Finite Group Databasebeta

← #876631

Subgroups — #876631

Order 512.

3786 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2699C212yesmaximal, complemented
2C2×#2682C212yesmaximal, complemented
3C2×#2543C212yesmaximal, complemented
4#17075C211yesmaximal, characteristic, complemented
5#17068C211yesmaximal, characteristic, complemented
6#17015C212yesmaximal, complemented
7#15549C214yesmaximal, complemented
8#15412C214yesmaximal, complemented
9#15163C212yesmaximal, complemented
10#14762C214yesmaximal, complemented
11#14700C212yesmaximal, complemented
12#13996C214yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic, complemented
14C22×#123C2214yes—
15C22×#121C2214yes—
16C22×#121C2212yes—
17C22×C4×D4C2211yescharacteristic
18#2699—216no—
19#2682—216no—
20#2543—216no—
21C2×#398C2212yes—
22C2×#398C2212yes—
23C2×#397C2212yes—
24C2×#396C2214yes—
25C2×#394C2212yes—
26C2×#394C2212yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#392C2212yes—
30C2×#392C2214yes—
31C2×#392C2214yes—
32C2×#392C2214yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#389C2212yes—
40C2×#388C2214yes—
41C2×#388C2214yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#386C2214yes—
45C2×#385C2214yes—
46C2×#381C2214yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2214yes—
49C22×#95C2212yes—
50C2×C43C2211yescharacteristic
51#1445C2214yes—
52#1445C2212yes—
53#1445C2214yes—
54#1444C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2212yes—
60#1438C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2211yescharacteristic
70C23×D4C2314yes—
71C23×D4C2312yes—
72C23×D4C2314yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2314yes—
77C24×C4C2312yes—
78C2×#123—216no—
79C2×#123—216no—
80C2×#123—48no—
81C2×#121—216no—
82C2×#121—216no—
83C2×#121—216no—
84C2×#121—28no—
85C2×#121—44no—
86C2×#121—44no—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C2×C4×D4D418yescomplemented
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#398—28no—
121#398—28no—
122#397—28no—
123#396—216no—
124#394—28no—
125#394—28no—
126#394—216no—
127#394—216no—
128#392—216no—
129#392—216no—
130#392—28no—
131#392—48no—
132#390—28no—
133#390—28no—
134#390—44no—
135#390—44no—
136#390D418yescomplemented
137#390D418yescomplemented
138#389—28no—
139#388—216no—
140#388—216no—
141#386—216no—
142#386—216no—
143#386—216no—
144#385—216no—
145#381—216no—
146C4×C4⋊C4—42no—
147C4×C22⋊C4—48no—
148C2×#95—28no—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C43—42no—
165C25C2414yes—
166C25C2412yes—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—28no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—44no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4C2×D414yes—
203C22×D4C2×D414yes—
204C22×D4C2×D414yes—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4C2×D418yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228#123—464no—
229#121—432no—
230#121—464no—
231C4×D4—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4#4118yes—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4C2×D4116yes—
302C2×C22⋊C4C2×D418yes—
303C2×C22⋊C4C2×D4116yes—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—41no—
311C2×C42—41no—
312C2×C42—42no—
313C2×C42—41no—
314C2×C42—41no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—44no—
335#95—42no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24C22×D414yes—
360C24C22×D414yes—
361C24C22×D414yes—
362C24C2511yescharacteristic
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—48no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—88no—
386C2×D4—88no—
387C2×D4—84no—
388C2×D4—88no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4—88no—
394C2×D4—88no—
395C2×D4—88no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—88no—
400C2×D4#127116yes—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—216no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—88no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C2×#4118yes—
476C22×C4C2×#4114yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)14yes—
480C4⋊C4—216no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—416no—
485C4⋊C4—48no—
486C4⋊C4—416no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C22⋊C4—232no—
491C22⋊C4—416no—
492C22⋊C4—48no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C42—28no—
505C42—44no—
506C42—44no—
507C42—48no—
508C42—48no—
509C42—84no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23—44no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23C2×#12718yes—
568C23C23×D411yescharacteristic
569C23C23×D411yescharacteristic
570C23C23×D411yescharacteristic
571C23C22×#4112yes—
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×H(5,𝔽2)12yes—
575C23#58511yescharacteristic
576C23#58511yescharacteristic
577C23#58512yes—
578C23#58512yes—
579D4—416no—
580D4—816no—
581D4—832no—
582D4—816no—
583D4—816no—
584D4—832no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—816no—
650C2×C4—832no—
651C2×C4C2×#12714yes—
652C2×C4#53414yes—
653C2×C4#54618yes—
654C22—216no—
655C22—28no—
656C22—44no—
657C22—44no—
658C22—48no—
659C22—48no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—48no—
665C22—84no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22C22×#12711yescharacteristic
681C22C2×#53411yescharacteristic
682C22#344911yescharacteristic
683C22#346012yes—
684C22C2×#54612yes—
685C22D4×#4112yes—
686C22D4×#4112yes—
687C22#348311yescharacteristic
688C22#349312yes—
689C22#349612yes—
690C22#349611yescharacteristic
691C22#349611yescharacteristic
692C22#349612yes—
693C22#349611yescharacteristic
694C22#349611yescharacteristic
695C22#350012yes—
696C22#351712yes—
697C22#352911yescharacteristic
698C22#353412yes—
699C22#354311yescharacteristic
700C22#354511yescharacteristic
701C22#356012yes—
702C22#356712yes—
703C4—24no—
704C4—24no—
705C4—28no—
706C4—44no—
707C4—44no—
708C4—44no—
709C4—48no—
710C4—42no—
711C4—44no—
712C4—84no—
713C4—84no—
714C2—48no—
715C2—164no—
716C2#3500111yescharacteristic
717C2#3626012yes—
718C2#3712911yescharacteristic
719C2#3843312yes—
720C2#3941512yes—
721C2#3945412yes—
722C2#4623611yescharacteristic
723C2#4628011yescharacteristic
724C2#5216711yescharacteristic
725C2#5941911yescharacteristic
726C2#5983411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2699C2semidirect
2. C2×#2682C2semidirect
3. C2×#2543C2semidirect
4. #17075C2semidirect
5. #17068C2semidirect
6. #17015C2semidirect
7. #15549C2semidirect
8. #15412C2semidirect
9. #15163C2semidirect
10. #14762C2semidirect
11. #14700C2semidirect
12. #13996C2semidirect
13. C4×#390C2semidirect
89. C2×C4×D4D4semidirect
136. #390D4semidirect
137. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26992
C2×#26822
C2×#25432
#170752
#170682
#170152
#155492
#154122
#151632
#147622
#147002
#139962
C4×#3902