Finite Group Databasebeta

← #13151985

Subgroups — #13151985

Order 1728.

1120 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C9×Dic3C211yesmaximal, characteristic, complemented
2C2×#88658C211yesmaximal, characteristic, complemented
3#11721874C211yesmaximal, characteristic, complemented
4C3×#63515C211yesmaximal, characteristic, complemented
5#11721277C212yesmaximal, complemented
6Dic3×#611C212yesmaximal, complemented
7#11721259C211yesmaximal, characteristic, complemented
8#11721251C211yesmaximal, characteristic, complemented
9#11721249C212yesmaximal, complemented
10#11721244C211yesmaximal, characteristic, complemented
11#11721240C212yesmaximal, complemented
12#10595498—31nomaximal
13#10589893—31nomaximal
14C24×C3×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×#6233C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C2×C3×#611C2212yes—
19C22×C9×Dic3—24no—
20C22×C9×Dic3C2212yes—
21C22×C9×Dic3C2212yes—
22C22×C9×Dic3C2211yescharacteristic
23C22×C9×Dic3C2211yescharacteristic
24C2×#6192C2211yescharacteristic, complemented
25C2×(C3⋊D9)⋊C4C2211yescharacteristic, complemented
26#88658—24no—
27C2×Dic3×D9C2211yescharacteristic, complemented
28C2×#6189C2211yescharacteristic, complemented
29#88551C2211yescharacteristic
30#88551C2212yes—
31C3×#3676C2212yes—
32C3×#3676C2211yescharacteristic
33#88445C2212yescomplemented
34(C2×D9)⋊Dic3C2212yescomplemented
35Dic3⋊Dic9C2212yescomplemented
36#88442C2212yescomplemented
37Dic9⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C23×C3×Dic3D311yescharacteristic
40C2×#3804—31no—
41#64162—31no—
42C3×#953—31no—
43#63978—32no—
44Dic3×#67—32no—
45#63966—31no—
46#63964—32no—
47#63961—31no—
48#63955—31no—
49#63952—32no—
50C23×C4×C9—31no—
51C2×#3697—31no—
52#63515—31no—
53#63515—61no—
54#63515D311yescharacteristic, complemented
55#63492—32no—
56#63491—31no—
57#63491—31no—
58C4×#611—32no—
59C4×#611—32no—
60#63488—31no—
61#63486—32no—
62C23×C3×C9—22no—
63C23×C3×C9—24no—
64C23×C3×C9C2311yescharacteristic
65C23×C3×C9C2312yes—
66C22×C3⋊D9C2311yescharacteristic
67C22×C3×D9C2311yescharacteristic
68#6233—22no—
69#6233—24no—
70C2×C3⋊Dic9C2311yescharacteristic
71C2×C3⋊Dic9C2312yes—
72C3×#611—22no—
73C3×#611—24no—
74C2×C9×Dic3—22no—
75C2×C9×Dic3—24no—
76C2×C9×Dic3—24no—
77C2×C9×Dic3D414yescomplemented
78C2×C9×Dic3C2312yes—
79C2×C9×Dic3C2311yescharacteristic
80C2×C9×Dic3C2311yescharacteristic
81C2×C3×Dic9C2311yescharacteristic
82C2×C3×Dic9C2312yes—
83#6192—22no—
84(C3⋊D9)⋊C4—22no—
85Dic3×D9—22no—
86#6189—22no—
87#5791—91no—
88#5586—91no—
89C24×C32C2×D311yescharacteristic
90C2×#638—31no—
91C2×#638—32no—
92C2×C3×#67—31no—
93C2×C3×#67—32no—
94C22×C3×Dic3—24no—
95C22×C3×Dic3C2×D312yes—
96C22×C3×Dic3C2×D311yescharacteristic
97C22×C3×Dic3C2×D312yes—
98C22×C3×Dic3C2×D311yescharacteristic
99C2×#627—31no—
100C2×#626—31no—
101C2×#624—31no—
102#3804—64no—
103C2×D3×Dic3—31no—
104C24×C9—21no—
105C24×C9C2×D311yescharacteristic
106#3757—31no—
107#3757—32no—
108C3×#217—32no—
109C3×#217—31no—
110#3724—32no—
111Dic3⋊Dic3—32no—
112Dic3⋊Dic3—32no—
113#3722—32no—
114(C2×D3)⋊Dic3—32no—
115Dic32—32no—
116C22×C4×C9—31no—
117C22×C4×C9—32no—
118C22×C4×C9—31no—
119C22×C4×C9—32no—
120C22×C4×C9—64no—
121C2×#611—32no—
