Finite Group Databasebeta

← #13151145

Subgroups — #13151145

Order 1728.

1078 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
1C2×D3×D4×C9C211yesmaximal, characteristic, complemented
2C2×#88638C211yesmaximal, characteristic, complemented
3#11721836C211yesmaximal, characteristic, complemented
4C3×#63500C211yesmaximal, characteristic, complemented
5#11721286C212yesmaximal, complemented
6#11721281C212yesmaximal, complemented
7#11721280C212yesmaximal, complemented
8D3×#3676C212yesmaximal, complemented
9C4×#6194C211yesmaximal, characteristic, complemented
10#11721025C211yesmaximal, characteristic, complemented
11#11720961C211yesmaximal, characteristic, complemented
12#10595228—31nomaximal
13#10589908—31nomaximal
14C23×D3×C9C2212yes—
15C2×C3×D4×C9C2211yescharacteristic
16C2×#6226C2211yescharacteristic, complemented
17C2×C9×#67C2212yes—
18C2×C3×#611C2212yescomplemented
19D3×D4×C9—24no—
20C2×C9×D12C2211yescharacteristic
21C2×C4×D3×C9C2211yescharacteristic
22C2×C3×C4×D9C2211yescharacteristic, complemented
23C2×#6194C2211yescharacteristic
24C2×#6194C2212yescomplemented
25C2×D3×Dic9C2212yescomplemented
26C2×D3×Dic9C2212yescomplemented
27#88638—24no—
28#88551C2212yescomplemented
29#88551C2212yescomplemented
30C4×C3⋊Dic9C2211yescharacteristic, complemented
31C3×#3676C2212yescomplemented
32C3×C4⋊Dic9C2211yescharacteristic, complemented
33(C2×D3)⋊Dic9C2212yes—
34(C2×D3)⋊Dic9C2211yescharacteristic
35(C2×D3)⋊Dic9C2212yescomplemented
36(C2×D9)⋊Dic3C2211yescharacteristic, complemented
37#88442C2211yescharacteristic, complemented
38Dic9⋊Dic3C2212yescomplemented
39C2×C3×D3×D4D311yescharacteristic
40C2×#3795—31no—
41#64145—31no—
42C3×#938—31no—
43#63976—32no—
44#63970—32no—
45#63969—32no—
46D3×#217—32no—
47C4×#625—31no—
48#63898—31no—
49#63857—31no—
50C22×D4×C9—31no—
51C2×#3699—31no—
52#63515—32no—
53C2×#3676—32no—
54#63501—32no—
55#63500D311yescharacteristic, complemented
56#63498—31no—
57#63498—61no—
58#63497—32no—
59D4×Dic9—31no—
60C4×#611—31no—
61#63486—31no—
62C23×C3×C9C2312yes—
63C22×D3×C9—24no—
64C22×D3×C9—24no—
65C22×D3×C9—42no—
66C22×D3×C9C2312yes—
67C22×D3×C9C2311yescharacteristic
68C22×C3×D9C2311yescharacteristic
69C3×D4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3⋊Dic9C2311yescharacteristic
72C2×C3⋊Dic9C2312yes—
73C2×C3⋊Dic9C2311yescharacteristic
74#6226—22no—
75C9×#67—24no—
76C3×#611—42no—
77C2×C9×Dic3C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2312yes—
80C9×D12D414yescomplemented
81C4×D3×C9—22no—
82C3×C4×D9—22no—
83#6194—24no—
84#6194—22no—
85D3×Dic9—24no—
86D3×Dic9—42no—
87#5806—91no—
88#5612—91no—
89C23×C3×D3C2×D312yes—
90C2×C32×D4C2×D311yescharacteristic
91C2×#634—31no—
