Finite Group Databasebeta

← #11017262

Subgroups — #11017262

Order 768.

3114 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#535C211yesmaximal, characteristic, complemented
2C2×#5830C211yesmaximal, characteristic, complemented
3#84086C211yesmaximal, characteristic, complemented
4#84049C212yesmaximal, complemented
5#83190C214yesmaximal, complemented
6#82991C214yesmaximal, complemented
7#82700C212yesmaximal, complemented
8#82676C212yesmaximal, complemented
9#82663C212yesmaximal, complemented
10#82653C212yesmaximal, complemented
11#82597C214yesmaximal, complemented
12#82594C214yesmaximal, complemented
13#82257C212yesmaximal, complemented
14#37297—31nomaximal, Sylow
15C22×C3×#41C2211yescharacteristic
16C23×C3×D4C2211yescharacteristic, complemented
17C23×D12C2211yescharacteristic, complemented
18C3×#535—28no—
19C2×C3×#127C2212yes—
20C2×C3×#124C2212yescomplemented
21C2×C3×#121C2212yes—
22C2×C3×#121C2212yescomplemented
23C2×C3×#120C2212yes—
24C2×C3×#120C2212yescomplemented
25C2×C3×#117C2211yescharacteristic, complemented
26C2×#953C2218yescomplemented
27#5830—28no—
28C2×#947C2212yescomplemented
29C2×#940C2212yes—
30C2×#940C2212yes—
31C2×#940C2212yes—
32C2×#939C2218yescomplemented
33C2×#937C2212yes—
34C2×#937C2212yescomplemented
35C2×#937C2218yescomplemented
36C2×#930C2212yes—
37C2×#930C2212yescomplemented
38C2×#926C2211yescharacteristic, complemented
39C2×#926C2211yescharacteristic, complemented
40C22×C4×Dic3C2211yescharacteristic, complemented
41#5229C2212yescomplemented
42#5223C2218yescomplemented
43#5223C2218yescomplemented
44#5222C2218yescomplemented
45#5221C2212yes—
46#5221C2212yescomplemented
47#5221C2212yescomplemented
48#5220C2212yescomplemented
49#5220C2212yescomplemented
50C4×#217C2212yescomplemented
51C4×#217C2212yescomplemented
52#4988C2212yescomplemented
53#4988C2212yescomplemented
54#4977C2212yescomplemented
55#4975C2212yescomplemented
56#4961C2218yescomplemented
57#4960C2218yescomplemented
58#4959C2218yescomplemented
59#4957C2218yescomplemented
60#4944C2218yescomplemented
61#4940C2218yescomplemented
62C2×#535—31no—
63C2×#535D311yescharacteristic, complemented
64#2856—32no—
65#2733—34no—
66#2679—34no—
67#2669—34no—
68#2669—34no—
69#2619—32no—
70#2500—32no—
71#2500—32no—
72#2433—32no—
73#2433—32no—
74#2297—31no—
75C25×C3C2312yes—
76C24×D3C2312yes—
77C2×C3×#41—42no—
78C2×C3×#41—42no—
79C2×C3×#41D418yescomplemented
80C22×C3×Q8C2311yescharacteristic
81C22×C3×D4—24no—
82C22×C3×D4—42no—
83C22×C3×D4—42no—
84C22×C3×D4C2312yes—
85C22×C3×D4C2312yes—
86C22×C3×D4C2314yes—
87C22×C3×D4C2314yes—
88C22×C3×D4C2311yescharacteristic
89C22×C3×D4C2311yescharacteristic
90C22×C3×D4C2311yescharacteristic
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2311yescharacteristic
93C23×C3×C4C2311yescharacteristic
94C22×#67C2318yes—
95C22×#67C2314yes—
96C22×#67C2314yes—
97C22×#67C2318yes—
98C23×Dic3C2312yes—
99C22×D12—24no—
100C22×D12—42no—
101C22×D12—42no—
102C22×D12C2312yes—