122C2×#611—31no—
123C2×#611—62no—
124C2×#611—61no—
125C2×#611C2×D312yes—
126C2×#611C2×D311yescharacteristic
127#3697—64no—
128C2×D36—31no—
129C2×C4×D9—31no—
130C2×C4×D9—31no—
131C2×Dic18—31no—
132#3676—31no—
133#3676—32no—
134#3676—62no—
135#3676—61no—
136#3676C2×D311yescharacteristic
137#3676C2×D312yes—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—32no—
140C4⋊Dic9—32no—
141Dic9⋊C4—32no—
142Dic9⋊C4—32no—
143C4×Dic9—32no—
144C22×C3×C9—28no—
145C22×C3×C9—22no—
146C22×C3×C9—22no—
147C22×C3×C9—22no—
148C22×C3×C9C2×D412yes—
149C22×C3×C9#4114yescomplemented
150C22×C3×C9C2411yescharacteristic
151C2×C3⋊D9—21no—
152C2×C3⋊D9—22no—
153C2×C3×D9—21no—
154C2×C3×D9—22no—
155C3⋊Dic9—21no—
156C3⋊Dic9—22no—
157C9×Dic3—22no—
158C9×Dic3C2×D412yes—
159C9×Dic3C2×D412yes—
160C3×Dic9—21no—
161C3×Dic9—22no—
162C23×C3×C4—31no—
163C23×Dic3D911yescharacteristic, complemented
164C2×#237—91no—
165C2×#235—91no—
166C3×#120—91no—
167#953—31no—
168#953—91no—
169#953—91no—
170#953—181no—
171#939—92no—
172#935—91no—
173D4×Dic3—92no—
174#930—92no—
175#929—91no—
176#929—91no—
177C4×#67—92no—
178C4×#67—92no—
179#926—91no—
180#924—92no—
181#886—91no—
182#67⋊C4—92no—
183#878—92no—
184#877—91no—
185C23×C32—24no—
186C23×C32—22no—
187C23×C32C22×D312yes—
188C23×C32C22×D311yescharacteristic
189C22×C3⋊D3—31no—
190C22×C3×D3—31no—
191#638—64no—
192#638—62no—
193C2×C3⋊Dic3—32no—
194C2×C3⋊Dic3—31no—
195C3×#67—64no—
196C3×#67—62no—
197C2×C3×Dic3—22no—
198C2×C3×Dic3—24no—
199C2×C3×Dic3—24no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3#6714yes—
203C2×C3×Dic3C22×D312yes—
204C2×C3×Dic3C22×D311yescharacteristic
205C2×C3×Dic3C22×D311yescharacteristic
206#627—62no—
207#626—62no—
208#624—62no—
209D3×Dic3—62no—
210C23×C9—22no—
211C23×C9—24no—
212C23×C9—22no—
213C23×C9—21no—
214C23×C9—44no—
215C23×C9—42no—
216C23×C9C22×D311yescharacteristic
217C23×C9C22×D312yes—
218C22×D9—31no—
219C22×D9—61no—
220C22×D9C22×D311yescharacteristic
221C2×C4×C9—34no—
222C2×C4×C9—31no—
223C2×C4×C9—31no—
224C2×C4×C9—32no—
225C2×C4×C9—64no—
226C2×C4×C9—64no—
227C2×C4×C9—62no—
228#611—22no—
229#611—24no—
230#611—62no—
231#611—64no—
232#611—124no—
233#611—122no—
234C2×Dic9—32no—
235C2×Dic9—31no—
236C2×Dic9—62no—
237C2×Dic9—61no—
238C2×Dic9C22×D312yes—
239C2×Dic9C22×D311yescharacteristic
240D36—62no—
241C4×D9—62no—
242C4×D9—62no—
243Dic18—62no—
244#525—271noSylow
245C2×C3×C9—22no—
246C2×C3×C9—24no—
247C2×C3×C9C22×D411yescharacteristic
248C2×C3×C9C2×#4112yes—
249C3⋊D9—41no—
250C3×D9—41no—
251C24×C3—21no—
252C24×C3C2×D911yescharacteristic
253C24×C3D3211yescharacteristic
254C2×C3×D4—92no—
255C2×C3×D4—91no—
256C22×C3×C4—31no—
257C22×C3×C4—32no—
258C22×C3×C4—32no—
259C22×C3×C4—31no—
260C22×C3×C4—64no—
261C2×#67—32no—
262C2×#67—31no—
263C2×#67—92no—
264C2×#67—91no—
265C2×#67—92no—
266C2×#67—91no—
267C2×#67—91no—
268C2×#67—182no—
269C2×#67—181no—
270C22×Dic3—24no—
271C22×Dic3—91no—
272C22×Dic3C2×D912yes—
273C22×Dic3C2×D912yes—
274C22×Dic3C2×D911yescharacteristic
275C22×Dic3C2×D911yescharacteristic
276#237—184no—
277#235—184no—
278C2×D12—91no—
279C2×C4×D3—91no—
280C2×C4×D3—91no—
281C2×C4×D3—91no—
282C2×Dic6—91no—
283C2×Dic6—91no—
284C3×C22⋊C4—91no—
285C3×C22⋊C4—92no—
286#217—32no—
287#217—31no—
288#217—91no—
