Finite Group Databasebeta

← #13150910

Subgroups — #13150910

Order 1728.

1044 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88658C212yesmaximal, complemented
2#11721840C211yesmaximal, characteristic, complemented
3C3×#63502C211yesmaximal, characteristic, complemented
4D4×C9×Dic3C211yesmaximal, characteristic, complemented
5#11721277C212yesmaximal, complemented
6#11721271C212yesmaximal, complemented
7#11721256C212yesmaximal, complemented
8#11721006C211yesmaximal, characteristic, complemented
9#11720998C211yesmaximal, characteristic, complemented
10#11720992C211yesmaximal, characteristic, complemented
11C4×#6189C211yesmaximal, characteristic, complemented
12#10595143—31nomaximal
13#10589655—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×#6233C2212yes—
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C2×C3×#611C2212yes—
19C2×C3×#611C2212yes—
20C22×C9×Dic3C2212yes—
21C2×C3×D36C2211yescharacteristic
22C2×#6192C2212yes—
23C2×#6192C2212yes—
24C2×(C3⋊D9)⋊C4C2212yes—
25#88658—28no—
26C2×Dic3×D9C2212yes—
27C2×#6189C2211yescharacteristic
28C4×C3⋊Dic9C2211yescharacteristic
29C9×#217C2212yes—
30C9×C4⋊Dic3C2211yescharacteristic
31C4×C9×Dic3C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33#88445C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic3⋊Dic9C2211yescharacteristic
36#88442C2211yescharacteristic
37Dic9⋊Dic3C2211yescharacteristic
38C2×#3804—32no—
39#64149—31no—
40C3×#940—31no—
41C3×D4×Dic3D311yescharacteristic
42#63978—32no—
43#63971—32no—
44#63958—32no—
45C4×#627—31no—
46#63911—31no—
47#63898—31no—
48#63894—31no—
49C2×#3697—32no—
50C4×D4×C9—31no—
51#63502—31no—
52#63502—61no—
53#63502D311yescharacteristic, complemented
54#63492—32no—
55#63455—31no—
56#63455—31no—
57#63446—32no—
58#63446—32no—
59#63436—31no—
60C4×Dic18—31no—
61C23×C3×C9C2312yes—
62C22×C3⋊D9C2312yes—
63C22×C3×D9C2312yes—
64C3×D4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66#6233—24no—
67#6233—42no—
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2311yescharacteristic
70C4⋊(C3⋊D9)—22no—
71C3×#611—24no—
72C3×#611—42no—
73C2×C9×Dic3—24no—
74C2×C9×Dic3C2312yes—
75C2×C9×Dic3C2311yescharacteristic
76C2×C9×Dic3C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C3×D36—22no—
80#6192—24no—
81#6192—42no—
82(C3⋊D9)⋊C4—24no—
83Dic3×D9—24no—
84#6189D414yescomplemented
85#5718—91no—
86#5553—91no—
87C2×C32×D4C2×D311yescharacteristic
88C2×#638—32no—
89C2×#638—32no—
90C2×C4⋊(C3⋊D3)—31no—
91C2×C3×#67—32no—
92C2×C3×#67—32no—
93C22×C3×Dic3C2×D312yes—
94C2×C3×D12—31no—
95C2×#627—31no—
96C2×#626—32no—
97C2×#626—32no—
98C2×#624—32no—
99#3804—68no—
100C2×D3×Dic3—32no—
101C4×C3⋊Dic3—31no—
102C3×#217C2×D312yes—
103C3×C4⋊Dic3C2×D311yescharacteristic
104C3×C4×Dic3—31no—
105C3×C4×Dic3C2×D311yescharacteristic
106#3724—31no—
107Dic3⋊Dic3—31no—
108Dic3⋊Dic3—31no—
109#3722—32no—
110(C2×D3)⋊Dic3—32no—
111C2×D4×C9—21no—
112C2×D4×C9C2×D311yescharacteristic
113C22×C4×C9—32no—
114C2×#611—32no—
115C2×#611—32no—
116C2×#611—62no—
117C2×#611—62no—
118C2×#611C2×D312yes—
119C2×#611C2×D312yes—
120#3697—68no—
