Finite Group Databasebeta

← #13177532

Subgroups — #13177532

Order 1728.

3088 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×#89039C212yesmaximal, complemented
2C2×D3×#638C212yesmaximal, complemented
3#11723978C211yesmaximal, characteristic, complemented
4C3×#64149C211yesmaximal, characteristic, complemented
5C32×#937C211yesmaximal, characteristic, complemented
6#11723623C212yesmaximal, complemented
7#11723615C212yesmaximal, complemented
8#11723541C212yesmaximal, complemented
9#11723461C212yesmaximal, complemented
10#10596061—31nomaximal
11#10595295—34nomaximal
12C23×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
13C23×C32×D3C2211yescharacteristic, complemented
14C22×D3×C3⋊D3C2212yescomplemented
15C2×C33×D4C2211yescharacteristic, complemented
16C2×#6312C2212yescomplemented
17C2×C3×#638C2212yescomplemented
18C2×C3×#638C2212yescomplemented
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
20C2×C32×#67C2212yescomplemented
21#89039—28no—
22D3×#638—28no—
23C2×#6292C2212yescomplemented
24C2×#6291C2212yes—
25C2×#6290C2212yescomplemented
26C2×#6289C2212yescomplemented
27C2×D3×C3⋊Dic3C2212yescomplemented
28#88864C2211yescharacteristic, complemented
29#88859C2212yescomplemented
30C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
31C32×#217C2211yescharacteristic, complemented
32C32×(C2×D3)⋊C4C2212yescomplemented
33#88807C2212yescomplemented
34#88803C2212yescomplemented
35C22×#638—32no—
36C2×D4×C3⋊D3—32no—
37C2×#3811—38no—
38C2×D3×#67—38no—
39C32×#120—31no—
40#64149D311yescharacteristic, complemented
41#64148—32no—
42#64146—31no—
43#64146—34no—
44#64146—64no—
45#64093—32no—
46#64091—32no—
47#64090—32no—
48C3×#940—34no—
49C3×#937D314yescomplemented
50#63970—38no—
51#63958—38no—
52#63922—38no—
53#63880—38no—
54C23×C33C2311yescharacteristic
55C23×C33C2311yescharacteristic
56C22×C3⋊(C3⋊D3)—24no—
57C22×C3⋊(C3⋊D3)—41no—
58C22×C3⋊(C3⋊D3)C2312yes—
59C22×C3×C3⋊D3C2312yes—
60C22×C32×D3—24no—
61C22×C32×D3—41no—
62C22×C32×D3C2312yes—
63C2×D3×C3⋊D3—24no—
64C2×D3×C3⋊D3—24no—
65C2×D3×C3⋊D3—42no—
66C33×D4—41no—
67C2×C33×C4C2311yescharacteristic
68#6312—24no—
69C2×C3⋊(C3⋊Dic3)C2312yes—
70C3×#638—24no—
71C3×#638—24no—
72C2×C3×C3⋊Dic3C2312yes—
73C3×C4⋊(C3⋊D3)—41no—
74C32×#67—24no—
75C2×C32×Dic3C2312yes—
76#6292—24no—
77#6291D418yescomplemented
78#6290—24no—
79#6289—24no—
80D3×C3⋊Dic3—24no—
81#5717—91no—
82#5590—94no—
83C24×C32—31no—
84C23×C3⋊D3—31no—
85C23×C3⋊D3—34no—
86C23×C3⋊D3—32no—
87C23×C3⋊D3—64no—
88C23×C3×D3C2×D314yescomplemented
89C22×D32—38no—
90C2×C32×D4—24no—
91C2×C32×D4—32no—
92C2×C32×D4C2×D314yescomplemented
93C2×C32×D4C2×D311yescharacteristic, complemented
94C2×#638—32no—
95C2×#638—38no—
96C2×#638—32no—
97C2×#638—32no—
98C2×#638—68no—
99C2×#638—68no—
100C2×#638C2×D312yescomplemented
101C2×#638C2×D312yescomplemented
102C22×C3⋊Dic3—32no—
