Finite Group Databasebeta

← #13177327

Subgroups — #13177327

Order 1728.

2452 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#638C212yesmaximal, complemented
2#11723971C211yesmaximal, characteristic, complemented
3C3×#64139C211yesmaximal, characteristic, complemented
4C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
5#11723632C212yesmaximal, complemented
6#11723542C212yesmaximal, complemented
7#11723541C212yesmaximal, complemented
8#11723480C212yesmaximal, complemented
9#11723479C212yesmaximal, complemented
10#10595964—31nomaximal
11#10595072—34nomaximal
12C23×C32×D3C2211yescharacteristic
13C22×D3×C3⋊D3C2212yescomplemented
14C22×C33×C4C2211yescharacteristic, complemented
15C2×#6312C2212yescomplemented
16C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
17C2×C3×#638C2212yescomplemented
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
19C22×C32×Dic3C2211yescharacteristic, complemented
20D3×#638—28no—
21C2×#6292C2212yescomplemented
22C2×#6291C2212yes—
23C2×#6290C2212yes—
24C2×D3×C3⋊Dic3C2212yescomplemented
25#88859C2212yescomplemented
26C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
27C3×#3752C2212yescomplemented
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
29C32×(C2×D3)⋊C4C2212yescomplemented
30C32×(C2×D3)⋊C4C2212yescomplemented
31Dic3⋊(C3⋊Dic3)C2212yescomplemented
32#88804C2212yescomplemented
33#88803C2212yescomplemented
34C22×#638—32no—
35C2×D3×#67—38no—
36C2×C32×C22⋊C4—31no—
37#64139—31no—
38#64139—32no—
39#64139—34no—
40#64139—64no—
41#64139D311yescharacteristic, complemented
42#64095—32no—
43#64095—32no—
44#64091—32no—
45#64091—32no—
46C3×#930—34no—
47C2×C3×(C2×D3)⋊C4D314yescomplemented
48#63978—38no—
49#63923—38no—
50#63922—38no—
51#63891—38no—
52#63890—38no—
53C23×C33C2311yescharacteristic
54C22×C3⋊(C3⋊D3)C2312yes—
55C22×C3×C3⋊D3C2312yes—
56C22×C32×D3—41no—
57C22×C32×D3—41no—
58C22×C32×D3D414yescomplemented
59C22×C32×D3C2312yes—
60C2×D3×C3⋊D3—24no—
61C2×D3×C3⋊D3—42no—
62C2×D3×C3⋊D3—42no—
63C2×C33×C4—41no—
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66#6312—24no—
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C4⋊(C3⋊(C3⋊D3))—41no—
69C3×#638—24no—
70C2×C3×C3⋊Dic3C2312yes—
71C3×C4⋊(C3⋊D3)—41no—
72C2×C32×Dic3—41no—
73C2×C32×Dic3C2312yes—
74#6292—42no—
75#6291—24no—
76#6290—24no—
77D3×C3⋊Dic3—24no—
78#5605—91no—
79#5493—94no—
80C24×C32—31no—
81C23×C3⋊D3—32no—
82C23×C3×D3C2×D314yes—
83C22×D32—38no—
84C22×C32×C4—24no—
85C22×C32×C4—31no—
86C22×C32×C4C2×D314yescomplemented
87C22×C32×C4C2×D311yescharacteristic, complemented
88C2×#638—32no—
89C2×#638—38no—
90C2×#638—32no—
91C2×#638—32no—
92C2×#638—68no—
93C2×#638—68no—
94C2×#638C2×D312yescomplemented
95C22×C3⋊Dic3—32no—
96C2×C4⋊(C3⋊D3)—32no—
97C2×C4⋊(C3⋊D3)—31no—
98C2×C4⋊(C3⋊D3)—34no—