121C2×D36—32no—
122C2×D36—31no—
123C2×D36—32no—
124C2×D36—61no—
125C2×D36C2×D311yescharacteristic
126C2×C4×D9—32no—
127C2×C4×D9—32no—
128C2×Dic18—31no—
129C9×C4⋊C4—31no—
130C9×C22⋊C4—32no—
131C42×C9—31no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4—32no—
134C4⋊Dic9—31no—
135Dic9⋊C4—31no—
136Dic9⋊C4—31no—
137C4×Dic9—31no—
138C4×Dic9—61no—
139C4×Dic9C2×D311yescharacteristic
140C22×C3×C9—22no—
141C22×C3×C9—22no—
142C22×C3×C9—22no—
143C22×C3×C9C2411yescharacteristic
144C2×C3⋊D9—22no—
145C2×C3⋊D9—22no—
146C2×C3⋊D9—22no—
147C2×C3×D9—22no—
148C2×C3×D9—22no—
149C2×C3×D9—22no—
150C3×C4×C9#4112yescomplemented
151C3⋊Dic9—21no—
152C3⋊Dic9C2×D412yes—
153C9×Dic3—22no—
154C9×Dic3—21no—
155C9×Dic3C2×D412yes—
156C3×Dic9—21no—
157C3×Dic9C2×D412yes—
158C2×#237—92no—
159C2×#235—92no—
160C3×#127—91no—
161C3×C4×D4—31no—
162#940—31no—
163#940—91no—
164#940—91no—
165#940—91no—
166#940—181no—
167#939—92no—
168#939—92no—
169#938—92no—
170D4×Dic3—91no—
171D4×Dic3D911yescharacteristic, complemented
172#930—92no—
173#893—91no—
174#893—91no—
175#884—92no—
176#884—92no—
177#874—91no—
178C4×D12—91no—
179C4×Dic6—91no—
180C4×Dic6—91no—
181C23×C32C22×D312yes—
182C22×C3⋊D3—32no—
183C22×C3×D3—32no—
184C32×D4—22no—
185C2×C32×C4C22×D311yescharacteristic
186#638—64no—
187#638—122no—
188C2×C3⋊Dic3—31no—
189C2×C3⋊Dic3—31no—
190C4⋊(C3⋊D3)—62no—
191C3×#67—64no—
192C3×#67—122no—
193C2×C3×Dic3—24no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3C22×D312yes—
197C2×C3×Dic3C22×D311yescharacteristic
198C2×C3×Dic3C22×D311yescharacteristic
199C3×D12—62no—
200#627—34no—
201#626—64no—
202#626—122no—
203#624—64no—
204D3×Dic3—64no—
205C23×C9—22no—
206C23×C9C22×D312yes—
207C22×D9—32no—
208C22×D9—62no—
209C22×D9C22×D312yes—
210D4×C9—22no—
211D4×C9—42no—
212C2×C4×C9—21no—
213C2×C4×C9—31no—
214C2×C4×C9—31no—
215C2×C4×C9—32no—
216C2×C4×C9—64no—
217C2×C4×C9C22×D311yescharacteristic
218#611—24no—
219#611—42no—
220#611—64no—
221#611—124no—
222#611—122no—
223#611—124no—
224C2×Dic9—31no—
225C2×Dic9—31no—
226C2×Dic9—61no—
227C2×Dic9—61no—
228C2×Dic9C22×D311yescharacteristic
229C2×Dic9C22×D311yescharacteristic
230D36—22no—
231D36—64no—
232D36—62no—
233D36—122no—
234D36—122no—
235C4×D9—64no—
236C4×D9—64no—
237Dic18—34no—
238#550—271noSylow
239C2×C3×C9—42no—
240C2×C3×C9C22×D411yescharacteristic
241C2×C3×C9C2×#4111yescharacteristic
242C2×C3×C9H(5,𝔽2)11yescharacteristic
243C3⋊D9—42no—
244C3×D9—42no—
245C2×C3×D4—21no—
246C2×C3×D4—92no—
247C2×C3×D4—92no—
248C2×C3×D4—91no—
249C2×C3×D4C2×D911yescharacteristic
250C2×C3×D4D3211yescharacteristic
251C22×C3×C4—32no—
252C2×#67—32no—
253C2×#67—32no—
254C2×#67—92no—
255C2×#67—92no—
256C2×#67—92no—
257C2×#67—92no—
258C2×#67—92no—
259C2×#67—92no—
260C2×#67—182no—
261C2×#67—182no—
262C22×Dic3—92no—
263C22×Dic3C2×D912yes—
264#237—188no—
265#235—188no—
266C2×D12—31no—
267C2×D12—92no—
268C2×D12—91no—
269C2×D12—91no—
270C2×D12—92no—
271C2×D12—181no—
272C2×C4×D3—92no—
273C2×C4×D3—92no—
274C2×C4×D3—92no—
275C2×Dic6—91no—
276C2×Dic6—91no—
277C3×C4⋊C4—31no—
278C3×C22⋊C4—32no—
279C3×C42—31no—