103D4×C3⋊D3—68no—
104C2×C4⋊(C3⋊D3)—32no—
105C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
106C2×C4×C3⋊D3—32no—
107C2×C3×#67—38no—
108C2×C3×#67—38no—
109C2×C3×#67C2×D318yescomplemented
110C2×C3×D12—34no—
111#3811—632no—
112D3×#67—632no—
113C2×#626—38no—
114C2×#626—38no—
115C2×#625—38no—
116C2×#624—38no—
117C2×D3×Dic3—38no—
118C32×C22⋊C4—31no—
119C32×C22⋊C4—32no—
120#3757—31no—
121#3757—34no—
122#3757—64no—
123#3752—32no—
124#3752—32no—
125#3752—38no—
126#3752—68no—
127#3750—32no—
128C4×C3⋊Dic3C2×D311yescharacteristic, complemented
129C3×#217C2×D314yescomplemented
130C3×(C2×D3)⋊C4C2×D318yescomplemented
131C3×C4×Dic3—34no—
132Dic3⋊Dic3—38no—
133(C2×D3)⋊Dic3—38no—
134C22×C33—22no—
135C22×C33—22no—
136C22×C33—21no—
137C22×C33—21no—
138C22×C33C2411yescharacteristic
139C2×C3⋊(C3⋊D3)—22no—
140C2×C3⋊(C3⋊D3)—22no—
141C2×C3⋊(C3⋊D3)—44no—
142C2×C3⋊(C3⋊D3)C2×D414yescomplemented
143C2×C3×C3⋊D3—22no—
144C2×C3×C3⋊D3—22no—
145C2×C3×C3⋊D3—22no—
146C2×C32×D3—22no—
147C2×C32×D3—22no—
148C2×C32×D3—44no—
149C2×C32×D3C2×D414yescomplemented
150D3×C3⋊D3—48no—
151C33×C4—21no—
152C3⋊(C3⋊Dic3)—22no—
153C3×C3⋊Dic3C2×D414yescomplemented
154C32×Dic3—22no—
155C22×#67—98no—
156C2×D3×D4—98no—
157C2×D3×D4—92no—
158C2×D3×D4—92no—
159C3×#127—91no—
160C3×#120—34no—
161#940—34no—
162#939—98no—
163#938—92no—
164#938—92no—
165#937—94no—
166#937—91no—
167#937—92no—
168#937—184no—
169#937C3⋊D311yescharacteristic, complemented
170#884—98no—
171#882—98no—
172#67⋊C4—98no—
173C4×D12—92no—
174C23×C32—24no—
175C23×C32—24no—
176C23×C32—32no—
177C23×C32—64no—
178C23×C32—121no—
179C23×C32C22×D314yes—
180C23×C32C22×D314yes—
181C23×C32C22×D311yescharacteristic
182C23×C32C22×D311yescharacteristic
183C22×C3⋊D3—32no—
184C22×C3⋊D3—38no—
185C22×C3⋊D3—68no—
186C22×C3⋊D3—64no—
187C22×C3⋊D3—64no—
188C22×C3⋊D3—64no—
189C22×C3⋊D3—616no—
190C22×C3⋊D3—1216no—
191C22×C3⋊D3—122no—
192C22×C3⋊D3—121no—
193C22×C3⋊D3—124no—
194C22×C3⋊D3—128no—
195C22×C3⋊D3C22×D312yes—
196C22×C3×D3—216no—
197C22×C3×D3—38no—
198C22×C3×D3—44no—
199C22×C3×D3C22×D318yes—
200C2×D32—616no—
201C2×D32—616no—
202C2×D32—128no—
203C32×D4—41no—
204C32×D4—44no—
205C32×D4—48no—
206C32×D4—64no—
207C2×C32×C4—24no—
208C2×C32×C4—32no—
209C2×C32×C4C22×D314yes—
210C2×C32×C4C22×D311yescharacteristic
211#638—24no—
212#638—24no—
213#638—38no—
214#638—64no—
215#638—64no—
216#638—616no—
217#638—1216no—
218C2×C3⋊Dic3—32no—
219C2×C3⋊Dic3—38no—
220C2×C3⋊Dic3—68no—
221C2×C3⋊Dic3—64no—
222C2×C3⋊Dic3C22×D312yes—
223C4⋊(C3⋊D3)—41no—
224C4⋊(C3⋊D3)—64no—
225C4×C3⋊D3—64no—
226C3×#67—216no—
227C3×#67—616no—
228C3×#67—616no—
229C2×C3×Dic3—38no—
230C2×C3×Dic3C22×D318yes—
231C3×D12—124no—
232#626—332no—
233#626—616no—
234#625—616no—
235#624—616no—
236D3×Dic3—616no—
237D42—271noSylow
238C2×C33—41no—
239C2×C33—41no—
240C2×C33C22×D412yes—