289#217—92no—
290#217—92no—
291#217—91no—
292#217—92no—
293#217—181no—
294#217—182no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298C4⋊Dic3—92no—
299C4⋊Dic3—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304C4×Dic3—92no—
305C4×Dic3—92no—
306C22×C32—28no—
307C22×C32—22no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32#23514yes—
311C22×C32C2×#6712yes—
312C22×C32C23×D311yescharacteristic
313C2×C3⋊D3—62no—
314C2×C3⋊D3—61no—
315C2×C3×D3—62no—
316C2×C3×D3—61no—
317C3⋊Dic3—62no—
318C3⋊Dic3—61no—
319C3×Dic3—22no—
320C3×Dic3—62no—
321C3×Dic3—61no—
322C3×Dic3C2×#6712yes—
323C3×Dic3C2×#6712yes—
324C22×C9—28no—
325C22×C9—22no—
326C22×C9—22no—
327C22×C9—22no—
328C22×C9—21no—
329C22×C9—24no—
330C22×C9—22no—
331C22×C9—42no—
332C22×C9—42no—
333C22×C9—42no—
334C22×C9—48no—
335C22×C9D3×D412yes—
336C22×C9#23714yescomplemented
337C22×C9C23×D311yescharacteristic
338C2×D9—21no—
339C2×D9—22no—
340C2×D9—61no—
341C2×D9—62no—
342C2×D9—122no—
343C2×D9—121no—
344C4×C9—32no—
345C4×C9—32no—
346C4×C9—62no—
347Dic9—21no—
348Dic9—22no—
349Dic9—61no—
350Dic9—62no—
351Dic9—122no—
352Dic9—121no—
353C2×#41—271no—
354C23×C4—31no—
355#123—271no—
356#123—271no—
357C22⋊Q8—272no—
358#121—272no—
359#120—91no—
360#120—271no—
361C4×D4—272no—
362C4×D4—272no—
363#117—271no—
364C3×C9#52511yescharacteristic, complemented, Sylow
365C23×C3—24no—
366C23×C3—22no—
367C23×C3—24no—
368C23×C3—22no—
369C23×C3—22no—
370C23×C3—21no—
371C23×C3—44no—
372C23×C3—42no—
373C23×C3—91no—
374C23×C3C22×D912yes—
375C23×C3C22×D911yescharacteristic
376C23×C3C2×D3212yes—
377C23×C3C2×D3211yescharacteristic
378C22×D3—31no—
379C22×D3—91no—
380C22×D3—91no—
381C22×D3—181no—
382C3×D4—184no—
383C3×D4—182no—
384C2×C3×C4—31no—
385C2×C3×C4—31no—
386C2×C3×C4—34no—
387C2×C3×C4—32no—
388C2×C3×C4—64no—
389C2×C3×C4—64no—
390C2×C3×C4—62no—
391C2×C3×C4—92no—
392C2×C3×C4—91no—
393#67—64no—
394#67—62no—
395#67—184no—
396#67—182no—
397#67—184no—
398#67—182no—
399#67—182no—
400#67—364no—
401#67—362no—
402C2×Dic3—22no—
403C2×Dic3—24no—
404C2×Dic3—24no—
405C2×Dic3—32no—
406C2×Dic3—31no—
407C2×Dic3—92no—
408C2×Dic3—91no—
409C2×Dic3—92no—
410C2×Dic3—91no—
411C2×Dic3—182no—
412C2×Dic3—181no—
413C2×Dic3—182no—
414C2×Dic3#61114yescomplemented
415C2×Dic3C22×D912yes—
416C2×Dic3C22×D911yescharacteristic
417C2×Dic3C22×D911yescharacteristic
418D12—182no—
419C4×D3—182no—
420C4×D3—182no—
421C4×D3—182no—
422Dic6—182no—
423Dic6—182no—
424C2×C32—24no—
425C2×C32—22no—
426C2×C32C2×#23512yes—
427C2×C32C22×#6711yescharacteristic
428C3⋊D3—121no—
429C3×D3—121no—
430C2×C9—22no—
431C2×C9—24no—
432C2×C9—22no—
433C2×C9—21no—
434C2×C9—44no—
435C2×C9—42no—
436C2×C9C2×D3×D411yescharacteristic
437C2×C9C2×#23712yes—
438D9—41no—
439D9—121no—
440D9—241no—
441C24D3×D911yescharacteristic
442#41—544no—
443C2×Q8—271no—
444C2×D4—92no—
445C2×D4—91no—
446C2×D4—271no—
447C2×D4—272no—
448C2×D4—271no—
449C22×C4—31no—
450C22×C4—32no—
451C22×C4—32no—
452C22×C4—31no—
453C22×C4—64no—
454C22×C4—271no—
455C22×C4—271no—
456C4⋊C4—272no—
457C4⋊C4—272no—
458C4⋊C4—272no—
459C22⋊C4—91no—
460C22⋊C4—92no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4—271no—
464C22⋊C4—272no—
465C42—272no—
466C22×C3—28no—