103C22×D12C2312yes—
104C3×#127—28no—
105C3×#124—28no—
106C3×#121—28no—
107C3×#121—28no—
108C3×#120—28no—
109C3×#120—28no—
110C3×#117—24no—
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C22⋊C4C2314yes—
113C2×C3×C22⋊C4C2314yes—
114C2×C3×C22⋊C4C2312yes—
115C2×C3×C42C2312yes—
116#953—416no—
117C2×#217C2318yes—
118C2×#217C2314yes—
119C2×#217C2314yes—
120C2×#217C2312yes—
121C2×#217C2318yes—
122C2×#217C2312yes—
123#947—28no—
124#940—28no—
125#940—44no—
126#940—44no—
127#939—416no—
128#937—28no—
129#937—28no—
130#937—416no—
131#930—28no—
132#930—28no—
133C2×(C2×D3)⋊C4C2314yes—
134C2×(C2×D3)⋊C4C2318yes—
135C2×(C2×D3)⋊C4C2314yes—
136C2×(C2×D3)⋊C4C2318yes—
137#926—24no—
138#926—42no—
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2312yes—
141C2×Dic3⋊C4C2318yes—
142C2×C4×Dic3—42no—
143C2×C4×Dic3C2312yes—
144C2×C4×Dic3C2312yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2312yes—
147C3⋊#95C2314yes—
148C3⋊#95C2318yes—
149C3⋊#95C2314yes—
150C22×#41C2×D311yescharacteristic
151C23×D4—31no—
152C23×D4C2×D311yescharacteristic, complemented
153#535—28no—
154#535—68no—
155C2×#127—32no—
156C2×#127—32no—
157C2×#127—32no—
158C2×#127C2×D312yes—
159C2×#124—32no—
160C2×#124C2×D312yescomplemented
161C2×#121—32no—
162C2×#121—32no—
163C2×#121—38no—
164C2×#121C2×D312yes—
165C2×#121C2×D312yescomplemented
166C2×#120—32no—
167C2×#120—32no—
168C2×#120—38no—
169C2×#120—38no—
170C2×#120C2×D312yes—
171C2×#120C2×D312yescomplemented
172C2×#117—31no—
173C2×#117—31no—
174C2×#117C2×D311yescharacteristic, complemented
175C22×C42—31no—
176#394—38no—
177#394—38no—
178#394—38no—
179#392—38no—
180#392—38no—
181#392—38no—
182#391—32no—
183#390—32no—
184#390—32no—
185#390—32no—
186#390—32no—
187#390—32no—
188#390—38no—
189#388—38no—
190#388—38no—
191#386—32no—
192#386—32no—
193#384—32no—
194#382—32no—
195C4×C22⋊C4—32no—
196C4×C22⋊C4—32no—
197C24×C3—24no—
198C24×C3—42no—
199C24×C3—42no—
200C24×C3—48no—
201C24×C3C2414yes—
202C24×C3C2411yescharacteristic
203C24×C3C2411yescharacteristic
204C24×C3C2411yescharacteristic
205C23×D3—24no—
206C23×D3—42no—
207C23×D3—42no—
208C23×D3—48no—
209C23×D3C2414yes—
210C3×#41—416no—
211C2×C3×Q8—41no—
212C2×C3×Q8—41no—
213C2×C3×Q8C2×D414yes—
214C2×C3×D4—28no—
215C2×C3×D4—28no—
216C2×C3×D4—24no—
217C2×C3×D4—24no—
218C2×C3×D4—24no—
219C2×C3×D4—24no—
220C2×C3×D4—42no—
221C2×C3×D4—42no—
222C2×C3×D4—41no—
223C2×C3×D4—41no—
224C2×C3×D4—41no—
225C2×C3×D4—41no—
226C2×C3×D4—41no—
227C2×C3×D4—41no—
228C2×C3×D4—416no—
229C2×C3×D4—48no—
230C2×C3×D4—48no—
231C2×C3×D4C2×D414yes—
232C2×C3×D4C2×D414yes—
233C2×C3×D4C2×D414yes—
234C22×C3×C4—22no—
235C22×C3×C4—22no—
236C22×C3×C4—22no—
237C22×C3×C4—22no—
238C22×C3×C4—22no—
239C22×C3×C4—22no—
240C22×C3×C4C2×D414yes—
241C22×C3×C4C2×D414yes—
242C22×C3×C4C2×D414yes—
243C22×C3×C4C2414yes—
244C22×C3×C4C2411yescharacteristic
245C22×C3×C4C2411yescharacteristic
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2411yescharacteristic
