Finite Group Databasebeta

← #13151149

Subgroups — #13151149

Order 1728.

1068 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C211yesmaximal, characteristic, complemented
2C22×D3×Dic9C211yesmaximal, characteristic, complemented
3#11721835C211yesmaximal, characteristic, complemented
4C9×#937C211yesmaximal, characteristic, complemented
5#11721286C211yesmaximal, characteristic, complemented
6#67×Dic9C212yesmaximal, complemented
7#11721281C212yesmaximal, complemented
8#11721260C211yesmaximal, characteristic, complemented
9#11721245C211yesmaximal, characteristic, complemented
10#11721018C212yesmaximal, complemented
11#11721012C212yesmaximal, complemented
12#10595154—31nomaximal
13#10589688—31nomaximal
14C23×D3×C9C2211yescharacteristic
15C2×C3×D4×C9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C9×#67C2212yes—
18C2×C3×#611C2211yescharacteristic
19C2×C3×#611C2211yescharacteristic, complemented
20C22×C3×Dic9C2211yescharacteristic
21C22×C3×Dic9C2211yescharacteristic, complemented
22C3×#3699—24no—
23C2×C3×C4×D9C2211yescharacteristic, complemented
24C2×C3×Dic18C2211yescharacteristic, complemented
25C2×#6194C2212yes—
26C2×D3×Dic9—24no—
27C2×D3×Dic9C2212yes—
28C2×D3×Dic9C2212yes—
29#88551C2211yescharacteristic
30#88551C2212yescomplemented
31#88534C2212yescomplemented
32C9×#217C2211yescharacteristic
33C9×(C2×D3)⋊C4C2212yes—
34(C2×D3)⋊Dic9C2212yes—
35(C2×D9)⋊Dic3C2212yescomplemented
36Dic3⋊Dic9C2212yescomplemented
37Dic9⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C2×C3×#237—31no—
40C22×D3×Dic3—31no—
41#64144—31no—
42C3×#937D311yescharacteristic
43#63976—31no—
44Dic3×#67—32no—
45#63970—32no—
46#63962—31no—
47#63955—31no—
48#63887—32no—
49#63879—32no—
50C23×Dic9—31no—
51C2×#3699D311yescharacteristic, complemented
52C9×#120—31no—
53#63515—31no—
54#63501—32no—
55#63497—31no—
56#63497—61no—
57D4×Dic9—32no—
58#63448—31no—
59#611⋊C4—32no—
60#63440—32no—
61#63439—31no—
62C23×C3×C9C2311yescharacteristic
63C23×C3×C9C2311yescharacteristic
64C22×D3×C9—22no—
65C22×D3×C9—24no—
66C22×D3×C9C2312yes—
67C22×C3×D9C2311yescharacteristic
68C3×D4×C9—22no—
69C2×C3×C4×C9C2311yescharacteristic
70C2×C3⋊Dic9—22no—
71C2×C3⋊Dic9C2312yes—
72C2×C3⋊Dic9C2312yes—
73C9×#67—24no—
74C3×#611—22no—
75C3×#611—22no—
76C2×C9×Dic3C2312yes—
77C2×C3×Dic9—22no—
78C2×C3×Dic9D414yescomplemented
79C2×C3×Dic9C2311yescharacteristic
80C2×C3×Dic9C2311yescharacteristic
81C2×C3×Dic9C2311yescharacteristic
82C3×C4×D9—22no—
83C3×Dic18—22no—
84#6194—24no—
85D3×Dic9—24no—
86D3×Dic9—24no—
87#5828—91no—
88#5586—91no—
89C23×C3×D3C2×D311yescharacteristic
90C2×C32×D4C2×D311yescharacteristic
91C22×C3⋊Dic3—31no—
92C2×C3×#67—31no—
93C2×C3×#67—31no—
94C2×C3×#67C2×D312yes—
95C22×C3×Dic3—31no—
96C22×C3×Dic3—31no—
97C3×#237—64no—
98C2×C3×C4×D3—31no—
99C2×C3×Dic6—31no—
100C2×#625—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—32no—
103C2×D3×Dic3—64no—
104C24×C9—31no—
105#3757—31no—
106#3757—32no—
107#3750—32no—
108C3×#217C2×D311yescharacteristic
109C3×(C2×D3)⋊C4C2×D312yes—
110Dic3⋊Dic3—32no—
111Dic3⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114Dic32—32no—
115C2×D4×C9—21no—
116C2×D4×C9—32no—
117C2×D4×C9C2×D311yescharacteristic
118C2×#611—32no—
119C2×#611C2×D311yescharacteristic