92C2×C3×#67—32no—
93C2×C3×#67C2×D312yes—
94C3×D3×D4—24no—
95C2×C3×D12C2×D311yescharacteristic
96C2×C3×C4×D3—31no—
97C2×C3×C4×D3C2×D311yescharacteristic
98C2×#625—32no—
99C2×#625—31no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102#3795—64no—
103C24×C9—32no—
104#3757—32no—
105#3757—32no—
106C4×C3⋊Dic3—31no—
107C3×#217—32no—
108C3×C4⋊Dic3—31no—
109#3724—31no—
110Dic3⋊Dic3—32no—
111(C2×D3)⋊Dic3—32no—
112(C2×D3)⋊Dic3—31no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—31no—
115C2×D4×C9—21no—
116C2×D4×C9—32no—
117C2×D4×C9—31no—
118C2×D4×C9—64no—
119C2×D4×C9C2×D311yescharacteristic
120C22×C4×C9—31no—
121C2×#611—31no—
122C2×#611—32no—
123C2×#611C2×D312yescomplemented
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—31no—
133#3676—32no—
134#3676—32no—
135#3676—62no—
136#3676—62no—
137#3676C2×D312yescomplemented
138(C2×D9)⋊C4—31no—
139C4⋊Dic9C2×D311yescharacteristic, complemented
140Dic9⋊C4—31no—
141Dic9⋊C4—32no—
142C4×Dic9—31no—
143C4×Dic9—61no—
144C22×C3×C9—22no—
145C22×C3×C9—22no—
146C22×C3×C9—22no—
147C22×C3×C9C2411yescharacteristic
148C2×D3×C9—21no—
149C2×D3×C9—22no—
150C2×D3×C9—22no—
151C2×D3×C9—21no—
152C2×D3×C9—48no—
153C2×D3×C9C2×D414yes—
154C2×D3×C9#4112yescomplemented
155C2×C3×D9—21no—
156C2×C3×D9—21no—
157C2×C3×D9—21no—
158C3×C4×C9C2×D412yes—
159C3⋊Dic9—22no—
160C3⋊Dic9—21no—
161C3⋊Dic9—21no—
162C9×Dic3—21no—
163C3×Dic9—21no—
164C3×Dic9—22no—
165C22×C3×D4—31no—
166C2×#237—91no—
167C2×D3×D4D911yescharacteristic, complemented
168C2×#235—91no—
169C3×#121—91no—
170#953—92no—
171C2×#217—92no—
172#939—92no—
173#938—31no—
174#938—92no—
175#937—92no—
176#936—91no—
177#936—91no—
178#936—181no—
179#935—92no—
180D4×Dic3—91no—
181C4×#67—91no—
182C4×#67—91no—
183#924—91no—
184#897—91no—
185#893—91no—
186#883—92no—
187D3×C22⋊C4—92no—
188C23×C32C22×D312yes—
189C22×C3×D3—24no—
190C22×C3×D3—24no—
191C22×C3×D3—31no—
192C22×C3×D3—42no—
193C22×C3×D3C22×D311yescharacteristic
194C22×C3×D3C22×D312yes—
195C32×D4—22no—
196C2×C32×C4C22×D311yescharacteristic
197C2×C3⋊Dic3—31no—
198C2×C3⋊Dic3—31no—
199C2×C3⋊Dic3—32no—
200#634—62no—
201C3×#67—24no—
202C3×#67—122no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3C22×D311yescharacteristic
206C3×D12#6714yes—
207C3×C4×D3—22no—
208C3×C4×D3—62no—
209#625—62no—
210#625—64no—
211D3×Dic3—64no—
212D3×Dic3—122no—
213C23×C9—22no—
214C23×C9—31no—
215C23×C9—32no—
216C23×C9—64no—
217C23×C9—64no—
218C23×C9—122no—
219C23×C9C22×D312yes—
220C22×D9C22×D311yescharacteristic
221D4×C9—22no—
222D4×C9—34no—
223D4×C9—42no—
224D4×C9—64no—
225C2×C4×C9—21no—
226C2×C4×C9—31no—
227C2×C4×C9—62no—
228C2×C4×C9C22×D311yescharacteristic
229#611—42no—
230#611—64no—
231#611—62no—
