Finite Group Databasebeta

← #13177509

Subgroups — #13177509

Order 1728.

3000 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×#67C212yesmaximal, complemented
2C2×D3×#638C212yesmaximal, complemented
3#11723981C211yesmaximal, characteristic, complemented
4C3×#64146C211yesmaximal, characteristic, complemented
5C32×#937C211yesmaximal, characteristic, complemented
6#11723624C212yesmaximal, complemented
7#11723616C212yesmaximal, complemented
8#11723542C212yesmaximal, complemented
9#11723508C212yesmaximal, complemented
10#10596061—31nomaximal
11#10595289—34nomaximal
12C23×C3×C3⋊D3C2211yescharacteristic, complemented
13C23×C32×D3C2211yescharacteristic, complemented
14C22×D3×C3⋊D3C2212yescomplemented
15C2×C33×D4C2211yescharacteristic, complemented
16C2×#6312C2212yescomplemented
17C2×#6312C2212yescomplemented
18C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
19C2×C3×#638C2212yescomplemented
20C2×C32×#67C2212yescomplemented
21C3⋊D3×#67—28no—
22D3×#638—28no—
23C2×#6292C2212yescomplemented
24C2×#6291C2212yescomplemented
25C2×#6290C2212yes—
26C2×Dic3×C3⋊D3C2212yescomplemented
27C2×D3×C3⋊Dic3C2212yescomplemented
28C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
29C3×#3757C2211yescharacteristic, complemented
30C3×#3752C2212yescomplemented
31C32×#217C2211yescharacteristic, complemented
32C32×(C2×D3)⋊C4C2212yescomplemented
33#88806C2212yescomplemented
34#88804C2212yescomplemented
35C22×#638—32no—
36C2×D4×C3⋊D3—32no—
37C2×D3×#67—38no—
38C2×D3×#67—38no—
39C32×#120—31no—
40#64149—31no—
41#64149—34no—
42#64149—64no—
43#64148—32no—
44#64146D311yescharacteristic, complemented
45#64093—32no—
46#64091—32no—
47#64090—32no—
48C3×#937—34no—
49C3×#937D314yescomplemented
50#63971—38no—
51#63971—38no—
52#63923—38no—
53#63902—38no—
54C23×C33C2311yescharacteristic
55C23×C33C2311yescharacteristic
56C22×C3⋊(C3⋊D3)C2312yes—
57C22×C3×C3⋊D3—24no—
58C22×C3×C3⋊D3—41no—
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—
69#6312—24no—
70C2×C3⋊(C3⋊Dic3)C2312yes—
71C4⋊(C3⋊(C3⋊D3))—41no—
72C3×#638—24no—
73C2×C3×C3⋊Dic3C2312yes—
74C32×#67—24no—
75C2×C32×Dic3C2312yes—
76#6292—24no—
77#6291—24no—
78#6290D418yescomplemented
79Dic3×C3⋊D3—24no—
80D3×C3⋊Dic3—24no—
81#5590—94no—
82#5590—91no—
83C24×C32—31no—
84C23×C3⋊D3—32no—
85C23×C3⋊D3C2×D311yescharacteristic, complemented
86C23×C3×D3—34no—
87C23×C3×D3C2×D314yescomplemented
88C22×D32—38no—
89C2×C32×D4—24no—
90C2×C32×D4—32no—
91C2×C32×D4C2×D314yescomplemented
92C2×C32×D4C2×D311yescharacteristic, complemented
93C2×#638—32no—
94C2×#638—32no—
95C2×#638—32no—
96C2×#638—38no—
97C2×#638—38no—
98C2×#638—32no—
99C2×#638—68no—
100C2×#638—68no—