99C2×C4⋊(C3⋊D3)—64no—
100C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
101C2×C3×#67—38no—
102C22×C3×Dic3C2×D314yescomplemented
103C2×C3×D12—34no—
104D3×#67—632no—
105C2×#626—38no—
106C2×#626—38no—
107C2×#625—38no—
108C2×D3×Dic3—38no—
109C32×C22⋊C4—32no—
110C32×C22⋊C4—32no—
111#3752—32no—
112#3752—32no—
113#3752—38no—
114#3752—32no—
115#3752—68no—
116#3752C2×D312yescomplemented
117C4⋊(C3⋊Dic3)—31no—
118C4⋊(C3⋊Dic3)—34no—
119C4⋊(C3⋊Dic3)—32no—
120C4⋊(C3⋊Dic3)—64no—
121C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
122C3×(C2×D3)⋊C4—38no—
123C3×(C2×D3)⋊C4C2×D318yescomplemented
124C3×(C2×D3)⋊C4C2×D318yescomplemented
125C3×C4⋊Dic3—34no—
126Dic3⋊Dic3—38no—
127#3722—38no—
128(C2×D3)⋊Dic3—38no—
129C22×C33—21no—
130C22×C33—21no—
131C22×C33C2×D412yes—
132C22×C33C2411yescharacteristic
133C2×C3⋊(C3⋊D3)—22no—
134C2×C3⋊(C3⋊D3)—22no—
135C2×C3⋊(C3⋊D3)—22no—
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—22no—
139C2×C32×D3—22no—
140C2×C32×D3—22no—
141C2×C32×D3—44no—
142C2×C32×D3C2×D414yes—
143D3×C3⋊D3—48no—
144C33×C4—21no—
145C33×C4—21no—
146C3⋊(C3⋊Dic3)—22no—
147C3×C3⋊Dic3—22no—
148C32×Dic3—22no—
149C22×#67—98no—
150C2×D3×D4—92no—
151C3×#121—91no—
152C2×C3×C22⋊C4—34no—
153#939—92no—
154#937—92no—
155#930—34no—
156#930—98no—
157#930—94no—
158#930—91no—
159#930—184no—
160C2×(C2×D3)⋊C4C3⋊D311yescharacteristic, complemented
161#894—92no—
162#886—98no—
163#886—98no—
164#883—92no—
165#882—98no—
166#882—98no—
167#882—92no—
168C23×C32—24no—
169C23×C32—32no—
170C23×C32—34no—
171C23×C32—121no—
172C23×C32—121no—
173C23×C32C22×D314yes—
174C23×C32C22×D311yescharacteristic
175C22×C3⋊D3—32no—
176C22×C3⋊D3—38no—
177C22×C3⋊D3—68no—
178C22×C3⋊D3—64no—
179C22×C3⋊D3—122no—
180C22×C3⋊D3—122no—
181C22×C3⋊D3C22×D312yes—
182C22×C3×D3—38no—
183C22×C3×D3—44no—
184C22×C3×D3—44no—
185C22×C3×D3D12116yescomplemented
186C22×C3×D3C22×D318yes—
187C2×D32—616no—
188C2×D32—128no—
189C2×D32—128no—
190C2×C32×C4—24no—
191C2×C32×C4—24no—
192C2×C32×C4—32no—
193C2×C32×C4—41no—
194C2×C32×C4—44no—
195C2×C32×C4—48no—
196C2×C32×C4—121no—
197C2×C32×C4C22×D314yes—
198C2×C32×C4C22×D314yes—
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201#638—24no—
202#638—64no—
203#638—64no—
204#638—64no—
205#638—616no—
206#638—1216no—
207C2×C3⋊Dic3—32no—
208C2×C3⋊Dic3—38no—
209C2×C3⋊Dic3—68no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3C22×D312yes—
212C4⋊(C3⋊D3)—41no—
213C4⋊(C3⋊D3)—121no—
214C4⋊(C3⋊D3)—124no—
215C4⋊(C3⋊D3)—128no—
216C4⋊(C3⋊D3)—122no—
217C3×#67—616no—
218C2×C3×Dic3—38no—
219C2×C3×Dic3—44no—
220C2×C3×Dic3C22×D318yes—
221C3×D12—124no—
222#626—616no—
223#626—128no—
224#625—616no—
225D3×Dic3—616no—
226#534—271noSylow
227C2×C33—22no—
228C2×C33C22×D411yescharacteristic
229C2×C33H(5,𝔽2)11yescharacteristic
230C2×C33H(5,𝔽2)11yescharacteristic
