—181no—
378C22×D3—184no—
379C22×D3—362no—
380C22×D3—362no—
381C22×D3—364no—
382C2×C3×C4—22no—
383C2×C3×C4—31no—
384C2×C3×C4—31no—
385C2×C3×C4—34no—
386C2×C3×C4—32no—
387C2×C3×C4—92no—
388C2×C3×C4—362no—
389C2×C3×C4C22×D912yes—
390C2×C3×C4C2×D3212yes—
391#67—94no—
392#67—94no—
393#67—94no—
394#67—94no—
395C2×Dic3—32no—
396C2×Dic3—92no—
397C2×Dic3—92no—
398C2×Dic3—182no—
399C2×Dic3—362no—
400C2×Dic3C4×D914yescomplemented
401C2×Dic3C22×D912yes—
402C2×Dic3C22×D911yescharacteristic
403C2×Dic3C22×D911yescharacteristic
404D12—94no—
405D12—94no—
406D12—94no—
407D12—94no—
408C4×D3—122no—
409C4×D3—362no—
410C4×D3—362no—
411C4×D3—362no—
412C4×D3—362no—
413C4×D3—362no—
414C4×D3—364no—
415C2×C32—22no—
416C2×C32C22×C4×D311yescharacteristic
417C2×C32#100912yes—
418C3⋊D3—62no—
419C3⋊D3—62no—
420C3×D3—62no—
421C3×D3—62no—
422C2×C9—22no—
423C2×C9—21no—
424C2×C9—22no—
425C2×C9—42no—
426C2×C9C22×C4×D311yescharacteristic
427C2×C9#100412yes—
428D9—22no—
429D9—22no—
430D9—64no—
431D9—62no—
432D9—62no—
433D9—122no—
434C24—91no—
435C24—271no—
436C2×D4—271no—
437C2×D4—271no—
438C2×D4—271no—
439C2×D4—271no—
440C2×D4—278no—
441C22×C4—31no—
442C22×C4—92no—
443C22×C4—272no—
444C22×C4—272no—
445C22×C4—272no—
446C4⋊C4—32no—
447C4⋊C4—272no—
448C22⋊C4—31no—
449C22⋊C4—91no—
450C22⋊C4—92no—
451C22⋊C4—272no—
452C22⋊C4—272no—
453C22⋊C4—271no—
454C22⋊C4—272no—
455C22⋊C4D3×D911yescharacteristic
456C42—32no—
457C42—272no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—21no—
461C22×C3—22no—
462C22×C3—42no—
463C22×C3—92no—
464C22×C3—92no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3—364no—
468C22×C3C2×C4×D912yes—
469C22×C3C23×D911yescharacteristic
470C22×C3C4×D3212yes—
471C22×C3C22×D3211yescharacteristic
472C2×D3—32no—
473C2×D3—32no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—92no—
477C2×D3—92no—
478C2×D3—92no—
479C2×D3—92no—
480C2×D3—124no—
481C2×D3—182no—
482C2×D3—182no—
483C2×D3—184no—
484C2×D3—182no—
485C2×D3—182no—
486C2×D3—182no—
487C2×D3—182no—
488C2×D3—362no—
489C2×D3—364no—
490C2×D3—364no—
491C2×D3—368no—
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—32no—
495C3×C4—32no—
496C3×C4—42no—
497C3×C4—62no—
498C3×C4—182no—
499Dic3—22no—
500Dic3—62no—
501Dic3—182no—
502Dic3—182no—
503Dic3—362no—
504Dic3C2×C4×D912yes—
505Dic3C2×C4×D912yes—
506C32#552111yescharacteristic
507C9—21no—
508C9#548611yescharacteristic, complemented
509C23—91no—
510C23—91no—
511C23—94no—
512C23—271no—
513C23—271no—
514C23—274no—
515C23—362no—
516C23—1082no—
517C23C2×D3×D911yescharacteristic
518D4—274no—
519D4—274no—
520D4—274no—
521D4—274no—
522C2×C4—31no—
523C2×C4—31no—
524C2×C4—34no—
525C2×C4—32no—
526C2×C4—92no—
527C2×C4—272no—
528C2×C4—362no—
529C2×C4—1082no—
530C2×C4—1082no—
531C2×C4—1082no—
532C2×C4C2×D3×D912yes—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—22no—
536C2×C3—21no—
537C2×C3—42no—
538C2×C3—182no—
539C2×C3—182no—
540C2×C3C22×C4×D911yescharacteristic
541C2×C3#6371512yes—
542C2×C3C2×C4×D3211yescharacteristic
543C2×C3#6431412yes—
544D3—62no—
545D3—62no—
546D3—184no—
547D3—182no—
548D3—182no—
549D3—182no—
550D3—182no—
551D3—362no—
552C22—22no—
553C22—92no—
554C22—92no—
555C22—182no—
556C22—182no—
557C22—272no—
558C22—272no—
559C22—364no—
560C22—542no—
561C22—542no—
562C22—1084no—
563C22C4×D3×D912yes—
564C22C22×D3×D911yescharacteristic
565C4—22no—
566C4—32no—
567C4—32no—
568C4—62no—
569C4—182no—
570C4—542no—
571C3—21no—
572C3#1058962311yescharacteristic, complemented
573C3#1059418211yescharacteristic
574C2—22no—
575C2—182no—
576C2—182no—
577C2—542no—
578C2—542no—
579C2C2×C4×D3×D911yescharacteristic
580C2#1172250112yes—
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC22×#6192C2
cosocleC22×C3×C9C24
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#518
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22⋊C4×C3⋊D9C2semidirect
3. C3×C22⋊C4×D9C2semidirect
4. C9×#877C2semidirect
5. #11721274C2semidirect
6. D9×#217C2semidirect
7. #11721244C2semidirect
8. #11721002C2semidirect
9. C4×#6192C2semidirect
10. #11720978C2semidirect
11. #11720977C2semidirect
16. C2×C4×C3⋊D9C22semidirect
18. C2×C3×C4×D9C22semidirect
19. C2×#6192C4semidirect
24. C2×(C3⋊D9)⋊C4C22semidirect
25. C2×Dic3×D9C22semidirect
28. #88536C22semidirect
31. C3×(C2×D9)⋊C4C22semidirect
32. C9×Dic3⋊C4C22semidirect
33. C4×C9×Dic3C22semidirect
34. #88445C22semidirect
35. (C2×D9)⋊Dic3C22semidirect
36. Dic3⋊Dic9C22semidirect
37. Dic3×Dic9C22semidirect
58. C22⋊C4×D9D3semidirect
63. C22×C3⋊D9C2×C4semidirect
67. C22×C3×D9C2×C4semidirect
73. C2×C9×Dic3C2×C4semidirect
79. #6192C2×C4semidirect
80. #6192C2×C4semidirect
81. #6192C2×C4semidirect
82. #6192C2×C4semidirect
124. C2×C4×D9C2×D3semidirect
137. (C2×D9)⋊C4C2×D3semidirect
179. #877D9semidirect
217. C22×D9C4×D3semidirect
295. Dic3⋊C4C2×D9semidirect
298. C4×Dic3C2×D9semidirect
358. C3×C9#518semidirect
400. C2×Dic3C4×D9semidirect
508. C9#5486semidirect
572. C3#10589623semidirect

Maximal subgroups

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

subgroupindex
C22×#61922
C22⋊C4×C3⋊D92
C3×C22⋊C4×D92
C9×#8772
#117212742
D9×#2172
#117212442
#117210022
C4×#61922
#117209782
#117209772
#105941823
#105896233