Finite Group Databasebeta

← #11069470

Subgroups — #11069470

Order 768.

2354 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5619C212yesmaximal, complemented
2C3×#2535C211yesmaximal, characteristic, complemented
3#82644C212yesmaximal, complemented
4#82415C211yesmaximal, characteristic, complemented
5#82333C211yesmaximal, characteristic, complemented
6C4×#940C211yesmaximal, characteristic, complemented
7#82319C211yesmaximal, characteristic, complemented
8#77595C214yesmaximal, complemented
9#76716C214yesmaximal, complemented
10#74895C214yesmaximal, complemented
11#74696C214yesmaximal, complemented
12#74228C211yesmaximal, characteristic, complemented
13#74220C211yesmaximal, characteristic, complemented
14#73955C212yesmaximal, complemented
15#73893C212yesmaximal, complemented
16#39372—31nomaximal, Sylow
17C2×C3×#121C2212yes—
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#940C2211yescharacteristic
21C2×#939C2214yes—
22C2×#938C2214yes—
23C2×#935C2214yes—
24C2×D4×Dic3C2212yes—
25C2×#930C2212yes—
26C2×C4×#67C2214yes—
27C2×#926C2212yes—
28#5619—216no—
29C2×#893C2212yes—
30C2×#890C2212yes—
31C2×#886C2214yes—
32C2×#883C2214yes—
33C2×#883C2214yes—
34C2×C4×D12C2211yescharacteristic
35C3×#397C2212yes—
36C3×#392C2212yes—
37C3×#386C2214yes—
38C3×#382C2211yescharacteristic
39C3×#380C2212yes—
40#5223C2212yes—
41#5222C2214yes—
42#5221C2212yes—
43#5221C2211yescharacteristic
44#5212C2212yes—
45#5212C2212yes—
46C4×#217C2212yes—
47C2×C3⋊#95C2212yes—
48#4992C2212yes—
49#4988C2211yescharacteristic
50#4984C2212yes—
51#4984C2212yes—
52#4975C2211yescharacteristic
53#4975C2212yes—
54#4972C2212yes—
55#4959C2214yes—
56#4957C2214yes—
57#4956C2214yes—
58#4952C2214yes—
59#4951C2212yes—
60#4951C2212yes—
61#4950C2214yes—
62#4948C2214yes—
63#4948C2214yes—
64#4946C2214yes—
65#4936C2211yescharacteristic
66C4×C4⋊Dic3C2211yescharacteristic
67C42×Dic3C2211yescharacteristic
68#4675C2214yes—
69#4668C2214yes—
70#4668C2214yes—
71#4667C2212yes—
72D3×#95C2212yes—
73#4661C2214yes—
74#4657C2214yes—
75#4647C2214yes—
76#4647C2214yes—
77C2×#540—32no—
78#2859—31no—
79#2724—34no—
80#2682—34no—
81#2629—32no—
82#2535—31no—
83#2535D311yescharacteristic, complemented
84#2425—34no—
85#2425—34no—
86#2422—32no—
87#2422—32no—
88#2349—31no—
89#2349—31no—
90C4×#127—31no—
91#2293—31no—
92C22×C3×D4C2311yescharacteristic
93C22×C3×D4C2311yescharacteristic
94C22×C3×D4C2314yes—
95C23×C3×C4C2312yes—
96C23×C3×C4C2312yes—
97C22×#67C2314yes—
98C22×#67C2314yes—
99C23×Dic3C2314yes—
100C22×D12C2311yescharacteristic
101C22×C4×D3C2312yes—
102C22×C4×D3C2314yes—
103C3×#121—28no—
104C3×C4×D4—24no—
105C3×C4×D4—24no—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4C2314yes—
110C2×C3×C22⋊C4C2312yes—
111C2×C3×C22⋊C4C2312yes—
112C2×C3×C22⋊C4C2314yes—
113C2×C3×C42C2311yescharacteristic
114C2×#217C2314yes—
115C2×#217C2312yes—
116C2×#217C2312yes—
117C2×#217C2314yes—
118C2×#217C2312yes—
119C2×#217C2312yes—
120C2×#217C2314yes—
121#940D418yescomplemented
122#939—216no—
123#938—216no—
124#935—48no—
125D4×Dic3—28no—
126#930—28no—
127C4×#67—48no—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132#926—28no—
133C2×C4⋊Dic3C2314yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2311yescharacteristic
138C2×C4⋊Dic3C2314yes—
