Finite Group Databasebeta

← #13177652

Subgroups — #13177652

Order 1728.

2304 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×#89031C212yesmaximal, complemented
2#11723981C211yesmaximal, characteristic, complemented
3C3×#64149C211yesmaximal, characteristic, complemented
4C32×D4×Dic3C211yesmaximal, characteristic, complemented
5#11723632C212yesmaximal, complemented
6#11723624C212yesmaximal, complemented
7#11723623C212yesmaximal, complemented
8C4×#6293C211yesmaximal, characteristic, complemented
9#11723517C211yesmaximal, characteristic, complemented
10#11723501C211yesmaximal, characteristic, complemented
11#11723494C211yesmaximal, characteristic, complemented
12#10596024—31nomaximal
13#10595143—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×#6312C2212yes—
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C2×C3×#638C2212yes—
19C2×C3×#638C2212yes—
20C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
21C22×C32×Dic3C2212yes—
22C2×#6293C2211yescharacteristic
23C2×#6292C2212yes—
24C2×#6292C2212yes—
25C2×#6289C2212yes—
26#89031—28no—
27C2×Dic3×C3⋊D3C2212yes—
28C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
29C3×C4×C3⋊Dic3C2211yescharacteristic
30C32×#217C2212yes—
31C32×C4⋊Dic3C2211yescharacteristic
32C32×C4×Dic3C2211yescharacteristic
33#88807C2211yescharacteristic
34#88806C2211yescharacteristic
35Dic3⋊(C3⋊Dic3)C2211yescharacteristic
36#88804C2212yes—
37#88803C2212yes—
38C2×#3828—32no—
39C2×#3804—38no—
40C32×C4×D4—31no—
41#64149—31no—
42#64149—34no—
43#64149—64no—
44#64149D311yescharacteristic, complemented
45#64139—32no—
46#64102—31no—
47#64102—31no—
48#64093—32no—
49#64093—32no—
50#64083—31no—
51C4×#634—31no—
52C3×#940—34no—
53C3×D4×Dic3D314yescomplemented
54#63978—38no—
55#63971—38no—
56#63958—38no—
57C4×#627—34no—
58#63911—34no—
59#63898—34no—
60#63894—34no—
61C23×C33C2312yes—
62C22×C3⋊(C3⋊D3)C2312yes—
63C22×C3×C3⋊D3C2312yes—
64C33×D4—22no—
65C2×C33×C4C2311yescharacteristic
66#6312—24no—
67#6312—42no—
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C4⋊(C3⋊(C3⋊D3))—22no—
71C3×#638—24no—
72C3×#638—42no—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C3×C4⋊(C3⋊D3)—22no—
76C2×C32×Dic3—24no—
77C2×C32×Dic3C2311yescharacteristic
78C2×C32×Dic3C2312yes—
79C2×C32×Dic3C2311yescharacteristic
80#6293D414yescomplemented
81#6292—24no—
82#6292—42no—
83#6289—24no—
84Dic3×C3⋊D3—24no—
85#5718—91no—
86#5553—94no—
87C2×C32×D4—24no—
88C2×C32×D4C2×D314yes—
89C2×C32×D4C2×D311yescharacteristic
90C22×C32×C4—32no—
91C2×#638—32no—
92C2×#638—38no—
93C2×#638—32no—
94C2×#638—38no—
95C2×#638—68no—
96C2×#638—68no—
97C2×#638C2×D312yes—
98C2×#638C2×D312yes—
99#3828—68no—
100C2×C4⋊(C3⋊D3)—32no—
101C2×C4⋊(C3⋊D3)—31no—
102C2×C4⋊(C3⋊D3)—34no—
103C2×C4⋊(C3⋊D3)—32no—
104C2×C4⋊(C3⋊D3)—64no—
105C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
106C2×C4×C3⋊D3—32no—
107C2×C4×C3⋊D3—32no—
108C2×#634—31no—
109C2×C3×#67—38no—
110C2×C3×#67—38no—
111C22×C3×Dic3C2×D318yes—
112C2×C3×D12—34no—
113C2×#627—34no—