467C22×C3—28no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—21no—
475C22×C3—24no—
476C22×C3—22no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—48no—
481C22×C3—182no—
482C22×C3—181no—
483C22×C3#369714yescomplemented
484C22×C3C2×#61112yes—
485C22×C3C23×D911yescharacteristic
486C22×C3#380414yes—
487C22×C3D3×#6712yes—
488C22×C3C22×D3211yescharacteristic
489C2×D3—62no—
490C2×D3—61no—
491C2×D3—182no—
492C2×D3—181no—
493C2×D3—182no—
494C2×D3—181no—
495C2×D3—362no—
496C2×D3—361no—
497C3×C4—32no—
498C3×C4—32no—
499C3×C4—62no—
500C3×C4—182no—
501C3×C4—181no—
502Dic3—22no—
503Dic3—62no—
504Dic3—61no—
505Dic3—182no—
506Dic3—181no—
507Dic3—182no—
508Dic3—181no—
509Dic3—362no—
510Dic3—361no—
511Dic3C2×#61112yes—
512Dic3C2×#61112yes—
513C32#579111yescharacteristic
514C9—21no—
515C9#558611yescharacteristic, complemented
516C23—24no—
517C23—22no—
518C23—91no—
519C23—271no—
520C23C2×D3×D912yes—
521C23C2×D3×D911yescharacteristic
522Q8—542no—
523D4—184no—
524D4—182no—
525D4—542no—
526D4—544no—
527D4—542no—
528C2×C4—31no—
529C2×C4—31no—
530C2×C4—34no—
531C2×C4—32no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—62no—
535C2×C4—92no—
536C2×C4—91no—
537C2×C4—272no—
538C2×C4—271no—
539C2×C4—542no—
540C2×C4—542no—
541C2×C3—24no—
542C2×C3—22no—
543C2×C3—24no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—21no—
547C2×C3—44no—
548C2×C3—42no—
549C2×C3—361no—
550C2×C3C2×#369712yes—
551C2×C3C22×#61111yescharacteristic
552C2×C3C2×#380412yes—
553C2×C3C2×D3×#6711yescharacteristic
554D3—121no—
555D3—361no—
556D3—361no—
557D3—721no—
558C22—28no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—182no—
563C22—181no—
564C22—542no—
565C22—541no—
566C22#8865814yescomplemented
567C22D3×#61112yes—
568C22C22×D3×D911yescharacteristic
569C4—32no—
570C4—32no—
571C4—62no—
572C4—182no—
573C4—181no—
574C4—542no—
575C4—541no—
576C3—21no—
577C3#1058989311yescharacteristic, complemented
578C3#1059549811yescharacteristic
579C2—24no—
580C2—22no—
581C2—361no—
582C2—1081no—
583C2C2×#8865812yes—
584C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC22×C32C23×D3
hypercenterC24D3×D9
hypocenterC3×C9#525
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C9×Dic3C2semidirect
2. C2×#88658C2semidirect
3. #11721874C2semidirect
4. C3×#63515C2semidirect
5. #11721277C2semidirect
6. Dic3×#611C2semidirect
7. #11721259C2semidirect
8. #11721251C2semidirect
9. #11721249C2semidirect
10. #11721244C2semidirect
11. #11721240C2semidirect
24. C2×#6192C22semidirect
25. C2×(C3⋊D9)⋊C4C22semidirect
27. C2×Dic3×D9C22semidirect
28. C2×#6189C22semidirect
33. #88445C22semidirect
34. (C2×D9)⋊Dic3C22semidirect
35. Dic3⋊Dic9C22semidirect
36. #88442C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
54. #63515D3semidirect
77. C2×C9×Dic3D4semidirect
149. C22×C3×C9#41semidirect
163. C23×Dic3D9semidirect
336. C22×C9#237semidirect
364. C3×C9#525semidirect
414. C2×Dic3#611semidirect
483. C22×C3#3697semidirect
515. C9#5586semidirect
566. C22#88658semidirect
577. C3#10589893semidirect

Maximal subgroups

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

subgroupindex
C23×C9×Dic32
C2×#886582
#117218742
C3×#635152
#117212772
Dic3×#6112
#117212592
#117212512
#117212492
#117212442
#117212402
#105954983
#105898933