120C2×#611C2×D311yescharacteristic, complemented
121C22×Dic9—31no—
122C22×Dic9—32no—
123C22×Dic9—32no—
124C22×Dic9—64no—
125C22×Dic9—61no—
126C22×Dic9C2×D311yescharacteristic
127C22×Dic9C2×D311yescharacteristic, complemented
128#3699—24no—
129C2×C4×D9C2×D311yescharacteristic, complemented
130C2×Dic18C2×D311yescharacteristic, complemented
131C9×C22⋊C4—31no—
132C9×C22⋊C4—32no—
133#3676—32no—
134#3676—31no—
135#3676—32no—
136#3676—61no—
137#3676—62no—
138(C2×D9)⋊C4—32no—
139C4⋊Dic9—32no—
140Dic9⋊C4—32no—
141Dic9⋊C4—32no—
142Dic9⋊C4—62no—
143C4×Dic9—32no—
144C22×C3×C9—22no—
145C22×C3×C9—21no—
146C22×C3×C9—22no—
147C22×C3×C9C2×D412yes—
148C22×C3×C9C2411yescharacteristic
149C2×D3×C9—28no—
150C2×D3×C9—22no—
151C2×D3×C9—22no—
152C2×D3×C9#4114yescomplemented
153C2×C3×D9—21no—
154C2×C3×D9—22no—
155C3×C4×C9—21no—
156C3⋊Dic9—22no—
157C3⋊Dic9—22no—
158C9×Dic3—22no—
159C3×Dic9—21no—
160C3×Dic9C2×D412yes—
161C3×Dic9C2×D412yes—
162C2×C3×#41—91no—
163C23×Dic3—91no—
164C2×#237—31no—
165C22×C4×D3—91no—
166C3×#120—31no—
167#953—91no—
168#946—92no—
169#939—92no—
170#938—92no—
171#937—91no—
172#937D911yescharacteristic, complemented
173#935—91no—
174#935—91no—
175#935—91no—
176#935—181no—
177D4×Dic3—92no—
178C4×#67—92no—
179C4×#67—92no—
180#926—91no—
181#886—91no—
182#67⋊C4—92no—
183#878—92no—
184#877—91no—
185C23×C32C22×D311yescharacteristic
186C23×C32C22×D311yescharacteristic
187C22×C3×D3—22no—
188C22×C3×D3—24no—
189C22×C3×D3—31no—
190C22×C3×D3C22×D312yes—
191C32×D4—22no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C3⋊Dic3—32no—
194C2×C3⋊Dic3—32no—
195C2×C3⋊Dic3—62no—
196C3×#67—24no—
197C3×#67—62no—
198C3×#67—62no—
199C2×C3×Dic3—34no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3C22×D312yes—
205C3×C4×D3—62no—
206C3×Dic6—62no—
207#625—64no—
208D3×Dic3—64no—
209D3×Dic3—64no—
210C23×C9—21no—
211C23×C9—21no—
212C23×C9—32no—
213C23×C9—64no—
214C23×C9—62no—
215C23×C9C22×D311yescharacteristic
216C23×C9C22×D311yescharacteristic
217C22×D9C22×D311yescharacteristic
218D4×C9—22no—
219D4×C9—42no—
220D4×C9—64no—
221C2×C4×C9—21no—
222C2×C4×C9—32no—
223C2×C4×C9C22×D311yescharacteristic
224#611—22no—
225#611—22no—
226#611—64no—
227C2×Dic9—22no—
228C2×Dic9—32no—
229C2×Dic9—32no—
230C2×Dic9—62no—
231C2×Dic9—62no—
232C2×Dic9—62no—
233C2×Dic9—64no—
234C2×Dic9—64no—
235C2×Dic9—122no—
236C2×Dic9#6714yescomplemented
237C2×Dic9C22×D311yescharacteristic
238C2×Dic9C22×D311yescharacteristic
239C2×Dic9C22×D311yescharacteristic
240C4×D9—22no—
241Dic18—22no—
242#525—271noSylow
243C2×C3×C9—22no—
244C2×C3×C9—41no—
245C2×C3×C9C22×D411yescharacteristic
246C2×C3×C9C2×#4112yes—
247D3×C9—24no—
248C3×D9—41no—
249C24×C3—31no—
250C23×D3C2×D911yescharacteristic
251C3×#41—184no—
252C2×C3×Q8—91no—
253C2×C3×D4—21no—
254C2×C3×D4—32no—
255C2×C3×D4—91no—
256C2×C3×D4—91no—
257C2×C3×D4C2×D911yescharacteristic
258C2×C3×D4D3211yescharacteristic
259C22×C3×C4—91no—
260C22×C3×C4—91no—
261C22×C3×C4—91no—
262C2×#67—31no—
263C2×#67—31no—
264C2×#67—92no—
265C2×#67—92no—
266C2×#67C2×D912yes—
267C22×Dic3—31no—
268C22×Dic3—31no—
269C22×Dic3—92no—
270C22×Dic3—92no—
271C22×Dic3—91no—
272C22×Dic3—91no—
273C22×Dic3—181no—
274C22×Dic3—184no—
275#237—64no—