248C2×#67—28no—
249C2×#67—28no—
250C2×#67—216no—
251C2×#67—48no—
252C2×#67—48no—
253C2×#67—48no—
254C2×#67—48no—
255C2×#67—48no—
256C2×#67—48no—
257C2×#67—48no—
258C22×Dic3—28no—
259C22×Dic3—42no—
260C22×Dic3C2414yes—
261C22×Dic3C2414yes—
262C22×Dic3C2414yes—
263C2×D12—24no—
264C2×D12—24no—
265C2×D12—24no—
266C2×D12—24no—
267C2×D12—42no—
268C2×D12—42no—
269C2×D12—416no—
270C3×C4⋊C4—24no—
271C3×C22⋊C4—24no—
272C3×C22⋊C4—28no—
273C3×C22⋊C4—28no—
274C3×C42—24no—
275#217—24no—
276#217—28no—
277#217—216no—
278#217—28no—
279#217—42no—
280#217—48no—
281(C2×D3)⋊C4—28no—
282(C2×D3)⋊C4—28no—
283(C2×D3)⋊C4—216no—
284(C2×D3)⋊C4—48no—
285C4⋊Dic3—24no—
286C4⋊Dic3—42no—
287Dic3⋊C4—48no—
288C4×Dic3—24no—
289C4×Dic3—24no—
290C4×Dic3—24no—
291C4×Dic3—42no—
292C25—32no—
293C25C22×D312yes—
294C2×#41—42no—
295C2×#41—42no—
296C2×#41#6718yescomplemented
297C22×Q8C22×D311yescharacteristic
298C22×D4—24no—
299C22×D4—34no—
300C22×D4—38no—
301C22×D4—34no—
302C22×D4—38no—
303C22×D4—32no—
304C22×D4—32no—
305C22×D4—42no—
306C22×D4—42no—
307C22×D4—64no—
308C22×D4—122no—
309C22×D4—122no—
310C22×D4C22×D312yes—
311C22×D4C22×D314yes—
312C22×D4C22×D314yes—
313C22×D4C22×D312yes—
314C22×D4C22×D311yescharacteristic
315C22×D4C22×D311yescharacteristic
316C22×D4C22×D311yescharacteristic
317C23×C4—32no—
318C23×C4C22×D311yescharacteristic
319C23×C4C22×D311yescharacteristic
320C23×C4C22×D311yescharacteristic
321#127—28no—
322#127—68no—
323#127—124no—
324#127—124no—
325#124—28no—
326#124—68no—
327#121—28no—
328#121—28no—
329#121—68no—
330#121—68no—
331#121—1216no—
332#120—28no—
333#120—28no—
334#120—68no—
335#120—68no—
336#120—1216no—
337#120—1216no—
338#117—24no—
339#117—64no—
340#117—122no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—38no—
344C2×C4⋊C4C22×D312yes—
345C2×C22⋊C4—32no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—38no—
349C2×C22⋊C4—34no—
350C2×C22⋊C4—34no—
351C2×C22⋊C4—38no—
352C2×C22⋊C4—32no—
353C2×C22⋊C4—38no—
354C2×C22⋊C4—38no—
355C2×C22⋊C4C22×D312yes—
356C2×C22⋊C4C22×D314yes—
357C2×C22⋊C4C22×D314yes—
358C2×C42—32no—
359C2×C42—32no—
360C2×C42—32no—
361C2×C42—32no—
362C2×C42—122no—
363C2×C42C22×D312yes—
364#95—34no—
365#95—34no—
366#95—38no—
367C23×C3—24no—
368C23×C3—24no—
369C23×C3—21no—
370C23×C3—24no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—24no—
377C23×C3—28no—
378C23×C3—28no—
379C23×C3—24no—
380C23×C3—24no—
381C23×C3—48no—
382C23×C3—88no—
383C23×C3—88no—
384C23×C3—88no—
385C23×C3C22×D412yes—
386C23×C3C22×D412yes—
387C23×C3C22×D412yes—
388C23×C3C2511yescharacteristic
389C22×D3—24no—
390C22×D3—28no—
391C22×D3—28no—
392C22×D3—24no—
393C22×D3—24no—
394C22×D3—48no—
395C22×D3—88no—
396C22×D3—88no—
397C22×D3—88no—
398C3×Q8—44no—
399C3×D4—44no—
400C3×D4—44no—
401C3×D4—44no—
402C3×D4—48no—
403C3×D4—48no—
404C3×D4—88no—
405C3×D4—88no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—28no—
413C2×C3×C4—28no—
414C2×C3×C4—28no—
415C2×C3×C4—24no—
416C2×C3×C4—28no—
