Finite Group Databasebeta

← #2811710

Subgroups — #2811710

Order 512.

3530 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic, complemented
2C2×#2500C211yesmaximal, characteristic, complemented
3#16551C212yesmaximal, complemented
4#16512C214yesmaximal, complemented
5#15941C212yesmaximal, complemented
6#15320C214yesmaximal, complemented
7#15310C214yesmaximal, complemented
8#15310C212yesmaximal, complemented
9#14886C212yesmaximal, complemented
10#14811C212yesmaximal, complemented
11#14454C211yesmaximal, characteristic, complemented
12#14415C214yesmaximal, complemented
13#14175C212yesmaximal, complemented
14C22×#127C2212yes—
15C22×#121C2214yes—
16C22×#121C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×#117C2211yescharacteristic
19#2537—28no—
20#2500—28no—
21C2×#394C2212yes—
22C2×#394C2212yes—
23C2×#394C2214yes—
24C2×#394C2214yes—
25C2×#392C2212yes—
26C2×#392C2212yes—
27C2×#392C2214yes—
28C2×#392C2214yes—
29C2×#392C2214yes—
30C2×#392C2212yes—
31C2×#392C2214yes—
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#390C2212yes—
35C2×#388C2214yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#386C2214yes—
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#380C2214yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44#1443C2214yes—
45#1442C2212yes—
46#1441C2212yes—
47#1441C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2212yes—
52#1439C2214yes—
53#1439C2212yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2212yes—
59#1438C2212yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2212yes—
64#1436C2212yes—
65#1436C2214yes—
66#1434C2212yes—
67#1434C2212yes—
68C22⋊#95C2214yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C23×D4C2312yes—
72C23×D4C2314yes—
73C23×D4C2312yes—
74C23×D4C2314yes—
75C23×D4C2312yes—
76C23×D4C2314yes—
77C24×C4C2311yescharacteristic
78C24×C4C2314yes—
79C2×#127—48no—
80C2×#127—44no—
81C2×#121—24no—
82C2×#121—216no—
83C2×#121—42no—
84C2×#121—48no—
85C2×#121—48no—
86C2×#121—42no—
87C2×C4×D4—28no—
88C2×C4×D4—48no—
89C2×#117—24no—
90C2×#117—42no—
91C2×#117—42no—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
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⋊C4C2314yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118#394—216no—
119#394—28no—
120#394—28no—
121#394—48no—
122#392—28no—
123#392—48no—
124#392—48no—
125#392—44no—
126#392—44no—
127#392—48no—
128#392—48no—
129#390—24no—
130#390—24no—
131#390—44no—
132#388—216no—
133#386—216no—
134#386—48no—
135#386—48no—
136#386—48no—
137#384D418yescomplemented
138#382—24no—
139#380—216no—
140C4×C22⋊C4—24no—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C25C2412yes—
159C25C2411yescharacteristic
160C25C2414yes—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—24no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—22no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—44no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—44no—
183C22×D4—42no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—44no—
188C22×D4—42no—
189C22×D4—44no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—42no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229#127—432no—
230#121—416no—
231#121—464no—
232C4×D4—432no—
233#117—416no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D4116yes—
249C2×C4⋊C4C2×D414yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
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—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
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×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42C2×D414yes—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—44no—
323#95#4118yes—
324C24—21no—
325C24—24no—
326C24—24no—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—21no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24H(5,𝔽2)18yes—
355C24H(5,𝔽2)12yes—
356C24C2511yescharacteristic
357C2×D4—216no—
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—48no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—48no—
364C2×D4—416no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—416no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—44no—
376C2×D4—48no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4—84no—
387C2×D4—88no—
388C2×D4—88no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4—84no—
394C2×D4—88no—
395C2×D4—84no—
396C2×D4—88no—
397C2×D4—88no—
398C2×D4—88no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—44no—
457C22×C4—416no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D414yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4114yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C4H(5,𝔽2)14yes—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—416no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—84no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—48no—
483C22⋊C4—416no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C42—28no—
494C42—48no—
495C42—44no—
496C42—48no—
497C23—22no—
498C23—28no—
499C23—28no—
500C23—24no—
501C23—28no—
502C23—24no—
503C23—28no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—44no—
509C23—41no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—41no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—41no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—41no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—42no—
569C23C23×D411yescharacteristic
570C23C22×#4112yes—
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)12yes—
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×H(5,𝔽2)14yes—
575C23C2×H(5,𝔽2)11yescharacteristic
576C23#58512yes—
577D4—432no—
578D4—88no—
579D4—816no—
580D4—816no—
581D4—832no—
582D4—816no—
583D4—832no—
584D4—832no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—416no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—832no—
641C2×C4#53514yes—
642C2×C4#55018yes—
643C22—48no—
644C22—42no—
645C22—42no—
646C22—48no—
647C22—48no—
648C22—42no—
649C22—48no—
650C22—48no—
651C22—48no—
652C22—48no—
653C22—48no—
654C22—84no—
655C22—82no—
656C22—84no—
657C22—84no—
658C22—82no—
659C22—84no—
660C22—82no—
661C22—84no—
662C22—84no—
663C22—82no—
664C22—84no—
665C22—84no—
666C22—82no—
667C22—82no—
668C22—84no—
669C22—84no—
670C22—82no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22C2×#53511yescharacteristic
676C22C2×#55012yes—
677C22#348311yescharacteristic
678C22#348411yescharacteristic
679C22#348412yes—
680C22#348711yescharacteristic
681C22#349414yes—
682C22#351912yes—
683C22#351914yes—
684C22#353212yes—
685C22#353412yes—
686C22#353412yes—
687C22#353712yes—
688C22#353714yes—
689C22#354214yes—
690C22#354911yescharacteristic
691C4—24no—
692C4—28no—
693C4—44no—
694C4—44no—
695C4—42no—
696C4—48no—
697C4—82no—
698C4—84no—
699C4—82no—
700C4—84no—
701C2—82no—
702C2—88no—
703C2—162no—
704C2#3671011yescharacteristic
705C2#3913212yes—
706C2#4348112yes—
707C2#4493312yes—
708C2#4810914yes—
709C2#5200411yescharacteristic
710C2#5492612yes—
711C2#5905611yescharacteristic
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×#2537C2semidirect
2. C2×#2500C2semidirect
3. #16551C2semidirect
4. #16512C2semidirect
5. #15941C2semidirect
6. #15320C2semidirect
7. #15310C2semidirect
8. #15310C2semidirect
9. #14886C2semidirect
10. #14811C2semidirect
11. #14454C2semidirect
12. #14415C2semidirect
13. #14175C2semidirect
137. #384D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#25002
#165512
#165122
#159412
#153202
#153102
#148862
#148112
#144542
#144152
#141752