241C2×C33H(5,𝔽2)11yescharacteristic
242C3⋊(C3⋊D3)—24no—
243C3×C3⋊D3—42no—
244C32×D3—24no—
245C24×C3—34no—
246C23×D3—98no—
247C23×D3—94no—
248C23×D3—91no—
249C23×D3—92no—
250C23×D3—184no—
251C23×D3C2×C3⋊D311yescharacteristic, complemented
252C2×C3×D4—24no—
253C2×C3×D4—38no—
254C2×C3×D4—92no—
255C2×C3×D4—92no—
256C2×C3×D4—91no—
257C2×C3×D4D3214yescomplemented
258C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
259C2×#67—38no—
260C2×#67—38no—
261C2×#67—98no—
262C2×#67—98no—
263C2×#67—98no—
264C2×#67—92no—
265C2×#67—92no—
266C2×#67—92no—
267C2×#67—188no—
268C2×#67—1832no—
269C2×#67C2×C3⋊D312yescomplemented
270C22×Dic3—98no—
271D3×D4—188no—
272D3×D4—188no—
273D3×D4—1832no—
274C2×D12—34no—
275C2×D12—92no—
276C2×D12—98no—
277C2×C4×D3—98no—
278C2×C4×D3—92no—
279C2×C4×D3—92no—
280C3×C22⋊C4—34no—
281C3×C22⋊C4—38no—
282C3×C42—91no—
283#217—94no—
284#217—91no—
285#217—92no—
286#217—184no—
287#217C2×C3⋊D311yescharacteristic, complemented
288(C2×D3)⋊C4—98no—
289(C2×D3)⋊C4—98no—
290(C2×D3)⋊C4—92no—
291(C2×D3)⋊C4—188no—
292(C2×D3)⋊C4C2×C3⋊D312yescomplemented
293C4⋊Dic3—92no—
294Dic3⋊C4—98no—
295C4×Dic3—34no—
296C22×C32—28no—
297C22×C32—22no—
298C22×C32—22no—
299C22×C32—24no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—24no—
303C22×C32—28no—
304C22×C32—28no—
305C22×C32—24no—
306C22×C32—34no—
307C22×C32—48no—
308C22×C32—48no—
309C22×C32—44no—
310C22×C32—62no—
311C22×C32—62no—
312C22×C32—124no—
313C22×C32C23×D314yes—
314C22×C32C23×D311yescharacteristic
315C2×C3⋊D3—22no—
316C2×C3⋊D3—22no—
317C2×C3⋊D3—22no—
318C2×C3⋊D3—34no—
319C2×C3⋊D3—316no—
320C2×C3⋊D3—616no—
321C2×C3⋊D3—62no—
322C2×C3⋊D3—62no—
323C2×C3⋊D3—68no—
324C2×C3⋊D3—68no—
325C2×C3⋊D3—128no—
326C2×C3⋊D3—128no—
327C2×C3⋊D3—128no—
328C2×C3⋊D3—124no—
329C2×C3⋊D3—1216no—
330C2×C3⋊D3—1232no—
331C2×C3×D3—28no—
332C2×C3×D3—28no—
333C2×C3×D3—416no—
334C2×C3×D3—68no—
335C2×C3×D3—68no—
336C2×C3×D3—68no—
337C2×C3×D3D3×D4116yescomplemented
338D32—1232no—
339C32×C4—21no—
340C32×C4—24no—
341C32×C4—44no—
342C32×C4—62no—
343C3⋊Dic3—62no—
344C3⋊Dic3—68no—
345C3⋊Dic3—128no—
346C3⋊Dic3D3×D414yescomplemented
347C3×Dic3—28no—
348C3×Dic3—316no—
349C22×D4—272no—
350C22×D4—272no—
351#127—91no—
352#121—272no—
353#121—272no—
354#120—31no—
355#120—272no—
356#120—271no—
357C4×D4—272no—
358C33D4211yescharacteristic, complemented, Sylow
359C23×C3—24no—
360C23×C3—24no—
361C23×C3—38no—
362C23×C3—616no—
363C23×C3—92no—
364C23×C3—124no—
365C23×C3C2×D3214yes—
366C23×C3C2×D3214yes—
367C23×C3C22×C3⋊D311yescharacteristic
368C23×C3C22×C3⋊D311yescharacteristic
369C22×D3—24no—
370C22×D3—38no—
371C22×D3—41no—
372C22×D3—98no—
373C22×D3—92no—
374C22×D3—188no—
375C22×D3—184no—
376C22×D3—1816no—
377C22×D3—1816no—
378C22×D3—184no—
379C22×D3—1816no—
380C22×D3—184no—
381C22×D3—3616no—
382C22×D3—362no—
383C22×D3—368no—
384C22×D3—364no—