231C3⋊(C3⋊D3)—42no—
232C3×C3⋊D3—42no—
233C32×D3—24no—
234C24×C3—34no—
235C23×D3—98no—
236C23×D3—92no—
237C23×D3C2×C3⋊D311yescharacteristic
238C2×C3×D4—92no—
239C2×C3×D4—91no—
240C22×C3×C4—24no—
241C22×C3×C4—34no—
242C22×C3×C4D3214yescomplemented
243C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
244C2×#67—38no—
245C2×#67—98no—
246C2×#67—98no—
247C2×#67—98no—
248C2×#67—92no—
249C2×#67—92no—
250C2×#67—92no—
251C2×#67—188no—
252C2×#67—1832no—
253C22×Dic3—98no—
254C22×Dic3C2×C3⋊D311yescharacteristic, complemented
255D3×D4—188no—
256C2×D12—34no—
257C2×D12—92no—
258C2×D12—98no—
259C2×D12—94no—
260C2×D12—91no—
261C2×D12—184no—
262C2×C4×D3—92no—
263C3×C4⋊C4—91no—
264C3×C22⋊C4—38no—
265C3×C22⋊C4—38no—
266C3×C22⋊C4—92no—
267#217—92no—
268(C2×D3)⋊C4—38no—
269(C2×D3)⋊C4—98no—
270(C2×D3)⋊C4—98no—
271(C2×D3)⋊C4—98no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—188no—
275(C2×D3)⋊C4C2×C3⋊D312yescomplemented
276(C2×D3)⋊C4C2×C3⋊D312yescomplemented
277C4⋊Dic3—34no—
278C4⋊Dic3—98no—
279C4⋊Dic3—94no—
280C4⋊Dic3—91no—
281C4⋊Dic3—184no—
282Dic3⋊C4—92no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—24no—
286C22×C32—28no—
287C22×C32—24no—
288C22×C32—24no—
289C22×C32—34no—
290C22×C32—44no—
291C22×C32—44no—
292C22×C32—62no—
293C22×C32—62no—
294C22×C32—124no—
295C22×C32C2×D1218yes—
296C22×C32D3×D412yes—
297C22×C32C23×D314yes—
298C22×C32C23×D311yescharacteristic
299C2×C3⋊D3—22no—
300C2×C3⋊D3—22no—
301C2×C3⋊D3—22no—
302C2×C3⋊D3—62no—
303C2×C3⋊D3—62no—
304C2×C3⋊D3—62no—
305C2×C3⋊D3—68no—
306C2×C3⋊D3—68no—
307C2×C3⋊D3—68no—
308C2×C3⋊D3—128no—
309C2×C3⋊D3—128no—
310C2×C3⋊D3—128no—
311C2×C3⋊D3—128no—
312C2×C3×D3—28no—
313C2×C3×D3—28no—
314C2×C3×D3—416no—
315C2×C3×D3—68no—
316C2×C3×D3—68no—
317C2×C3×D3—68no—
318C2×C3×D3C2×D12116yes—
319D32—1232no—
320C32×C4—28no—
321C32×C4—21no—
322C32×C4—24no—
323C32×C4—21no—
324C32×C4—24no—
325C32×C4—44no—
326C32×C4—62no—
327C3⋊Dic3—22no—
328C3⋊Dic3—62no—
329C3⋊Dic3—68no—
330C3⋊Dic3—128no—
331C3×Dic3—28no—
332C3×Dic3—68no—
333C22×D4—272no—
334#123—272no—
335#123—272no—
336#121—91no—
337#121—272no—
338#121—271no—
339#120—272no—
340#120—272no—
341C2×C22⋊C4—31no—
342C33#53411yescharacteristic, complemented, Sylow
343C23×C3—24no—
344C23×C3—38no—
345C23×C3—316no—
346C23×C3—92no—
347C23×C3—124no—
348C23×C3—124no—
349C23×C3C2×D3214yes—
350C23×C3C22×C3⋊D311yescharacteristic
351C22×D3—38no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—98no—
355C22×D3—92no—
356C22×D3—188no—
357C22×D3—1816no—
358C22×D3—184no—
359C22×D3—362no—
360C22×D3—362no—
361C22×D3—368no—
362C22×D3—368no—
363C22×D3C4⋊(C3⋊D3)14yescomplemented
364C22×D3C22×C3⋊D312yes—
365C3×D4—184no—
366C3×D4—361no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—38no—
370C2×C3×C4—44no—
371C2×C3×C4—41no—