101C2×#638—68no—
102C2×#638C2×D312yescomplemented
103C22×C3⋊Dic3—32no—
104D4×C3⋊D3—68no—
105C2×C4⋊(C3⋊D3)—32no—
106C2×C4⋊(C3⋊D3)—31no—
107C2×C4⋊(C3⋊D3)—34no—
108C2×C4⋊(C3⋊D3)—64no—
109C2×C4×C3⋊D3—32no—
110C2×C3×#67—38no—
111C2×C3×#67C2×D318yescomplemented
112D3×#67—632no—
113D3×#67—632no—
114C2×#626—38no—
115C2×#626—38no—
116C2×#625—38no—
117C2×D3×Dic3—38no—
118C2×D3×Dic3—38no—
119C32×C22⋊C4—31no—
120C32×C22⋊C4—32no—
121#3757C2×D311yescharacteristic, complemented
122#3752—32no—
123#3752C2×D312yescomplemented
124#3750—32no—
125C4×C3⋊Dic3—31no—
126C4×C3⋊Dic3—34no—
127C4×C3⋊Dic3—64no—
128C3×#217—34no—
129C3×#217C2×D314yescomplemented
130C3×(C2×D3)⋊C4—38no—
131C3×(C2×D3)⋊C4C2×D318yescomplemented
132#3724—38no—
133#3722—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)—22no—
142C2×C3×C3⋊D3—22no—
143C2×C3×C3⋊D3—22no—
144C2×C3×C3⋊D3—44no—
145C2×C3×C3⋊D3C2×D414yescomplemented
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)C2×D414yescomplemented
153C3×C3⋊Dic3—22no—
154C32×Dic3—22no—
155C22×#67—98no—
156C22×#67—92no—
157C2×D3×D4—98no—
158C2×D3×D4—92no—
159C3×#120—34no—
160C3×#120—91no—
161#940—94no—
162#940—91no—
163#940—184no—
164#939—98no—
165#939—92no—
166#937—34no—
167#937C3⋊D311yescharacteristic, complemented
168#884—98no—
169#884—92no—
170#882—98no—
171#882—92no—
172#67⋊C4—98no—
173#67⋊C4—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—24no—
184C22×C3⋊D3—32no—
185C22×C3⋊D3—38no—
186C22×C3⋊D3—41no—
187C22×C3⋊D3—68no—
188C22×C3⋊D3—64no—
189C22×C3⋊D3—64no—
190C22×C3⋊D3—122no—
191C22×C3⋊D3C22×D312yes—
192C22×C3×D3—216no—
193C22×C3×D3—38no—
194C22×C3×D3—44no—
195C22×C3×D3—616no—
196C22×C3×D3—124no—
197C22×C3×D3C22×D318yes—
198C2×D32—616no—
199C2×D32—616no—
200C2×D32—128no—
201C32×D4—41no—
202C32×D4—44no—
203C32×D4—48no—
204C32×D4—64no—
205C2×C32×C4—24no—
206C2×C32×C4—32no—
207C2×C32×C4C22×D314yes—
208C2×C32×C4C22×D311yescharacteristic
209#638—24no—
210#638—38no—
211#638—64no—
212#638—64no—
213#638—64no—
214#638—616no—
215#638—616no—
216#638—1216no—
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)—64no—
224C4⋊(C3⋊D3)—121no—
225C4⋊(C3⋊D3)—124no—
226C4⋊(C3⋊D3)—128no—
227C4×C3⋊D3—64no—
228C3×#67—216no—
229C3×#67—616no—
230C2×C3×Dic3—38no—
231C2×C3×Dic3C22×D318yes—
232#626—616no—
233#626—616no—
234#625—332no—
235D3×Dic3—616no—
236D3×Dic3—616no—
237D42—271noSylow
238C2×C33—41no—
239C2×C33—41no—
240C2×C33C22×D412yes—
241C2×C33H(5,𝔽2)11yescharacteristic
242C3⋊(C3⋊D3)—42no—
243C3×C3⋊D3—24no—
244C32×D3—24no—
245C24×C3—34no—
246C24×C3—91no—
247C23×D3—34no—
248C23×D3—98no—
249C23×D3—92no—
250C23×D3C2×C3⋊D311yescharacteristic, complemented
251C2×C3×D4—24no—
252C2×C3×D4—38no—