280C3×C42—91no—
281#217—92no—
282#217C2×D912yes—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—92no—
286C4⋊Dic3—91no—
287C4⋊Dic3—91no—
288C4⋊Dic3C2×D911yescharacteristic
289Dic3⋊C4—91no—
290Dic3⋊C4—91no—
291Dic3⋊C4—91no—
292Dic3⋊C4—91no—
293C4×Dic3—31no—
294C4×Dic3—91no—
295C4×Dic3—91no—
296C4×Dic3—181no—
297C4×Dic3C2×D911yescharacteristic
298C22×C32—22no—
299C22×C32—22no—
300C22×C32—22no—
301C22×C32C23×D311yescharacteristic
302C2×C3⋊D3—62no—
303C2×C3⋊D3—62no—
304C2×C3⋊D3—62no—
305C2×C3×D3—62no—
306C2×C3×D3—62no—
307C2×C3×D3—62no—
308C32×C4#23512yes—
309C3⋊Dic3—32no—
310C3⋊Dic3—61no—
311C3×Dic3—22no—
312C3×Dic3—21no—
313C3×Dic3—32no—
314C3×Dic3—61no—
315C3×Dic3D3×D412yes—
316C22×C9—22no—
317C22×C9—22no—
318C22×C9—22no—
319C22×C9—21no—
320C22×C9—42no—
321C22×C9—42no—
322C22×C9—42no—
323C22×C9C23×D311yescharacteristic
324C2×D9—22no—
325C2×D9—22no—
326C2×D9—22no—
327C2×D9—62no—
328C2×D9—62no—
329C2×D9—62no—
330C2×D9—122no—
331C2×D9—122no—
332C2×D9—122no—
333C4×C9—22no—
334C4×C9—32no—
335C4×C9—62no—
336C4×C9—61no—
337C4×C9#23712yescomplemented
338Dic9—21no—
339Dic9—32no—
340Dic9—62no—
341Dic9—62no—
342Dic9—61no—
343Dic9D3×D412yes—
344C2×#41—272no—
345#127—91no—
346#127—271no—
347#127—271no—
348#121—272no—
349#121—272no—
350#121—272no—
351C4×Q8—271no—
352C4×D4—31no—
353C4×D4—271no—
354C4×D4—271no—
355C3×C9#55011yescharacteristic, complemented, Sylow
356C23×C3—22no—
357C23×C3—92no—
358C23×C3C22×D912yes—
359C23×C3C2×D3212yes—
360C22×D3—32no—
361C22×D3—92no—
362C22×D3—92no—
363C22×D3—182no—
364C3×D4—22no—
365C3×D4—22no—
366C3×D4—42no—
367C3×D4—184no—
368C3×D4—182no—
369C3×D4—362no—
370C2×C3×C4—21no—
371C2×C3×C4—31no—
372C2×C3×C4—31no—
373C2×C3×C4—32no—
374C2×C3×C4—64no—
375C2×C3×C4—91no—
376C2×C3×C4—91no—
377C2×C3×C4C22×D911yescharacteristic
378C2×C3×C4C2×D3211yescharacteristic
379#67—64no—
380#67—122no—
381#67—184no—
382#67—184no—
383#67—184no—
384#67—364no—
385#67—362no—
386#67—362no—
387#67—362no—
388#67—364no—
389C2×Dic3—24no—
390C2×Dic3—31no—
391C2×Dic3—31no—
392C2×Dic3—91no—
393C2×Dic3—91no—
394C2×Dic3—91no—
395C2×Dic3—91no—
396C2×Dic3—181no—
397C2×Dic3—181no—
398C2×Dic3—184no—
399C2×Dic3C22×D911yescharacteristic
400C2×Dic3C22×D912yes—
401C2×Dic3C22×D911yescharacteristic
402D12—62no—
403D12—184no—
404D12—182no—
405D12—182no—
406D12—362no—
407D12—362no—
408C4×D3—184no—
409C4×D3—184no—
410C4×D3—184no—
411Dic6—94no—
412Dic6—94no—
413C2×C32—42no—
414C2×C32C2×#23511yescharacteristic
415C2×C32C2×D3×D411yescharacteristic
416C2×C32#100911yescharacteristic
417C3⋊D3—122no—
418C3×D3—122no—
419C2×C9—21no—
420C2×C9—21no—
421C2×C9—21no—
422C2×C9—42no—
423C2×C9—82no—
424C2×C9C2×D3×D411yescharacteristic
425C2×C9C2×#23711yescharacteristic
426C2×C9#100411yescharacteristic
427D9—42no—
428D9—122no—
429D9—242no—
430#41—548no—
431C2×Q8—271no—
432C2×D4—92no—
433C2×D4—92no—
434C2×D4—91no—
435C2×D4—272no—
436C2×D4—272no—
437C2×D4—272no—
438C2×D4—272no—
439C2×D4—271no—
440C2×D4D3×D911yescharacteristic
441C22×C4—32no—
442C22×C4—272no—
443C22×C4—272no—
444C4⋊C4—31no—
445C4⋊C4—271no—