232C2×Dic9—31no—
233C2×Dic9—31no—
234C2×Dic9—32no—
235C2×Dic9—61no—
236C2×Dic9—61no—
237C2×Dic9—62no—
238C2×Dic9—64no—
239C2×Dic9—122no—
240C2×Dic9C22×D312yes—
241C2×Dic9C22×D311yescharacteristic
242C4×D9—22no—
243Dic18—62no—
244Dic18—122no—
245#546—271noSylow
246C2×C3×C9—42no—
247C2×C3×C9C22×D411yescharacteristic
248C2×C3×C9C2×#4111yescharacteristic
249C2×C3×C9H(5,𝔽2)11yescharacteristic
250D3×C9—22no—
251D3×C9—24no—
252C3×D9—41no—
253C24×C3—32no—
254C23×D3C2×D912yes—
255C2×C3×D4—21no—
256C2×C3×D4—32no—
257C2×C3×D4—31no—
258C2×C3×D4—64no—
259C2×C3×D4—92no—
260C2×C3×D4C2×D911yescharacteristic
261C2×C3×D4D3211yescharacteristic
262C22×C3×C4—31no—
263C22×C3×C4—91no—
264C2×#67—32no—
265C2×#67—91no—
266C2×#67—92no—
267C2×#67—91no—
268C2×#67—92no—
269C2×#67C2×D912yes—
270C22×Dic3—92no—
271C22×Dic3—92no—
272#237—184no—
273D3×D4—24no—
274#235—184no—
275C2×D12C2×D911yescharacteristic
276C2×C4×D3—31no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3C2×D911yescharacteristic
280C2×Dic6—91no—
281C2×Dic6—91no—
282C2×Dic6—181no—
283C3×C4⋊C4—91no—
284C3×C22⋊C4—92no—
285#217—32no—
286#217—92no—
287#217—91no—
288#217—92no—
289#217—92no—
290#217—92no—
291#217—92no—
292#217—92no—
293#217—91no—
294#217—182no—
295#217—182no—
296(C2×D3)⋊C4—91no—
297(C2×D3)⋊C4—91no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300C4⋊Dic3—31no—
301C4⋊Dic3—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—91no—
304Dic3⋊C4—91no—
305C4×Dic3—91no—
306C4×Dic3—91no—
307C4×Dic3—181no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32C23×D311yescharacteristic
312C2×C3×D3—22no—
313C2×C3×D3—22no—
314C2×C3×D3—21no—
315C2×C3×D3—21no—
316C2×C3×D3—48no—
317C2×C3×D3—61no—
318C2×C3×D3—61no—
319C2×C3×D3—61no—
320C2×C3×D3#23712yes—
321C2×C3×D3C2×#6714yes—
322C32×C4C2×#6712yes—
323C3⋊Dic3—61no—
324C3⋊Dic3—61no—
325C3⋊Dic3—62no—
326C3×Dic3—21no—
327C3×Dic3—62no—
328C3×Dic3—61no—
329C22×C9—22no—
330C22×C9—22no—
331C22×C9—22no—
332C22×C9—21no—
333C22×C9—34no—
334C22×C9—32no—
335C22×C9—42no—
336C22×C9—42no—
337C22×C9—42no—
338C22×C9—61no—
339C22×C9—62no—
340C22×C9—61no—
341C22×C9—62no—
342C22×C9—128no—
343C22×C9C23×D311yescharacteristic
344C2×D9—21no—
345C2×D9—21no—
346C2×D9—21no—
347C4×C9—22no—
348C4×C9—61no—
349C4×C9D3×D412yes—
350Dic9—21no—
351Dic9—22no—
352Dic9—62no—
353Dic9—61no—
354Dic9—61no—
355Dic9—122no—
356Dic9—121no—
357Dic9—121no—
358C2×#41—271no—
359C22×D4—31no—
360#124—271no—
361#123—272no—
362C22⋊Q8—271no—
363#121—91no—
364#121—272no—
365#120—272no—
366C4×D4—271no—
367C4×D4—271no—
368C2×C22⋊C4—272no—
369C3×C9#54611yescharacteristic, complemented, Sylow
370C23×C3—22no—
371C23×C3—31no—
372C23×C3—32no—
373C23×C3—64no—
374C23×C3—64no—