139C2×C4⋊Dic3C2312yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2314yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2311yescharacteristic
147C2×C4×Dic3C2312yes—
148C2×C4×Dic3C2312yes—
149C2×C4×Dic3C2311yescharacteristic
150#893—28no—
151#890—28no—
152#886—216no—
153#883—216no—
154#883—48no—
155C4×D12—24no—
156C3×#95C2312yes—
157C3×#95C2312yes—
158C3×#95C2314yes—
159C3⋊#95C2312yes—
160C3⋊#95C2312yes—
161C3⋊#95C2314yes—
162C3⋊#95C2314yes—
163C3⋊#95C2314yes—
164C3⋊#95C2311yescharacteristic
165C3⋊#95C2311yescharacteristic
166C3⋊#95C2312yes—
167C3⋊#95C2312yes—
168#540—616no—
169C2×#127—31no—
170C2×#125—32no—
171C2×#123—34no—
172C2×#123—34no—
173C2×#123—34no—
174C2×#123—34no—
175C2×#121—34no—
176C2×#121—34no—
177C2×#121—32no—
178C2×#121C2×D312yes—
179C2×C4×D4—32no—
180C2×C4×D4—34no—
181C2×C4×D4—32no—
182C2×C4×D4—31no—
183C2×C4×D4C2×D311yescharacteristic
184C2×C4×D4C2×D311yescharacteristic
185C2×#117—32no—
186#398—34no—
187#398—34no—
188#397—32no—
189#397—34no—
190#397—34no—
191#397C2×D312yes—
192#396—34no—
193#396—34no—
194#394—34no—
195#394—34no—
196#392—32no—
197#392C2×D312yes—
198#390—31no—
199#390—32no—
200#390—31no—
201#390—34no—
202#389—32no—
203#389—32no—
204#389—32no—
205#389—34no—
206#389—34no—
207#388—32no—
208#388—32no—
209#386—34no—
210#386C2×D314yes—
211#385—34no—
212#385—34no—
213#385—34no—
214#385—34no—
215#384—31no—
216#384—32no—
217#382—31no—
218#382C2×D311yescharacteristic
219#380—32no—
220#380—32no—
221#380—32no—
222#380C2×D312yes—
223C4×C4⋊C4—31no—
224C4×C22⋊C4—32no—
225C2×#95—32no—
226C2×#95—32no—
227C43—31no—
228C24×C3C2412yes—
229C24×C3C2412yes—
230C23×D3C2412yes—
231C2×C3×D4—28no—
232C2×C3×D4—22no—
233C2×C3×D4—22no—
234C2×C3×D4—22no—
235C2×C3×D4—42no—
236C2×C3×D4—44no—
237C2×C3×D4—44no—
238C2×C3×D4C2×D414yes—
239C22×C3×C4—24no—
240C22×C3×C4—24no—
241C22×C3×C4—24no—
242C22×C3×C4—24no—
243C22×C3×C4—44no—
244C22×C3×C4C2412yes—
245C22×C3×C4C2412yes—
246C22×C3×C4C2414yes—
247C22×C3×C4C2411yescharacteristic
248C22×C3×C4C2411yescharacteristic
249C22×C3×C4C2411yescharacteristic
250C2×#67—28no—
251C2×#67—44no—
252C2×#67—44no—
253C2×#67—44no—
254C2×#67—44no—
255C2×#67C2×D4116yes—
256C22×Dic3—28no—
257C22×Dic3—44no—
258C22×Dic3—44no—
259C22×Dic3C2412yes—
260C22×Dic3C2412yes—
261C22×Dic3C2412yes—
262C22×Dic3C2414yes—
263C22×Dic3C2412yes—
264C22×Dic3C2412yes—
265C2×D12—22no—
266C2×D12—22no—
267C2×D12C2×D414yes—
268C2×C4×D3—28no—
269C2×C4×D3—24no—
270C2×C4×D3—44no—
271C2×C4×D3—44no—
272C2×C4×D3—44no—
273C3×C4⋊C4—24no—
274C3×C4⋊C4—22no—
275C3×C4⋊C4—22no—
276C3×C22⋊C4—28no—
277C3×C22⋊C4—24no—
278C3×C22⋊C4—24no—
279C3×C22⋊C4—44no—
280C3×C42#4114yes—
281#217—28no—
282#217—24no—
283#217—24no—
284#217—24no—
285#217—28no—
286#217—24no—
287#217—44no—
288(C2×D3)⋊C4—28no—
289(C2×D3)⋊C4—28no—
290(C2×D3)⋊C4—24no—
291(C2×D3)⋊C4—28no—
292C4⋊Dic3—24no—
293C4⋊Dic3—22no—
294C4⋊Dic3—24no—
295C4⋊Dic3—24no—
296C4⋊Dic3—28no—
297C4⋊Dic3—44no—
298C4⋊Dic3—42no—
299Dic3⋊C4—28no—
300Dic3⋊C4—28no—
301Dic3⋊C4—44no—
302Dic3⋊C4—44no—
303Dic3⋊C4—44no—
304C4×Dic3—24no—
305C4×Dic3—22no—
306C4×Dic3—24no—
307C4×Dic3—42no—
308C4×Dic3C2×D414yes—
309C22×D4—34no—
310C22×D4—34no—
311C22×D4—31no—
312C22×D4C22×D311yescharacteristic
313C22×D4C22×D311yescharacteristic
314C22×D4C22×D314yes—
315C23×C4—34no—
316C23×C4—32no—
317C23×C4—34no—
318C23×C4C22×D312yes—