372C2×C3×C4—48no—
373C2×C3×C4—92no—
374C2×C3×C4—124no—
375C2×C3×C4C2×D3214yes—
376C2×C3×C4C2×D3214yes—
377C2×C3×C4C22×C3⋊D311yescharacteristic
378C2×C3×C4C22×C3⋊D311yescharacteristic
379#67—616no—
380#67—1816no—
381#67—1816no—
382#67—1816no—
383#67—184no—
384#67—184no—
385#67—3616no—
386#67—362no—
387C2×Dic3—38no—
388C2×Dic3—41no—
389C2×Dic3—98no—
390C2×Dic3—92no—
391C2×Dic3—188no—
392C2×Dic3—1816no—
393C2×Dic3C22×C3⋊D312yes—
394D12—124no—
395D12—184no—
396D12—364no—
397D12—361no—
398D12—368no—
399D12—368no—
400C4×D3—184no—
401C2×C32—22no—
402C2×C32—24no—
403C2×C32—24no—
404C2×C32—24no—
405C2×C32—28no—
406C2×C32—48no—
407C2×C32—64no—
408C2×C32C22×D1214yes—
409C2×C32C2×D3×D411yescharacteristic
410C2×C32#100414yes—
411C2×C32#100414yes—
412C2×C32#100411yescharacteristic
413C2×C32#100911yescharacteristic
414C3⋊D3—42no—
415C3⋊D3—122no—
416C3⋊D3—128no—
417C3⋊D3—248no—
418C3×D3—216no—
419C3×D3—128no—
420C24—31no—
421C24—272no—
422C2×D4—92no—
423C2×D4—91no—
424C2×D4—272no—
425C2×D4—272no—
426C2×D4—272no—
427C2×D4—272no—
428C2×D4—271no—
429C2×D4—548no—
430C22×C4—31no—
431C22×C4—272no—
432C22×C4D3×C3⋊D311yescharacteristic, complemented
433C4⋊C4—91no—
434C4⋊C4—272no—
435C4⋊C4—271no—
436C22⋊C4—32no—
437C22⋊C4—32no—
438C22⋊C4—92no—
439C22⋊C4—272no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C22×C3—24no—
443C22×C3—24no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—24no—
447C22×C3—28no—
448C22×C3—316no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—68no—
452C22×C3—68no—
453C22×C3—1216no—
454C22×C3—182no—
455C22×C3—182no—
456C22×C3—182no—
457C22×C3D3×D1218yes—
458C22×C3C2×C4⋊(C3⋊D3)12yes—
459C22×C3C22×D3214yes—
460C22×C3C23×C3⋊D311yescharacteristic
461C2×D3—22no—
462C2×D3—22no—
463C2×D3—44no—
464C2×D3—68no—
465C2×D3—68no—
466C2×D3—68no—
467C2×D3—188no—
468C2×D3—188no—
469C2×D3—188no—
470C2×D3—182no—
471C2×D3—182no—
472C2×D3—182no—
473C2×D3—368no—
474C2×D3—368no—
475C2×D3—368no—
476C2×D3—368no—
477C2×D3—3632no—
478C2×D3C2×C4⋊(C3⋊D3)14yes—
479C3×C4—28no—
480C3×C4—24no—
481C3×C4—24no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—44no—
485C3×C4—68no—
486C3×C4—182no—
487Dic3—22no—
488Dic3—68no—
489Dic3—188no—
490Dic3—182no—
491Dic3—368no—
492C32—24no—
493C32#549314yescomplemented
494C32#560511yescharacteristic, complemented
495C23—32no—
496C23—34no—
497C23—92no—
498C23—121no—
499C23—121no—
500C23—272no—
501C23—544no—
502C23—1082no—
503C23—1082no—
504C23C2×D3×C3⋊D311yescharacteristic
505D4—184no—
506D4—361no—
507D4—544no—
508D4—544no—
509D4—544no—
510D4—1081no—
511D4—1082no—
512C2×C4—32no—
513C2×C4—41no—
514C2×C4—92no—
515C2×C4—121no—
516C2×C4—272no—
517C2×C4—544no—
518C2×C4C2×D3×C3⋊D311yescharacteristic
519C2×C4C2×D3×C3⋊D311yescharacteristic
520C2×C3—28no—
521C2×C3—22no—