114C2×#626—38no—
115C2×#626—38no—
116C2×#624—38no—
117#3804—632no—
118C2×D3×Dic3—38no—
119C32×C4⋊C4—31no—
120C32×C22⋊C4—32no—
121C32×C42—31no—
122#3752—32no—
123#3752—32no—
124C4⋊(C3⋊Dic3)—31no—
125#3750—31no—
126#3750—31no—
127C4×C3⋊Dic3—31no—
128C4×C3⋊Dic3—34no—
129C4×C3⋊Dic3—64no—
130C4×C3⋊Dic3C2×D311yescharacteristic
131C3×#217C2×D318yes—
132C3×C4⋊Dic3C2×D314yes—
133C3×C4×Dic3—34no—
134C3×C4×Dic3C2×D314yes—
135#3724—34no—
136Dic3⋊Dic3—34no—
137Dic3⋊Dic3—34no—
138#3722—38no—
139(C2×D3)⋊Dic3—38no—
140C22×C33—22no—
141C22×C33—22no—
142C22×C33—22no—
143C22×C33C2411yescharacteristic
144C2×C3⋊(C3⋊D3)—22no—
145C2×C3⋊(C3⋊D3)—22no—
146C2×C3⋊(C3⋊D3)—22no—
147C2×C3×C3⋊D3—22no—
148C2×C3×C3⋊D3—22no—
149C2×C3×C3⋊D3—22no—
150C33×C4#4112yescomplemented
151C3⋊(C3⋊Dic3)—21no—
152C3⋊(C3⋊Dic3)C2×D412yes—
153C3×C3⋊Dic3—21no—
154C3×C3⋊Dic3C2×D412yes—
155C32×Dic3—22no—
156C32×Dic3—21no—
157C32×Dic3C2×D412yes—
158C2×#237—92no—
159C2×#235—98no—
160C3×#127—91no—
161C3×C4×D4—34no—
162#940—34no—
163#940—94no—
164#940—91no—
165#940—91no—
166#940—184no—
167#939—92no—
168#939—92no—
169#938—92no—
170D4×Dic3—91no—
171D4×Dic3C3⋊D311yescharacteristic, complemented
172#930—98no—
173#893—94no—
174#893—94no—
175#884—98no—
176#884—98no—
177#874—94no—
178C4×D12—91no—
179C4×Dic6—94no—
180C4×Dic6—91no—
181C23×C32—28no—
182C23×C32C22×D318yes—
183C23×C32C22×D312yes—
184C22×C3⋊D3—32no—
185C22×C3⋊D3—38no—
186C22×C3⋊D3—68no—
187C22×C3⋊D3C22×D312yes—
188C22×C3×D3—38no—
189C32×D4—22no—
190C32×D4—28no—
191C32×D4—48no—
192C2×C32×C4—24no—
193C2×C32×C4—31no—
194C2×C32×C4—31no—
195C2×C32×C4—32no—
196C2×C32×C4—64no—
197C2×C32×C4C22×D314yes—
198C2×C32×C4C22×D311yescharacteristic
199#638—24no—
200#638—42no—
201#638—64no—
202#638—616no—
203#638—1216no—
204#638—122no—
205#638—128no—
206#638—1216no—
207C2×C3⋊Dic3—31no—
208C2×C3⋊Dic3—31no—
209C2×C3⋊Dic3—34no—
210C2×C3⋊Dic3—34no—
211C2×C3⋊Dic3—64no—
212C2×C3⋊Dic3—64no—
213C2×C3⋊Dic3C22×D311yescharacteristic
214C2×C3⋊Dic3C22×D311yescharacteristic
215C4⋊(C3⋊D3)—22no—
216C4⋊(C3⋊D3)—64no—
217C4⋊(C3⋊D3)—62no—
218C4⋊(C3⋊D3)—68no—
219C4⋊(C3⋊D3)—128no—
220C4⋊(C3⋊D3)—122no—
221C4×C3⋊D3—64no—
222C4×C3⋊D3—64no—
223#634—34no—
224C3×#67—616no—
225C3×#67—128no—
226C2×C3×Dic3—216no—
227C2×C3×Dic3—34no—
228C2×C3×Dic3—34no—
229C2×C3×Dic3C22×D318yes—
230C2×C3×Dic3C22×D314yes—
231C2×C3×Dic3C22×D314yes—
232C3×D12—68no—
233#627—316no—
234#626—616no—
235#626—128no—
236#624—616no—
237D3×Dic3—616no—
238#550—271noSylow
239C2×C33—42no—
240C2×C33C22×D411yescharacteristic
241C2×C33C2×#4111yescharacteristic
242C2×C33H(5,𝔽2)11yescharacteristic
243C3⋊(C3⋊D3)—42no—
244C3×C3⋊D3—42no—
245C2×C3×D4—24no—
246C2×C3×D4—92no—
247C2×C3×D4—92no—
248C2×C3×D4—91no—
249C2×C3×D4D3214yes—
250C2×C3×D4C2×C3⋊D311yescharacteristic
251C22×C3×C4—38no—
252C2×#67—38no—
253C2×#67—38no—
254C2×#67—98no—
255C2×#67—92no—
256C2×#67—98no—
257C2×#67—92no—
258C2×#67—92no—
259C2×#67—92no—
260C2×#67—188no—
261C2×#67—188no—
262C22×Dic3—92no—
263C22×Dic3C2×C3⋊D312yes—