319C23×C4C22×D312yes—
320#127—38no—
321#125—68no—
322#123—616no—
323#123—616no—
324#123—128no—
325#123—128no—
326#121—28no—
327#121—68no—
328#121—616no—
329#121—616no—
330C4×D4—24no—
331C4×D4—24no—
332C4×D4—68no—
333C4×D4—68no—
334C4×D4—64no—
335C4×D4—128no—
336#117—68no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—32no—
339C2×C4⋊C4—32no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—34no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4—34no—
344C2×C4⋊C4—34no—
345C2×C4⋊C4—34no—
346C2×C4⋊C4—34no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4C22×D312yes—
350C2×C4⋊C4C22×D311yescharacteristic
351C2×C4⋊C4C22×D311yescharacteristic
352C2×C22⋊C4—32no—
353C2×C22⋊C4—32no—
354C2×C22⋊C4—34no—
355C2×C22⋊C4—34no—
356C2×C22⋊C4—32no—
357C2×C22⋊C4—32no—
358C2×C22⋊C4—34no—
359C2×C22⋊C4—34no—
360C2×C22⋊C4—32no—
361C2×C22⋊C4—34no—
362C2×C22⋊C4—34no—
363C2×C22⋊C4C22×D312yes—
364C2×C22⋊C4C22×D312yes—
365C2×C22⋊C4C22×D314yes—
366C2×C22⋊C4C22×D314yes—
367C2×C42—31no—
368C2×C42—32no—
369C2×C42—32no—
370C2×C42—31no—
371C2×C42—32no—
372C2×C42C22×D311yescharacteristic
373#95—31no—
374#95—31no—
375#95—32no—
376#95—34no—
377#95—34no—
378#95—32no—
379#95—32no—
380#95—34no—
381#95—32no—
382#95C22×D312yes—
383#95C22×D314yes—
384#95C22×D312yes—
385C23×C3—22no—
386C23×C3—24no—
387C23×C3—22no—
388C23×C3—24no—
389C23×C3—22no—
390C23×C3—22no—
391C23×C3—22no—
392C23×C3—22no—
393C23×C3—22no—
394C23×C3—22no—
395C23×C3—22no—
396C23×C3C22×D414yes—
397C23×C3C2511yescharacteristic
398C22×D3—22no—
399C22×D3—24no—
400C22×D3—22no—
401C22×D3—24no—
402C22×D3C22×D414yes—
403C3×D4—28no—
404C3×D4—44no—
405C3×D4—88no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—24no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—24no—
413C2×C3×C4—24no—
414C2×C3×C4—22no—
415C2×C3×C4—24no—
416C2×C3×C4—48no—
417C2×C3×C4—84no—
418C2×C3×C4C22×D412yes—
419C2×C3×C4C2×#4112yes—
420C2×C3×C4C2×#4112yes—
421C2×C3×C4C2×#4112yes—
422C2×C3×C42−1+412yes—
423#67—416no—
424#67—88no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—24no—
432C2×Dic3—22no—
433C2×Dic3—24no—
434C2×Dic3—24no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—24no—
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—88no—
441C2×Dic3C22×D414yes—
442C2×Dic3H(5,𝔽2)14yes—
443D12—28no—
444C4×D3—84no—
445C4×D3—88no—
446C24—32no—
447C24C23×D312yes—
448C24C23×D312yes—
449C2×D4—28no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—316no—
454C2×D4—34no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—68no—
459C2×D4—62no—
460C2×D4—62no—
461C2×D4—124no—
462C2×D4—124no—
463C2×D4—124no—
464C2×D4—124no—
465C2×D4D3×D414yes—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—32no—
471C22×C4—32no—
472C22×C4—32no—
473C22×C4—32no—
474C22×C4—32no—
475C22×C4—34no—
476C22×C4—44no—
477C22×C4—68no—
478C22×C4—68no—
479C22×C4—64no—
480C22×C4—124no—
481C22×C4—124no—
482C22×C4—124no—
483C22×C4—124no—
484C22×C4—124no—
485C22×C4C23×D312yes—
486C22×C4C23×D312yes—
487C22×C4C23×D314yes—
488C22×C4C23×D311yescharacteristic
489C22×C4C23×D311yescharacteristic
490C22×C4C23×D311yescharacteristic
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—64no—
495C4⋊C4—64no—
496C4⋊C4—68no—
497C4⋊C4—68no—
498C4⋊C4—68no—
499C4⋊C4—64no—
500C4⋊C4—62no—
501C4⋊C4—124no—
502C4⋊C4—124no—