522C2×C3—24no—
523C2×C3—24no—
524C2×C3—24no—
525C2×C3—48no—
526C2×C3—616no—
527C2×C3—362no—
528C2×C3C2×D3×D1214yes—
529C2×C3#6430714yes—
530C2×C3#6433014yes—
531C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
532C2×C3#6436111yescharacteristic
533C2×C3#6436111yescharacteristic
534D3—24no—
535D3—128no—
536D3—368no—
537D3—362no—
538D3—728no—
539C22—21no—
540C22—21no—
541C22—34no—
542C22—62no—
543C22—62no—
544C22—124no—
545C22—182no—
546C22—182no—
547C22—182no—
548C22—542no—
549C22—542no—
550C22—542no—
551C22—1088no—
552C22D3×C4⋊(C3⋊D3)12yes—
553C22C22×D3×C3⋊D311yescharacteristic
554C4—21no—
555C4—21no—
556C4—62no—
557C4—182no—
558C4—542no—
559C3—24no—
560C3#1059507214yescomplemented
561C3#1059596411yescharacteristic, complemented
562C2—22no—
563C2—64no—
564C2—362no—
565C2—1082no—
566C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
567C2#1172457911yescharacteristic
568C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#534
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×#638C2semidirect
2. #11723971C2semidirect
3. C3×#64139C2semidirect
4. C2×C32×(C2×D3)⋊C4C2semidirect
5. #11723632C2semidirect
6. #11723542C2semidirect
7. #11723541C2semidirect
8. #11723480C2semidirect
9. #11723479C2semidirect
13. C22×D3×C3⋊D3C22semidirect
14. C22×C33×C4C22semidirect
15. C2×#6312C22semidirect
16. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
17. C2×C3×#638C22semidirect
18. C2×C3×C4⋊(C3⋊D3)C22semidirect
19. C22×C32×Dic3C22semidirect
21. C2×#6292C22semidirect
24. C2×D3×C3⋊Dic3C22semidirect
25. #88859C22semidirect
26. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
27. C3×#3752C22semidirect
28. C3×C4⋊(C3⋊Dic3)C22semidirect
29. C32×(C2×D3)⋊C4C22semidirect
30. C32×(C2×D3)⋊C4C22semidirect
31. Dic3⋊(C3⋊Dic3)C22semidirect
32. #88804C22semidirect
33. #88803C22semidirect
41. #64139D3semidirect
47. C2×C3×(C2×D3)⋊C4D3semidirect
58. C22×C32×D3D4semidirect
86. C22×C32×C4C2×D3semidirect
87. C22×C32×C4C2×D3semidirect
94. C2×#638C2×D3semidirect
100. C2×C4⋊(C3⋊D3)C2×D3semidirect
102. C22×C3×Dic3C2×D3semidirect
116. #3752C2×D3semidirect
121. C4⋊(C3⋊Dic3)C2×D3semidirect
123. C3×(C2×D3)⋊C4C2×D3semidirect
124. C3×(C2×D3)⋊C4C2×D3semidirect
160. C2×(C2×D3)⋊C4C3⋊D3semidirect
185. C22×C3×D3D12semidirect
242. C22×C3×C4D32semidirect
243. C22×C3×C4C2×C3⋊D3semidirect
254. C22×Dic3C2×C3⋊D3semidirect
275. (C2×D3)⋊C4C2×C3⋊D3semidirect
276. (C2×D3)⋊C4C2×C3⋊D3semidirect
342. C33#534semidirect
363. C22×D3C4⋊(C3⋊D3)semidirect
432. C22×C4D3×C3⋊D3semidirect
493. C32#5493semidirect
494. C32#5605semidirect
560. C3#10595072semidirect
561. C3#10595964semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6382
#117239712
C3×#641392
C2×C32×(C2×D3)⋊C42
#117236322
#117235422
#117235412
#117234802
#117234792
#105959643
#105950723