264#237—188no—
265#235—1832no—
266C2×D12—34no—
267C2×D12—98no—
268C2×D12—94no—
269C2×D12—91no—
270C2×D12—98no—
271C2×D12—184no—
272C2×C4×D3—98no—
273C2×C4×D3—98no—
274C2×C4×D3—92no—
275C2×Dic6—94no—
276C2×Dic6—91no—
277C3×C4⋊C4—34no—
278C3×C22⋊C4—38no—
279C3×C42—34no—
280C3×C42—91no—
281#217—92no—
282#217C2×C3⋊D312yes—
283(C2×D3)⋊C4—98no—
284(C2×D3)⋊C4—98no—
285(C2×D3)⋊C4—92no—
286C4⋊Dic3—94no—
287C4⋊Dic3—91no—
288C4⋊Dic3C2×C3⋊D311yescharacteristic
289Dic3⋊C4—94no—
290Dic3⋊C4—94no—
291Dic3⋊C4—91no—
292Dic3⋊C4—91no—
293C4×Dic3—34no—
294C4×Dic3—94no—
295C4×Dic3—91no—
296C4×Dic3—184no—
297C4×Dic3C2×C3⋊D311yescharacteristic
298C22×C32—22no—
299C22×C32—22no—
300C22×C32—22no—
301C22×C32—24no—
302C22×C32—28no—
303C22×C32—28no—
304C22×C32—28no—
305C22×C32—48no—
306C22×C32—48no—
307C22×C32—48no—
308C22×C32C23×D314yes—
309C22×C32C23×D311yescharacteristic
310C2×C3⋊D3—22no—
311C2×C3⋊D3—22no—
312C2×C3⋊D3—22no—
313C2×C3⋊D3—62no—
314C2×C3⋊D3—62no—
315C2×C3⋊D3—62no—
316C2×C3⋊D3—68no—
317C2×C3⋊D3—68no—
318C2×C3⋊D3—68no—
319C2×C3⋊D3—128no—
320C2×C3⋊D3—128no—
321C2×C3⋊D3—128no—
322C2×C3×D3—68no—
323C2×C3×D3—68no—
324C2×C3×D3—68no—
325C32×C4—28no—
326C32×C4—32no—
327C32×C4—62no—
328C32×C4—61no—
329C32×C4#23518yescomplemented
330C32×C4#23712yescomplemented
331C3⋊Dic3—21no—
332C3⋊Dic3—32no—
333C3⋊Dic3—38no—
334C3⋊Dic3—68no—
335C3⋊Dic3—68no—
336C3⋊Dic3—61no—
337C3⋊Dic3—64no—
338C3⋊Dic3D3×D412yes—
339C3×Dic3—28no—
340C3×Dic3—24no—
341C3×Dic3—38no—
342C3×Dic3—64no—
343C3×Dic3D3×D418yes—
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—
355C33#55011yescharacteristic, complemented, Sylow
356C23×C3—28no—
357C23×C3—92no—
358C23×C3C2×D3218yes—
359C23×C3C22×C3⋊D312yes—
360C22×D3—38no—
361C22×D3—98no—
362C22×D3—92no—
363C22×D3—188no—
364C3×D4—28no—
365C3×D4—22no—
366C3×D4—48no—
367C3×D4—184no—
368C3×D4—182no—
369C3×D4—362no—
370C2×C3×C4—24no—
371C2×C3×C4—34no—
372C2×C3×C4—34no—
373C2×C3×C4—38no—
374C2×C3×C4—616no—
375C2×C3×C4—91no—
376C2×C3×C4—91no—
377C2×C3×C4C2×D3214yes—
378C2×C3×C4C22×C3⋊D311yescharacteristic
379#67—616no—
380#67—128no—
381#67—184no—
382#67—1816no—
383#67—184no—
384#67—3616no—
385#67—362no—
386#67—368no—
387#67—362no—
388#67—3616no—
389C2×Dic3—24no—
390C2×Dic3—34no—
391C2×Dic3—34no—
392C2×Dic3—94no—
393C2×Dic3—91no—
394C2×Dic3—94no—
395C2×Dic3—91no—
396C2×Dic3—184no—
397C2×Dic3—184no—
398C2×Dic3—184no—
399C2×Dic3C22×C3⋊D311yescharacteristic
400C2×Dic3C22×C3⋊D312yes—
401C2×Dic3C22×C3⋊D311yescharacteristic
402D12—68no—
403D12—1816no—
404D12—188no—
405D12—182no—
406D12—368no—
407D12—368no—
408C4×D3—184no—
409C4×D3—1816no—
410C4×D3—1816no—
411Dic6—916no—
412Dic6—94no—
413C2×C32—24no—
414C2×C32—24no—
415C2×C32—24no—
416C2×C32—42no—
417C2×C32—48no—
418C2×C32—88no—
419C2×C32C2×#23514yes—
420C2×C32C2×D3×D414yes—
421C2×C32C2×D3×D411yescharacteristic
422C2×C32C2×#23711yescharacteristic
423C2×C32#100411yescharacteristic
424C2×C32#100914yes—
425C3⋊D3—42no—
426C3⋊D3—122no—
427C3⋊D3—128no—
428C3⋊D3—248no—