503C4⋊C4—124no—
504C4⋊C4—124no—
505C4⋊C4—122no—
506C22⋊C4—28no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—44no—
510C22⋊C4—68no—
511C22⋊C4—68no—
512C22⋊C4—68no—
513C22⋊C4—64no—
514C22⋊C4—64no—
515C22⋊C4—68no—
516C22⋊C4—64no—
517C22⋊C4—64no—
518C22⋊C4—68no—
519C22⋊C4—64no—
520C22⋊C4—124no—
521C42—34no—
522C42—64no—
523C42—64no—
524C42—62no—
525C42—122no—
526C42#23714yes—
527C22×C3—24no—
528C22×C3—28no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—44no—
532C22×C3—42no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—42no—
538C22×C3—42no—
539C22×C3C23×D411yescharacteristic
540C22×C3C22×#4111yescharacteristic
541C22×C3C2×H(5,𝔽2)12yes—
542C22×C3C2×2−1+411yescharacteristic
543C22×C3#58512yes—
544C2×D3—24no—
545C2×D3—28no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—44no—
549C3×C4—22no—
550C3×C4—22no—
551C3×C4—42no—
552C3×C4—44no—
553C3×C4—42no—
554C3×C4#54714yes—
555Dic3—24no—
556Dic3—24no—
557Dic3—42no—
558Dic3—42no—
559Dic3—44no—
560Dic3—42no—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—24no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—34no—
573C23—62no—
574C23—62no—
575C23—64no—
576C23—64no—
577C23C2×D3×D414yes—
578C23C24×D311yescharacteristic
579D4—28no—
580D4—44no—
581D4—68no—
582D4—88no—
583D4—1216no—
584D4—248no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—34no—
596C2×C4—34no—
597C2×C4—48no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—64no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—64no—
613C2×C4—84no—
614C2×C4—244no—
615C2×C4—248no—
616C2×C4—248no—
617C2×C4C2×D3×D412yes—
618C2×C4C2×#23712yes—
619C2×C4C2×#23712yes—
620C2×C4C2×#23712yes—
621C2×C4#101012yes—
622C2×C3—44no—
623C2×C3—82no—
624C2×C3#345511yescharacteristic
625C2×C3C2×#54711yescharacteristic
626C2×C3#349612yes—
627C2×C3#350811yescharacteristic
628C2×C3#355912yes—
629D3—44no—
630C22—28no—
631C22—24no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—68no—
643C22—64no—
644C22—122no—
645C22—122no—
646C22—124no—
647C22C22×D3×D411yescharacteristic
648C22C22×#23711yescharacteristic
649C22C2×#101011yescharacteristic
650C22#596412yes—
651C22D3×H(5,𝔽2)12yes—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—64no—
658C4—64no—
659C4—122no—
660C4—122no—
661C4—122no—
662C4—124no—
663C4#561114yes—
664C3#3937211yescharacteristic, complemented, Sylow
665C2—44no—
666C2—82no—
667C2—124no—
668C2C2×#561111yescharacteristic
669C2#8601911yescharacteristic
670C2#8631911yescharacteristic
671C2#8666512yes—
672C2#8691912yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2535C2
hypercenter#2535D3
hypocenterC3#39372
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. C2×#5619C2semidirect
2. C3×#2535C2semidirect
3. #82644C2semidirect
4. #82415C2semidirect
5. #82333C2semidirect
6. C4×#940C2semidirect
7. #82319C2semidirect
8. #77595C2semidirect
9. #76716C2semidirect
10. #74895C2semidirect
11. #74696C2semidirect
12. #74228C2semidirect
13. #74220C2semidirect
14. #73955C2semidirect
15. #73893C2semidirect
83. #2535D3semidirect
121. #940D4semidirect
664. C3#39372semidirect

Maximal subgroups

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

subgroupindex
C2×#56192
C3×#25352
#826442
#824152
#823332
C4×#9402
#823192
#775952
#767162
#748952
#746962
#742282
#742202
#739552
#738932
#393723