417C2×C3×C4C22×D412yes—
418C2×C3×C4C22×D412yes—
419C2×C3×C4C22×D412yes—
420C2×C3×C4C22×D412yes—
421C2×C3×C4H(5,𝔽2)12yes—
422C2×C3×C4H(5,𝔽2)12yes—
423#67—816no—
424#67—88no—
425#67—88no—
426#67—816no—
427C2×Dic3—24no—
428C2×Dic3—24no—
429C2×Dic3—24no—
430C2×Dic3—24no—
431C2×Dic3—28no—
432C2×Dic3—24no—
433C2×Dic3—24no—
434C2×Dic3—48no—
435C2×Dic3H(5,𝔽2)18yes—
436D12—48no—
437D12—48no—
438C24—24no—
439C24—34no—
440C24—42no—
441C24—42no—
442C24—48no—
443C24—64no—
444C24—122no—
445C24—122no—
446C24—128no—
447C24C23×D314yes—
448C24C23×D311yescharacteristic
449C24C23×D311yescharacteristic
450C24C23×D311yescharacteristic
451#41—416no—
452C2×Q8—41no—
453C2×Q8—41no—
454C2×Q8C2×#6714yes—
455C2×D4—28no—
456C2×D4—28no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—416no—
470C2×D4—48no—
471C2×D4—48no—
472C2×D4—616no—
473C2×D4—68no—
474C2×D4—68no—
475C2×D4—64no—
476C2×D4—64no—
477C2×D4—64no—
478C2×D4—64no—
479C2×D4—122no—
480C2×D4—122no—
481C2×D4—1216no—
482C2×D4—128no—
483C2×D4—128no—
484C2×D4—128no—
485C2×D4—128no—
486C2×D4—128no—
487C2×D4—128no—
488C2×D4—128no—
489C2×D4C2×#6714yes—
490C2×D4C2×#6714yes—
491C2×D4C2×#6714yes—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—34no—
499C22×C4—34no—
500C22×C4—34no—
501C22×C4—68no—
502C22×C4—122no—
503C22×C4C2×#6714yes—
504C22×C4C2×#6714yes—
505C22×C4C2×#6714yes—
506C22×C4C23×D314yes—
507C22×C4C23×D311yescharacteristic
508C22×C4C23×D311yescharacteristic
509C22×C4C23×D311yescharacteristic
510C22×C4C23×D311yescharacteristic
511C4⋊C4—24no—
512C4⋊C4—64no—
513C4⋊C4—122no—
514C4⋊C4—128no—
515C22⋊C4—24no—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—616no—
519C22⋊C4—68no—
520C22⋊C4—68no—
521C22⋊C4—616no—
522C22⋊C4—64no—
523C22⋊C4—68no—
524C22⋊C4—68no—
525C22⋊C4—128no—
526C22⋊C4—122no—
527C22⋊C4—128no—
528C42—24no—
529C42—64no—
530C42—64no—
531C42—64no—
532C42—122no—
533C22×C3—22no—
534C22×C3—22no—
535C22×C3—22no—
536C22×C3—24no—
537C22×C3—24no—
538C22×C3—24no—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3—42no—
542C22×C3—42no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—44no—
546C22×C3—44no—
547C22×C3—44no—
548C22×C3—48no—
549C22×C3—48no—
550C22×C3—816no—
551C22×C3C23×D411yescharacteristic
552C22×C3C2×H(5,𝔽2)11yescharacteristic
553C22×C3C2×H(5,𝔽2)11yescharacteristic
554C22×C3C2×H(5,𝔽2)14yes—
555C2×D3—44no—
556C2×D3—44no—
557C2×D3—44no—
558C2×D3—48no—
559C2×D3—48no—
560C2×D3—816no—
561C3×C4—22no—
562C3×C4—22no—
563C3×C4—22no—
564C3×C4—44no—
565C3×C4#53414yes—
566Dic3—28no—
567Dic3—44no—
568Dic3—44no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—28no—
581C23—28no—
582C23—24no—
583C23—48no—
584C23—64no—
585C23—64no—
586C23—68no—
587C23—68no—
588C23—64no—
589C23—88no—
590C23—88no—
591C23—88no—
592C23—128no—
593C23—248no—
594C23—248no—
595C23—248no—
596C23C22×#6712yes—
597C23C22×#6712yes—
598C23C22×#6712yes—
599C23C24×D311yescharacteristic
600Q8—44no—
601D4—44no—
602D4—44no—
603D4—44no—
604D4—48no—
605D4—48no—