276C2×C4×D3—31no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—184no—
280C2×Dic6—31no—
281C3×C22⋊C4—31no—
282C3×C22⋊C4—32no—
283#217—92no—
284#217—91no—
285#217—92no—
286#217—91no—
287#217—92no—
288#217—92no—
289#217—181no—
290#217—182no—
291#217C2×D911yescharacteristic
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4C2×D912yes—
295C4⋊Dic3—92no—
296C4⋊Dic3—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—182no—
302C4×Dic3—92no—
303C4×Dic3—92no—
304C22×C32—22no—
305C22×C32—21no—
306C22×C32—22no—
307C22×C32D3×D412yes—
308C22×C32C23×D311yescharacteristic
309C2×C3×D3—28no—
310C2×C3×D3—22no—
311C2×C3×D3—22no—
312C2×C3×D3—61no—
313C2×C3×D3—62no—
314C2×C3×D3#23714yes—
315C32×C4—21no—
316C3⋊Dic3—62no—
317C3⋊Dic3—62no—
318C3×Dic3—22no—
319C3×Dic3—32no—
320C3×Dic3—32no—
321C3×Dic3—61no—
322C22×C9—22no—
323C22×C9—22no—
324C22×C9—21no—
325C22×C9—21no—
326C22×C9—22no—
327C22×C9—34no—
328C22×C9—42no—
329C22×C9—41no—
330C22×C9—42no—
331C22×C9—68no—
332C22×C9—62no—
333C22×C9—62no—
334C22×C9C2×#6712yes—
335C22×C9C23×D311yescharacteristic
336C2×D9—22no—
337C2×D9—21no—
338C4×C9—21no—
339C4×C9—41no—
340C4×C9—62no—
341Dic9—21no—
342Dic9—62no—
343Dic9—62no—
344Dic9—122no—
345Dic9—122no—
346Dic9C2×#6712yes—
347Dic9C2×#6712yes—
348C2×#41—91no—
349C23×C4—271no—
350#123—271no—
351#123—271no—
352C22⋊Q8—272no—
353#121—272no—
354#120—31no—
355#120—271no—
356C4×D4—272no—
357C4×D4—272no—
358#117—271no—
359C3×C9#52511yescharacteristic, complemented, Sylow
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—32no—
363C23×C3—64no—
364C23×C3—62no—
365C23×C3—91no—
366C23×C3C22×D911yescharacteristic
367C23×C3C22×D911yescharacteristic
368C23×C3C2×D3211yescharacteristic
369C23×C3C2×D3211yescharacteristic
370C22×D3—22no—
371C22×D3—24no—
372C22×D3—31no—
373C22×D3C22×D912yes—
374C3×Q8—182no—
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—42no—
378C3×D4—64no—
379C3×D4—182no—
380C3×D4—182no—
381C2×C3×C4—21no—
382C2×C3×C4—32no—
383C2×C3×C4—94no—
384C2×C3×C4—91no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4—182no—
388C2×C3×C4—182no—
389C2×C3×C4C22×D911yescharacteristic
390C2×C3×C4C2×D3211yescharacteristic
391#67—24no—
392#67—62no—
393#67—62no—
394#67—184no—
395#67—184no—
396C2×Dic3—31no—
397C2×Dic3—31no—
398C2×Dic3—34no—
399C2×Dic3—31no—
400C2×Dic3—62no—
401C2×Dic3—92no—
402C2×Dic3—92no—
403C2×Dic3—92no—
404C2×Dic3—92no—
405C2×Dic3—182no—
406C2×Dic3—182no—
407C2×Dic3—184no—
408C2×Dic3—184no—
409C2×Dic3—182no—
410C2×Dic3—182no—
411C2×Dic3—362no—
412C2×Dic3C22×D912yes—
413C4×D3—62no—
414C4×D3—184no—
415C4×D3—184no—
416Dic6—62no—
417C2×C32—22no—
418C2×C32—41no—
419C2×C32C2×D3×D411yescharacteristic
420C2×C32C2×#23712yes—
421C3×D3—24no—
422C3×D3—121no—
423C2×C9—22no—
424C2×C9—22no—
425C2×C9—21no—
426C2×C9—42no—
427C2×C9—41no—
428C2×C9—64no—
429C2×C9—81no—
430C2×C9D3×#4112yes—
431C2×C9C22×#6711yescharacteristic
432D9—41no—
433C24—31no—
434#41—184no—
435C2×Q8—91no—
436C2×D4—32no—
437C2×D4—91no—
438C2×D4—91no—
439C2×D4—272no—
440C2×D4D3×D911yescharacteristic
441C22×C4—91no—
442C22×C4—91no—
443C22×C4—91no—
444C22×C4—272no—
445C22×C4—272no—
446C22×C4—271no—
447C22×C4—544no—
448C4⋊C4—272no—
449C4⋊C4—272no—
450C4⋊C4—272no—
451C22⋊C4—31no—
452C22⋊C4—32no—
453C22⋊C4—272no—