375C23×C3—91no—
376C23×C3—122no—
377C23×C3C22×D912yes—
378C23×C3C2×D3212yes—
379C22×D3—24no—
380C22×D3—24no—
381C22×D3—31no—
382C22×D3—42no—
383C22×D3C22×D911yescharacteristic
384C22×D3C22×D912yes—
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—34no—
388C3×D4—42no—
389C3×D4—64no—
390C3×D4—362no—
391C2×C3×C4—21no—
392C2×C3×C4—31no—
393C2×C3×C4—62no—
394C2×C3×C4—91no—
395C2×C3×C4—92no—
396C2×C3×C4—182no—
397C2×C3×C4C22×D911yescharacteristic
398C2×C3×C4C2×D3211yescharacteristic
399#67—24no—
400#67—122no—
401#67—184no—
402#67—182no—
403#67—182no—
404#67—184no—
405C2×Dic3—32no—
406C2×Dic3—31no—
407C2×Dic3—91no—
408C2×Dic3—91no—
409C2×Dic3—92no—
410C2×Dic3—91no—
411C2×Dic3—91no—
412C2×Dic3—92no—
413C2×Dic3—181no—
414C2×Dic3—181no—
415C2×Dic3—182no—
416C2×Dic3—184no—
417C2×Dic3—362no—
418C2×Dic3C22×D911yescharacteristic
419D12#61114yescomplemented
420C4×D3—22no—
421C4×D3—62no—
422C4×D3—184no—
423C4×D3—362no—
424Dic6—182no—
425Dic6—182no—
426Dic6—362no—
427C2×C32—42no—
428C2×C32C2×#23711yescharacteristic
429C2×C32#100411yescharacteristic
430C2×C32C22×#6711yescharacteristic
431C3×D3—22no—
432C3×D3—24no—
433C3×D3—121no—
434C2×C9—21no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—42no—
438C2×C9—64no—
439C2×C9—62no—
440C2×C9—82no—
441C2×C9C2×D3×D411yescharacteristic
442C2×C9#100411yescharacteristic
443C2×C9D3×#4111yescharacteristic
444D9—41no—
445C24—32no—
446#41—544no—
447C2×Q8—271no—
448C2×D4—32no—
449C2×D4—31no—
450C2×D4—64no—
451C2×D4—92no—
452C2×D4—271no—
453C2×D4—272no—
454C2×D4D3×D911yescharacteristic
455C22×C4—31no—
456C22×C4—91no—
457C22×C4—272no—
458C22×C4—272no—
459C4⋊C4—91no—
460C4⋊C4—271no—
461C4⋊C4—272no—
462C22⋊C4—92no—
463C22⋊C4—271no—
464C22⋊C4—271no—
465C22⋊C4—272no—
466C22⋊C4—272no—
467C22⋊C4—272no—
468C22⋊C4—272no—
469C42—271no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—21no—
477C22×C3—32no—
478C22×C3—34no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—62no—
483C22×C3—62no—
484C22×C3—61no—
485C22×C3—61no—
486C22×C3—128no—
487C22×C3—181no—
488C22×C3—181no—
489C22×C3—181no—
490C22×C3C23×D911yescharacteristic
491C22×C3C22×D3211yescharacteristic
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—21no—
495C2×D3—21no—
496C2×D3—48no—
497C2×D3—61no—
498C2×D3—61no—
499C2×D3—61no—
500C2×D3#369912yescomplemented
501C2×D3C2×#61114yes—
502C3×C4—22no—
503C3×C4—61no—
504C3×C4—182no—
505C3×C4—181no—
506C3×C4C2×#61112yes—
507C3×C4D3×#6712yes—
508Dic3—21no—
509Dic3—62no—
510Dic3—61no—
511Dic3—181no—
512Dic3—181no—
513Dic3—182no—
514Dic3—181no—
515Dic3—181no—
516Dic3—182no—
517Dic3—362no—
518Dic3—361no—
519Dic3—361no—
520C32#580611yescharacteristic
521C9—21no—