253C2×C3×D4—92no—
254C2×C3×D4D3214yescomplemented
255C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
256C2×#67—38no—
257C2×#67—98no—
258C2×#67—98no—
259C2×#67—92no—
260C2×#67—98no—
261C2×#67—98no—
262C2×#67—92no—
263C2×#67—92no—
264C2×#67—92no—
265C2×#67—188no—
266C2×#67—188no—
267C2×#67—188no—
268C2×#67—1832no—
269C2×#67C2×C3⋊D312yescomplemented
270C22×Dic3—98no—
271C22×Dic3—92no—
272D3×D4—1832no—
273D3×D4—188no—
274C2×D12—98no—
275C2×D12—92no—
276C2×D12—94no—
277C2×D12—91no—
278C2×D12—184no—
279C2×C4×D3—98no—
280C2×C4×D3—92no—
281C3×C22⋊C4—34no—
282C3×C22⋊C4—38no—
283C3×C22⋊C4—91no—
284C3×C22⋊C4—92no—
285#217—34no—
286#217C2×C3⋊D311yescharacteristic, complemented
287(C2×D3)⋊C4—38no—
288(C2×D3)⋊C4—98no—
289(C2×D3)⋊C4—92no—
290(C2×D3)⋊C4C2×C3⋊D312yescomplemented
291Dic3⋊C4—98no—
292Dic3⋊C4—92no—
293C4×Dic3—94no—
294C4×Dic3—91no—
295C4×Dic3—184no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—24no—
301C22×C32—28no—
302C22×C32—28no—
303C22×C32—24no—
304C22×C32—24no—
305C22×C32—34no—
306C22×C32—48no—
307C22×C32—48no—
308C22×C32—44no—
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—44no—
318C2×C3⋊D3—62no—
319C2×C3⋊D3—62no—
320C2×C3⋊D3—62no—
321C2×C3⋊D3—68no—
322C2×C3⋊D3—68no—
323C2×C3⋊D3—68no—
324C2×C3⋊D3—128no—
325C2×C3⋊D3—128no—
326C2×C3⋊D3—128no—
327C2×C3⋊D3—128no—
328C2×C3⋊D3D3×D414yescomplemented
329C2×C3×D3—28no—
330C2×C3×D3—28no—
331C2×C3×D3—316no—
332C2×C3×D3—416no—
333C2×C3×D3—68no—
334C2×C3×D3—68no—
335C2×C3×D3—1216no—
336C2×C3×D3D3×D4116yescomplemented
337D32—1232no—
338C32×C4—28no—
339C32×C4—24no—
340C32×C4—21no—
341C32×C4—62no—
342C3⋊Dic3—22no—
343C3⋊Dic3—34no—
344C3⋊Dic3—316no—
345C3⋊Dic3—616no—
346C3×Dic3—28no—
347C3×Dic3—68no—
348C22×D4—272no—
349C22×D4—272no—
350#127—271no—
351#121—272no—
352#121—272no—
353#120—31no—
354#120—91no—
355#120—272no—
356C4×D4—272no—
357C33D4211yescharacteristic, complemented, Sylow
358C23×C3—24no—
359C23×C3—24no—
360C23×C3—38no—
361C23×C3—616no—
362C23×C3—92no—
363C23×C3—124no—
364C23×C3—184no—
365C23×C3—361no—
366C23×C3C2×D3214yes—
367C23×C3C2×D3214yes—
368C23×C3C22×C3⋊D311yescharacteristic
369C23×C3C22×C3⋊D311yescharacteristic
370C22×D3—24no—
371C22×D3—38no—
372C22×D3—41no—
373C22×D3—616no—
374C22×D3—98no—
375C22×D3—92no—
376C22×D3—124no—
377C22×D3—188no—
378C22×D3—184no—
379C22×D3—1816no—
380C22×D3—1816no—
381C22×D3—184no—
382C22×D3—362no—
383C22×D3—368no—
384C22×D3C22×C3⋊D312yes—
385C3×D4—44no—
386C3×D4—41no—
387C3×D4—48no—
388C3×D4—616no—
389C3×D4—184no—
390C2×C3×C4—24no—
391C2×C3×C4—38no—
392C2×C3×C4—92no—
393C2×C3×C4C2×D3214yes—
394C2×C3×C4C22×C3⋊D311yescharacteristic
395#67—24no—
396#67—616no—