429C3×D3—128no—
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×C3⋊D311yescharacteristic
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—28no—
455C22×C3—28no—
456C22×C3—28no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—24no—
461C22×C3—48no—
462C22×C3—48no—
463C22×C3—48no—
464C22×C3—182no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3C22×D3214yes—
468C22×C3C23×C3⋊D311yescharacteristic
469C2×D3—68no—
470C2×D3—68no—
471C2×D3—68no—
472C2×D3—188no—
473C2×D3—188no—
474C2×D3—188no—
475C2×D3—182no—
476C2×D3—182no—
477C2×D3—182no—
478C2×D3—368no—
479C2×D3—368no—
480C2×D3—368no—
481C3×C4—28no—
482C3×C4—38no—
483C3×C4—68no—
484C3×C4—64no—
485C3×C4—92no—
486C3×C4—181no—
487C3×C4#380418yescomplemented
488C3×C4#382812yescomplemented
489Dic3—22no—
490Dic3—21no—
491Dic3—38no—
492Dic3—64no—
493Dic3—98no—
494Dic3—92no—
495Dic3—188no—
496Dic3—188no—
497Dic3—184no—
498Dic3—181no—
499Dic3D4×C3⋊D312yes—
500C32—24no—
501C32#555314yescomplemented
502C32#571811yescharacteristic, complemented
503C23—92no—
504C23—272no—
505C23C2×D3×C3⋊D312yes—
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×C3⋊D311yescharacteristic
527C2×C3—24no—
528C2×C3—24no—
529C2×C3—24no—
530C2×C3—48no—
531C2×C3—42no—
532C2×C3—88no—
533C2×C3—362no—
534C2×C3D32×D414yes—
535C2×C3#6431414yes—
536C2×C3C2×#380414yes—
537C2×C3C2×#382811yescharacteristic
538C2×C3C2×D4×C3⋊D311yescharacteristic
539C2×C3#6436611yescharacteristic
540D3—128no—
541D3—368no—
542D3—362no—
543D3—728no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—182no—
548C22—182no—
549C22—182no—
550C22—542no—
551C22—542no—
552C22—542no—
553C22C22×D3×C3⋊D311yescharacteristic
554C4—32no—
555C4—62no—
556C4—61no—
557C4—92no—
558C4—181no—
559C4—272no—
560C4—541no—
561C4#8903112yescomplemented
562C3—24no—
563C3#1059514314yescomplemented
564C3#1059602411yescharacteristic, complemented
565C2—42no—
566C2—362no—
567C2—1082no—
568C2D3×D4×C3⋊D311yescharacteristic
569C2#1172458911yescharacteristic
570C2C2×#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#550
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89031C2semidirect
2. #11723981C2semidirect
3. C3×#64149C2semidirect
4. C32×D4×Dic3C2semidirect
5. #11723632C2semidirect
6. #11723624C2semidirect
7. #11723623C2semidirect
8. C4×#6293C2semidirect
9. #11723517C2semidirect
10. #11723501C2semidirect
11. #11723494C2semidirect
44. #64149D3semidirect
53. C3×D4×Dic3D3semidirect
80. #6293D4semidirect
150. C33×C4#41semidirect
171. D4×Dic3C3⋊D3semidirect
329. C32×C4#235semidirect
330. C32×C4#237semidirect
355. C33#550semidirect
487. C3×C4#3804semidirect
488. C3×C4#3828semidirect
501. C32#5553semidirect
502. C32#5718semidirect
561. C4#89031semidirect
563. C3#10595143semidirect
564. C3#10596024semidirect

Maximal subgroups

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

subgroupindex
C2×#890312
#117239812
C3×#641492
C32×D4×Dic32
#117236322
#117236242
#117236232
C4×#62932
#117235172
#117235012
#117234942
#105960243
#105951433