446C4⋊C4—271no—
447C4⋊C4—271no—
448C22⋊C4—32no—
449C22⋊C4—272no—
450C22⋊C4—272no—
451C42—31no—
452C42—91no—
453C42—271no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—21no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—182no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3C23×D911yescharacteristic
468C22×C3C22×D3211yescharacteristic
469C2×D3—62no—
470C2×D3—62no—
471C2×D3—62no—
472C2×D3—182no—
473C2×D3—182no—
474C2×D3—182no—
475C2×D3—182no—
476C2×D3—182no—
477C2×D3—182no—
478C2×D3—362no—
479C2×D3—362no—
480C2×D3—362no—
481C3×C4—22no—
482C3×C4—32no—
483C3×C4—62no—
484C3×C4—61no—
485C3×C4—92no—
486C3×C4—181no—
487C3×C4#369712yescomplemented
488C3×C4#380412yes—
489Dic3—22no—
490Dic3—21no—
491Dic3—32no—
492Dic3—61no—
493Dic3—92no—
494Dic3—92no—
495Dic3—182no—
496Dic3—182no—
497Dic3—181no—
498Dic3—181no—
499Dic3D4×D912yes—
500C32#555311yescharacteristic
501C9—21no—
502C9#571811yescharacteristic, complemented
503C23—92no—
504C23—272no—
505C23C2×D3×D912yes—
506Q8—274no—
507D4—22no—
508D4—184no—
509D4—182no—
510D4—362no—
511D4—544no—
512D4—544no—
513D4—542no—
514D4—1082no—
515D4—1082no—
516C2×C4—31no—
517C2×C4—31no—
518C2×C4—32no—
519C2×C4—64no—
520C2×C4—91no—
521C2×C4—91no—
522C2×C4—271no—
523C2×C4—271no—
524C2×C4—544no—
525C2×C4—544no—
526C2×C4C2×D3×D911yescharacteristic
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—42no—
531C2×C3—42no—
532C2×C3—82no—
533C2×C3—362no—
534C2×C3C2×#369711yescharacteristic
535C2×C3C2×D4×D911yescharacteristic
536C2×C3#6371511yescharacteristic
537C2×C3D32×D411yescharacteristic
538C2×C3#6431411yescharacteristic
539C2×C3C2×#380411yescharacteristic
540D3—122no—
541D3—362no—
542D3—362no—
543D3—722no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—182no—
548C22—182no—
549C22—182no—
550C22—542no—
551C22—542no—
552C22—542no—
553C22C22×D3×D911yescharacteristic
554C4—32no—
555C4—61no—
556C4—62no—
557C4—92no—
558C4—181no—
559C4—272no—
560C4—541no—
561C4#8865812yescomplemented
562C3—21no—
563C3#1058965511yescharacteristic, complemented
564C3#1059514311yescharacteristic
565C2—42no—
566C2—362no—
567C2—1082no—
568C2D3×D4×D911yescharacteristic
569C2#1172250111yescharacteristic
570C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#550
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×#88658C2semidirect
2. #11721840C2semidirect
3. C3×#63502C2semidirect
4. D4×C9×Dic3C2semidirect
5. #11721277C2semidirect
6. #11721271C2semidirect
7. #11721256C2semidirect
8. #11721006C2semidirect
9. #11720998C2semidirect
10. #11720992C2semidirect
11. C4×#6189C2semidirect
53. #63502D3semidirect
84. #6189D4semidirect
150. C3×C4×C9#41semidirect
171. D4×Dic3D9semidirect
337. C4×C9#237semidirect
355. C3×C9#550semidirect
487. C3×C4#3697semidirect
502. C9#5718semidirect
561. C4#88658semidirect
563. C3#10589655semidirect

Maximal subgroups

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

subgroupindex
C2×#886582
#117218402
C3×#635022
D4×C9×Dic32
#117212772
#117212712
#117212562
#117210062
#117209982
#117209922
C4×#61892
#105951433
#105896553