397#67—932no—
398#67—98no—
399#67—184no—
400#67—1816no—
401#67—184no—
402#67—184no—
403#67—1816no—
404#67—1816no—
405#67—3616no—
406#67—3616no—
407C2×Dic3—38no—
408C2×Dic3—98no—
409C2×Dic3—92no—
410C2×Dic3—188no—
411C2×Dic3—184no—
412C2×Dic3—1816no—
413C2×Dic3C22×C3⋊D312yes—
414D12—1816no—
415D12—184no—
416D12—364no—
417D12—361no—
418D12—368no—
419C4×D3—1816no—
420C4×D3—184no—
421C2×C32—28no—
422C2×C32—24no—
423C2×C32—41no—
424C2×C32—41no—
425C2×C32—44no—
426C2×C32—44no—
427C2×C32—64no—
428C2×C32—84no—
429C2×C32—84no—
430C2×C32C2×D3×D418yes—
431C2×C32C2×D3×D412yes—
432C2×C32#100414yes—
433C2×C32#100411yescharacteristic
434C3⋊D3—24no—
435C3⋊D3—122no—
436C3⋊D3—128no—
437C3⋊D3—248no—
438C3×D3—216no—
439C3×D3—616no—
440C24—31no—
441C24—91no—
442C24—272no—
443C2×D4—32no—
444C2×D4—92no—
445C2×D4—272no—
446C2×D4—272no—
447C2×D4—272no—
448C2×D4—272no—
449C2×D4—272no—
450C2×D4—271no—
451C2×D4—548no—
452C2×D4—548no—
453C2×D4D3×C3⋊D311yescharacteristic, complemented
454C22×C4—272no—
455C22×C4—272no—
456C4⋊C4—272no—
457C22⋊C4—31no—
458C22⋊C4—32no—
459C22⋊C4—91no—
460C22⋊C4—92no—
461C22⋊C4—272no—
462C42—271no—
463C22×C3—28no—
464C22×C3—28no—
465C22×C3—24no—
466C22×C3—24no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—24no—
472C22×C3—316no—
473C22×C3—48no—
474C22×C3—48no—
475C22×C3—44no—
476C22×C3—44no—
477C22×C3—68no—
478C22×C3—68no—
479C22×C3—94no—
480C22×C3—1216no—
481C22×C3—182no—
482C22×C3—182no—
483C22×C3—364no—
484C22×C3C22×D3214yes—
485C22×C3C23×C3⋊D311yescharacteristic
486C2×D3—22no—
487C2×D3—22no—
488C2×D3—316no—
489C2×D3—44no—
490C2×D3—68no—
491C2×D3—68no—
492C2×D3—1216no—
493C2×D3—188no—
494C2×D3—188no—
495C2×D3—188no—
496C2×D3—182no—
497C2×D3—182no—
498C2×D3—182no—
499C2×D3—368no—
500C2×D3—368no—
501C2×D3—368no—
502C2×D3—368no—
503C2×D3—3632no—
504C2×D3D4×C3⋊D314yescomplemented
505C3×C4—28no—
506C3×C4—24no—
507C3×C4—21no—
508C3×C4—68no—
509C3×C4—182no—
510Dic3—22no—
511Dic3—68no—
512Dic3—916no—
513Dic3—94no—
514Dic3—1816no—
515C32—24no—
516C32#559014yescomplemented
517C32#559011yescharacteristic, complemented
518C23—32no—
519C23—64no—
520C23—92no—
521C23—121no—
522C23—184no—
523C23—272no—
524C23—361no—
525C23—544no—
526C23—544no—
527C23—1082no—
528C23C2×D3×C3⋊D311yescharacteristic
529C23C2×D3×C3⋊D311yescharacteristic
530D4—41no—
531D4—64no—
532D4—184no—
533D4—278no—
534D4—544no—
535D4—544no—
536D4—544no—
537D4—544no—
538D4—1081no—
539C2×C4—32no—
540C2×C4—92no—
541C2×C4—272no—
542C2×C4—544no—
543C2×C4—544no—
544C2×C4C2×D3×C3⋊D311yescharacteristic
545C2×C3—28no—
546C2×C3—24no—
547C2×C3—44no—
548C2×C3—44no—
549C2×C3—41no—
550C2×C3—41no—
551C2×C3—616no—
552C2×C3—84no—