606D4—88no—
607D4—88no—
608D4—128no—
609D4—128no—
610D4—2416no—
611D4—248no—
612D4—248no—
613D4—2416no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—38no—
626C2×C4—64no—
627C2×C4—64no—
628C2×C4—64no—
629C2×C4—64no—
630C2×C4—64no—
631C2×C4—68no—
632C2×C4—64no—
633C2×C4—128no—
634C2×C4#100412yes—
635C2×C4#100412yes—
636C2×C4C22×#6712yes—
637C2×C4C22×#6712yes—
638C2×C4C22×#6712yes—
639C2×C4C22×#6712yes—
640C2×C3—42no—
641C2×C3—42no—
642C2×C3—42no—
643C2×C3—84no—
644C2×C3C2×#53411yescharacteristic
645C2×C3#349812yes—
646C2×C3#353712yes—
647C2×C3#354912yes—
648D3—84no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—48no—
662C22—48no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—816no—
667C22—128no—
668C22—128no—
669C22—124no—
670C22—124no—
671C22—124no—
672C22—2416no—
673C22C2×#100411yescharacteristic
674C22C2×#100411yescharacteristic
675C22D3×H(5,𝔽2)14yes—
676C22C23×#6711yescharacteristic
677C4—22no—
678C4—22no—
679C4—22no—
680C4—44no—
681C4—68no—
682C4—124no—
683C4—124no—
684C4#580414yes—
685C3#3729711yescharacteristic, complemented, Sylow
686C2—42no—
687C2—42no—
688C2—42no—
689C2—84no—
690C2—244no—
691C2#8632612yes—
692C2#8633412yes—
693C2C2×#580411yescharacteristic
694C2#8728212yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#535C2
hypercenterC2×#535D3
hypocenterC3#37297
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
1. C2×C3×#535C2semidirect
2. C2×#5830C2semidirect
3. #84086C2semidirect
4. #84049C2semidirect
5. #83190C2semidirect
6. #82991C2semidirect
7. #82700C2semidirect
8. #82676C2semidirect
9. #82663C2semidirect
10. #82653C2semidirect
11. #82597C2semidirect
12. #82594C2semidirect
13. #82257C2semidirect
16. C23×C3×D4C22semidirect
17. C23×D12C22semidirect
20. C2×C3×#124C22semidirect
22. C2×C3×#121C22semidirect
24. C2×C3×#120C22semidirect
25. C2×C3×#117C22semidirect
26. C2×#953C22semidirect
28. C2×#947C22semidirect
32. C2×#939C22semidirect
34. C2×#937C22semidirect
35. C2×#937C22semidirect
37. C2×#930C22semidirect
38. C2×#926C22semidirect
39. C2×#926C22semidirect
40. C22×C4×Dic3C22semidirect
41. #5229C22semidirect
42. #5223C22semidirect
43. #5223C22semidirect
44. #5222C22semidirect
46. #5221C22semidirect
47. #5221C22semidirect
48. #5220C22semidirect
49. #5220C22semidirect
50. C4×#217C22semidirect
51. C4×#217C22semidirect
52. #4988C22semidirect
53. #4988C22semidirect
54. #4977C22semidirect
55. #4975C22semidirect
56. #4961C22semidirect
57. #4960C22semidirect
58. #4959C22semidirect
59. #4957C22semidirect
60. #4944C22semidirect
61. #4940C22semidirect
63. C2×#535D3semidirect
79. C2×C3×#41D4semidirect
152. C23×D4C2×D3semidirect
160. C2×#124C2×D3semidirect
165. C2×#121C2×D3semidirect
171. C2×#120C2×D3semidirect
174. C2×#117C2×D3semidirect
296. C2×#41#67semidirect
685. C3#37297semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5352
C2×#58302
#840862
#840492
#831902
#829912
#827002
#826762
#826632
#826532
#825972
#825942
#822572
#372973