522C9#561211yescharacteristic, complemented
523C23—31no—
524C23—32no—
525C23—64no—
526C23—64no—
527C23—91no—
528C23—122no—
529C23C2×D3×D912yes—
530Q8—542no—
531D4—22no—
532D4—34no—
533D4—64no—
534D4—362no—
535D4—544no—
536D4—542no—
537C2×C4—31no—
538C2×C4—62no—
539C2×C4—91no—
540C2×C4—92no—
541C2×C4—182no—
542C2×C4—271no—
543C2×C4—271no—
544C2×C4—272no—
545C2×C4—544no—
546C2×C4—1082no—
547C2×C4C2×D3×D911yescharacteristic
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—42no—
552C2×C3—42no—
553C2×C3—62no—
554C2×C3—64no—
555C2×C3—82no—
556C2×C3—361no—
557C2×C3C2×#369911yescharacteristic
558C2×C3#6371011yescharacteristic
559C2×C3C22×#61111yescharacteristic
560C2×C3D3×#23711yescharacteristic
561C2×C3#6431311yescharacteristic
562C2×C3C2×D3×#6711yescharacteristic
563D3—22no—
564D3—24no—
565D3—121no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—32no—
570C22—34no—
571C22—62no—
572C22—62no—
573C22—61no—
574C22—61no—
575C22—128no—
576C22—181no—
577C22—181no—
578C22—181no—
579C22C22×D3×D911yescharacteristic
580C4—61no—
581C4—182no—
582C4—181no—
583C4—541no—
584C4—541no—
585C4—542no—
586C4D3×#61112yes—
587C3—21no—
588C3#1058990811yescharacteristic, complemented
589C3#1059522811yescharacteristic
590C2—42no—
591C2—62no—
592C2—64no—
593C2—361no—
594C2D3×#369911yescharacteristic
595C2#1172249911yescharacteristic
596C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#546
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D3×D4×C9C2semidirect
2. C2×#88638C2semidirect
3. #11721836C2semidirect
4. C3×#63500C2semidirect
5. #11721286C2semidirect
6. #11721281C2semidirect
7. #11721280C2semidirect
8. D3×#3676C2semidirect
9. C4×#6194C2semidirect
10. #11721025C2semidirect
11. #11720961C2semidirect
16. C2×#6226C22semidirect
18. C2×C3×#611C22semidirect
22. C2×C3×C4×D9C22semidirect
24. C2×#6194C22semidirect
25. C2×D3×Dic9C22semidirect
26. C2×D3×Dic9C22semidirect
28. #88551C22semidirect
29. #88551C22semidirect
30. C4×C3⋊Dic9C22semidirect
31. C3×#3676C22semidirect
32. C3×C4⋊Dic9C22semidirect
35. (C2×D3)⋊Dic9C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. #88442C22semidirect
38. Dic9⋊Dic3C22semidirect
55. #63500D3semidirect
80. C9×D12D4semidirect
123. C2×#611C2×D3semidirect
127. C2×C4×D9C2×D3semidirect
137. #3676C2×D3semidirect
139. C4⋊Dic9C2×D3semidirect
154. C2×D3×C9#41semidirect
167. C2×D3×D4D9semidirect
369. C3×C9#546semidirect
419. D12#611semidirect
500. C2×D3#3699semidirect
522. C9#5612semidirect
588. C3#10589908semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D4×C92
C2×#886382
#117218362
C3×#635002
#117212862
#117212812
#117212802
D3×#36762
C4×#61942
#117210252
#117209612
#105952283
#105899083