553C2×C3—84no—
554C2×C3—184no—
555C2×C3D32×D418yes—
556C2×C3#6433014yes—
557C2×C3C2×D4×C3⋊D312yes—
558C2×C3#6436111yescharacteristic
559D3—24no—
560D3—616no—
561D3—368no—
562D3—362no—
563D3—728no—
564C22—22no—
565C22—22no—
566C22—21no—
567C22—21no—
568C22—34no—
569C22—62no—
570C22—62no—
571C22—94no—
572C22—124no—
573C22—182no—
574C22—182no—
575C22—364no—
576C22—542no—
577C22—542no—
578C22—542no—
579C22—1088no—
580C22C22×D3×C3⋊D311yescharacteristic
581C4—21no—
582C4—62no—
583C4—182no—
584C4—274no—
585C3—24no—
586C3#1059528914yescomplemented
587C3#1059606111yescharacteristic, complemented
588C2—41no—
589C2—41no—
590C2—64no—
591C2—184no—
592C2—1082no—
593C2D3×D4×C3⋊D312yes—
594C2#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×C3⋊D3×#67C2semidirect
2. C2×D3×#638C2semidirect
3. #11723981C2semidirect
4. C3×#64146C2semidirect
5. C32×#937C2semidirect
6. #11723624C2semidirect
7. #11723616C2semidirect
8. #11723542C2semidirect
9. #11723508C2semidirect
12. C23×C3×C3⋊D3C22semidirect
13. C23×C32×D3C22semidirect
14. C22×D3×C3⋊D3C22semidirect
15. C2×C33×D4C22semidirect
16. C2×#6312C22semidirect
17. C2×#6312C22semidirect
18. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
19. C2×C3×#638C22semidirect
20. C2×C32×#67C22semidirect
23. C2×#6292C22semidirect
24. C2×#6291C22semidirect
26. C2×Dic3×C3⋊D3C22semidirect
27. C2×D3×C3⋊Dic3C22semidirect
28. C4×C3⋊(C3⋊Dic3)C22semidirect
29. C3×#3757C22semidirect
30. C3×#3752C22semidirect
31. C32×#217C22semidirect
32. C32×(C2×D3)⋊C4C22semidirect
33. #88806C22semidirect
34. #88804C22semidirect
44. #64146D3semidirect
49. C3×#937D3semidirect
78. #6290D4semidirect
85. C23×C3⋊D3C2×D3semidirect
87. C23×C3×D3C2×D3semidirect
91. C2×C32×D4C2×D3semidirect
92. C2×C32×D4C2×D3semidirect
102. C2×#638C2×D3semidirect
111. C2×C3×#67C2×D3semidirect
121. #3757C2×D3semidirect
123. #3752C2×D3semidirect
129. C3×#217C2×D3semidirect
131. C3×(C2×D3)⋊C4C2×D3semidirect
145. C2×C3×C3⋊D3C2×D4semidirect
149. C2×C32×D3C2×D4semidirect
152. C3⋊(C3⋊Dic3)C2×D4semidirect
167. #937C3⋊D3semidirect
250. C23×D3C2×C3⋊D3semidirect
254. C2×C3×D4D32semidirect
255. C2×C3×D4C2×C3⋊D3semidirect
269. C2×#67C2×C3⋊D3semidirect
286. #217C2×C3⋊D3semidirect
290. (C2×D3)⋊C4C2×C3⋊D3semidirect
328. C2×C3⋊D3D3×D4semidirect
336. C2×C3×D3D3×D4semidirect
357. C33D42semidirect
453. C2×D4D3×C3⋊D3semidirect
504. C2×D3D4×C3⋊D3semidirect
516. C32#5590semidirect
517. C32#5590semidirect
586. C3#10595289semidirect
587. C3#10596061semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×#672
C2×D3×#6382
#117239812
C3×#641462
C32×#9372
#117236242
#117236162
#117235422
#117235082
#105960613
#105952893