385C22×D3—361no—
386C22×D3—368no—
387C22×D3C22×C3⋊D312yes—
388C3×D4—44no—
389C3×D4—41no—
390C3×D4—48no—
391C3×D4—616no—
392C3×D4—184no—
393C3×D4—184no—
394C3×D4—361no—
395C2×C3×C4—24no—
396C2×C3×C4—38no—
397C2×C3×C4—92no—
398C2×C3×C4C2×D3214yes—
399C2×C3×C4C22×C3⋊D311yescharacteristic
400#67—24no—
401#67—616no—
402#67—616no—
403#67—932no—
404#67—184no—
405#67—1816no—
406#67—184no—
407#67—1816no—
408#67—184no—
409#67—3616no—
410C2×Dic3—38no—
411C2×Dic3—98no—
412C2×Dic3—92no—
413C2×Dic3—188no—
414C2×Dic3—1816no—
415C2×Dic3C22×C3⋊D312yes—
416D12—98no—
417D12—124no—
418D12—1816no—
419C4×D3—184no—
420C4×D3—1816no—
421C4×D3—184no—
422C2×C32—28no—
423C2×C32—24no—
424C2×C32—48no—
425C2×C32—41no—
426C2×C32—41no—
427C2×C32—44no—
428C2×C32—44no—
429C2×C32—64no—
430C2×C32—84no—
431C2×C32C2×D3×D418yes—
432C2×C32C2×D3×D412yes—
433C2×C32#100414yes—
434C2×C32#100411yescharacteristic
435C3⋊D3—42no—
436C3⋊D3—64no—
437C3⋊D3—616no—
438C3⋊D3—1216no—
439C3×D3—216no—
440C3×D3—128no—
441C24—31no—
442C24—272no—
443C24—271no—
444C2×D4—32no—
445C2×D4—92no—
446C2×D4—92no—
447C2×D4—91no—
448C2×D4—272no—
449C2×D4—272no—
450C2×D4—272no—
451C2×D4—272no—
452C2×D4—548no—
453C2×D4—548no—
454C2×D4D3×C3⋊D311yescharacteristic, complemented
455C22×C4—272no—
456C22×C4—272no—
457C4⋊C4—272no—
458C22⋊C4—31no—
459C22⋊C4—32no—
460C22⋊C4—272no—
461C22⋊C4—271no—
462C22⋊C4—272no—
463C42—91no—
464C22×C3—28no—
465C22×C3—28no—
466C22×C3—28no—
467C22×C3—24no—
468C22×C3—24no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—24no—
474C22×C3—316no—
475C22×C3—48no—
476C22×C3—48no—
477C22×C3—44no—
478C22×C3—68no—
479C22×C3—68no—
480C22×C3—1216no—
481C22×C3—182no—
482C22×C3—182no—
483C22×C3—182no—
484C22×C3C22×D3214yes—
485C22×C3C23×C3⋊D311yescharacteristic
486C2×D3—22no—
487C2×D3—22no—
488C2×D3—44no—
489C2×D3—68no—
490C2×D3—68no—
491C2×D3—68no—
492C2×D3—916no—
493C2×D3—94no—
494C2×D3—1816no—
495C2×D3—188no—
496C2×D3—188no—
497C2×D3—182no—
498C2×D3—182no—
499C2×D3—368no—
500C2×D3—368no—
501C2×D3—368no—
502C2×D3—3632no—
503C2×D3—364no—
504C2×D3—3616no—
505C2×D3—3632no—
506C2×D3D4×C3⋊D314yescomplemented
507C3×C4—24no—
508C3×C4—21no—
509C3×C4—44no—
510C3×C4—68no—
511C3×C4—94no—
512Dic3—22no—
513Dic3—316no—
514Dic3—188no—
515Dic3—182no—
516Dic3—368no—
517C32—24no—
518C32#559014yescomplemented
519C32#571711yescharacteristic, complemented
520C23—32no—
521C23—64no—
522C23—92no—
523C23—121no—
524C23—272no—
525C23—544no—
526C23—544no—
527C23—544no—
528C23—1081no—
529C23—1082no—
530C23C2×D3×C3⋊D311yescharacteristic
531C23C2×D3×C3⋊D311yescharacteristic
532D4—41no—
533D4—64no—
534D4—184no—
535D4—184no—
536D4—278no—
537D4—361no—
538D4—544no—
539D4—544no—
540D4—544no—
541C2×C4—32no—
542C2×C4—92no—
543C2×C4—272no—
544C2×C4—544no—
545C2×C4—544no—
546C2×C4C2×D3×C3⋊D311yescharacteristic
547C2×C3—28no—
548C2×C3—24no—