454C22⋊C4—272no—
455C22⋊C4—271no—
456C22⋊C4—272no—
457C42—272no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—22no—
464C22×C3—21no—
465C22×C3—22no—
466C22×C3—34no—
467C22×C3—42no—
468C22×C3—41no—
469C22×C3—42no—
470C22×C3—68no—
471C22×C3—62no—
472C22×C3—62no—
473C22×C3—181no—
474C22×C3—182no—
475C22×C3D4×D912yes—
476C22×C3C23×D911yescharacteristic
477C22×C3D3×#6712yes—
478C22×C3C22×D3211yescharacteristic
479C2×D3—28no—
480C2×D3—22no—
481C2×D3—22no—
482C2×D3—62no—
483C2×D3—61no—
484C2×D3#369914yescomplemented
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—41no—
488C3×C4—62no—
489C3×C4—92no—
490C3×C4—92no—
491C3×C4—181no—
492Dic3—22no—
493Dic3—32no—
494Dic3—32no—
495Dic3—61no—
496Dic3—182no—
497Dic3—182no—
498Dic3—182no—
499Dic3—182no—
500Dic3—362no—
501Dic3—362no—
502C32#558611yescharacteristic
503C9—21no—
504C9#582811yescharacteristic, complemented
505C23—32no—
506C23—64no—
507C23—62no—
508C23—91no—
509C23C2×D3×D911yescharacteristic
510C23C2×D3×D911yescharacteristic
511Q8—182no—
512D4—22no—
513D4—64no—
514D4—182no—
515D4—182no—
516D4—544no—
517C2×C4—32no—
518C2×C4—91no—
519C2×C4—91no—
520C2×C4—94no—
521C2×C4—91no—
522C2×C4—182no—
523C2×C4—182no—
524C2×C4—272no—
525C2×C4—272no—
526C2×C4—544no—
527C2×C4—544no—
528C2×C4—542no—
529C2×C4C2×D3×D911yescharacteristic
530C2×C3—22no—
531C2×C3—22no—
532C2×C3—22no—
533C2×C3—21no—
534C2×C3—42no—
535C2×C3—41no—
536C2×C3—41no—
537C2×C3—64no—
538C2×C3—81no—
539C2×C3—361no—
540C2×C3C2×D4×D911yescharacteristic
541C2×C3C2×#369912yes—
542C2×C3D3×#23712yes—
543C2×C3C2×D3×#6711yescharacteristic
544D3—24no—
545D3—121no—
546C22—22no—
547C22—22no—
548C22—21no—
549C22—34no—
550C22—68no—
551C22—62no—
552C22—62no—
553C22—182no—
554C22—181no—
555C22D9×#6712yes—
556C22C22×D3×D911yescharacteristic
557C4—21no—
558C4—62no—
559C4—92no—
560C4—92no—
561C4—181no—
562C4—542no—
563C4—542no—
564C3—21no—
565C3#1058968811yescharacteristic, complemented
566C3#1059515411yescharacteristic
567C2—22no—
568C2—41no—
569C2—64no—
570C2—361no—
571C2D3×#369912yes—
572C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#525
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×C3×#3699C2semidirect
2. C22×D3×Dic9C2semidirect
3. #11721835C2semidirect
4. C9×#937C2semidirect
5. #11721286C2semidirect
6. #67×Dic9C2semidirect
7. #11721281C2semidirect
8. #11721260C2semidirect
9. #11721245C2semidirect
10. #11721018C2semidirect
11. #11721012C2semidirect
19. C2×C3×#611C22semidirect
21. C22×C3×Dic9C22semidirect
23. C2×C3×C4×D9C22semidirect
24. C2×C3×Dic18C22semidirect
30. #88551C22semidirect
31. #88534C22semidirect
35. (C2×D9)⋊Dic3C22semidirect
36. Dic3⋊Dic9C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
51. C2×#3699D3semidirect
78. C2×C3×Dic9D4semidirect
120. C2×#611C2×D3semidirect
127. C22×Dic9C2×D3semidirect
129. C2×C4×D9C2×D3semidirect
130. C2×Dic18C2×D3semidirect
152. C2×D3×C9#41semidirect
172. #937D9semidirect
236. C2×Dic9#67semidirect
359. C3×C9#525semidirect
484. C2×D3#3699semidirect
504. C9#5828semidirect
565. C3#10589688semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C22×D3×Dic92
#117218352
C9×#9372
#117212862
#67×Dic92
#117212812
#117212602
#117212452
#117210182
#117210122
#105951543
#105896883