549C2×C3—48no—
550C2×C3—41no—
551C2×C3—41no—
552C2×C3—44no—
553C2×C3—44no—
554C2×C3—616no—
555C2×C3—84no—
556C2×C3—362no—
557C2×C3D32×D418yes—
558C2×C3#6433014yes—
559C2×C3C2×D4×C3⋊D312yes—
560C2×C3#6436111yescharacteristic
561D3—24no—
562D3—128no—
563D3—184no—
564D3—1816no—
565D3—3616no—
566C22—22no—
567C22—22no—
568C22—21no—
569C22—21no—
570C22—34no—
571C22—62no—
572C22—62no—
573C22—124no—
574C22—182no—
575C22—182no—
576C22—182no—
577C22—274no—
578C22—542no—
579C22—542no—
580C22—1084no—
581C22—1088no—
582C22C22×D3×C3⋊D311yescharacteristic
583C4—21no—
584C4—62no—
585C4—94no—
586C4—542no—
587C3—24no—
588C3#1059529514yescomplemented
589C3#1059606111yescharacteristic, complemented
590C2—41no—
591C2—41no—
592C2—64no—
593C2—362no—
594C2—544no—
595C2D3×D4×C3⋊D312yes—
596C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33D42
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89039C2semidirect
2. C2×D3×#638C2semidirect
3. #11723978C2semidirect
4. C3×#64149C2semidirect
5. C32×#937C2semidirect
6. #11723623C2semidirect
7. #11723615C2semidirect
8. #11723541C2semidirect
9. #11723461C2semidirect
12. C23×C3⋊(C3⋊D3)C22semidirect
13. C23×C32×D3C22semidirect
14. C22×D3×C3⋊D3C22semidirect
15. C2×C33×D4C22semidirect
16. C2×#6312C22semidirect
17. C2×C3×#638C22semidirect
18. C2×C3×#638C22semidirect
19. C2×C3×C4⋊(C3⋊D3)C22semidirect
20. C2×C32×#67C22semidirect
23. C2×#6292C22semidirect
25. C2×#6290C22semidirect
26. C2×#6289C22semidirect
27. C2×D3×C3⋊Dic3C22semidirect
28. #88864C22semidirect
29. #88859C22semidirect
30. C3×C4×C3⋊Dic3C22semidirect
31. C32×#217C22semidirect
32. C32×(C2×D3)⋊C4C22semidirect
33. #88807C22semidirect
34. #88803C22semidirect
40. #64149D3semidirect
49. C3×#937D3semidirect
77. #6291D4semidirect
88. C23×C3×D3C2×D3semidirect
92. C2×C32×D4C2×D3semidirect
93. C2×C32×D4C2×D3semidirect
100. C2×#638C2×D3semidirect
101. C2×#638C2×D3semidirect
105. C2×C4⋊(C3⋊D3)C2×D3semidirect
109. C2×C3×#67C2×D3semidirect
128. C4×C3⋊Dic3C2×D3semidirect
129. C3×#217C2×D3semidirect
130. C3×(C2×D3)⋊C4C2×D3semidirect
142. C2×C3⋊(C3⋊D3)C2×D4semidirect
149. C2×C32×D3C2×D4semidirect
153. C3×C3⋊Dic3C2×D4semidirect
169. #937C3⋊D3semidirect
251. C23×D3C2×C3⋊D3semidirect
257. C2×C3×D4D32semidirect
258. C2×C3×D4C2×C3⋊D3semidirect
269. C2×#67C2×C3⋊D3semidirect
287. #217C2×C3⋊D3semidirect
292. (C2×D3)⋊C4C2×C3⋊D3semidirect
337. C2×C3×D3D3×D4semidirect
346. C3⋊Dic3D3×D4semidirect
358. C33D42semidirect
454. C2×D4D3×C3⋊D3semidirect
506. C2×D3D4×C3⋊D3semidirect
518. C32#5590semidirect
519. C32#5717semidirect
588. C3#10595295semidirect
589. C3#10596061semidirect

Maximal subgroups

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

subgroupindex
C2×#890392
C2×D3×#6382
#117239782
C3×#641492
C32×#9372
#117236232
#117236152
#117235412
#117234612
#105960613
#105952953