Finite Group Databasebeta

← #876625

Subgroups — #876625

Order 512.

3106 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2610C212yesmaximal, complemented
2#17072C211yesmaximal, characteristic, complemented
3#17068C211yesmaximal, characteristic, complemented
4#16878C212yesmaximal, complemented
5#15476C214yesmaximal, complemented
6#15320C214yesmaximal, complemented
7#14928C214yesmaximal, complemented
8#14886C212yesmaximal, complemented
9#14698C212yesmaximal, complemented
10#14589C212yesmaximal, complemented
11#14445C212yesmaximal, complemented
12#14074C214yesmaximal, complemented
13#13756C211yesmaximal, characteristic, complemented
14C22×#124C2212yes—
15C22×#121C2212yes—
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2610—216no—
20C2×#395C2212yes—
21C2×#395C2214yes—
22C2×#394C2212yes—
23C2×#394C2212yes—
24C2×#394C2214yes—
25C2×#393C2212yes—
26C2×#393C2214yes—
27C2×#392C2212yes—
28C2×#392C2212yes—
29C2×#392C2214yes—
30C2×#390C2214yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#380C2212yes—
40C2×#376C2212yes—
41#1442C2214yes—
42#1441C2212yes—
43#1441C2214yes—
44#1440C2214yes—
45#1439C2214yes—
46#1439C2212yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1438C2214yes—
51#1438C2214yes—
52#1438C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64#1434C2212yes—
65#1434C2214yes—
66C22⋊#95C2214yes—
67C4×#95C2214yes—
68C23×Q8C2312yes—
69C23×D4C2314yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C2×#124—28no—
76C2×#124—44no—
77C2×#124—44no—
78C2×#121—28no—
79C2×#121—44no—
80C2×#121—44no—
81C2×#117—24no—
82C2×#117—42no—
83C2×#117—42no—
84C2×#117—42no—
85C2×#117—42no—
86C2×#117D418yescomplemented
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
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⋊C4C2312yes—
110C22×C42—24no—
111C22×C42—42no—
112C22×C42—42no—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#395—28no—
120#395—48no—
121#394—28no—
122#394—28no—
123#394—48no—
124#393—28no—
125#393—48no—
126#392—28no—
127#392—48no—
128#392—44no—
129#390—216no—
130#390—28no—
131#390—44no—
132#390—44no—
133#388—216no—
134#388—216no—
135#388—216no—
136#386—216no—
137#386—48no—
138#380—44no—
139#376—28no—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C25C2414yes—
158C25C2411yescharacteristic
159C22×Q8—24no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—24no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—44no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—42no—
187C22×D4—44no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4C2×D414yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213#124—432no—
214#121—432no—
215#117—416no—
216#117—416no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4#4114yes—
242C2×C4⋊C4#4118yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—42no—
295C2×C42—41no—
296C2×C42—41no—
297C2×C42—42no—
298C2×C42—41no—
299C2×C42—41no—
300C2×C42—44no—
301C2×C42—416no—
302C2×C42C2×D418yes—
303C2×C42#4114yes—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95—44no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24C22×D412yes—
345C24H(5,𝔽2)12yes—
346C242−1+412yes—
347C24C2511yescharacteristic
348C2×Q8—48no—
349C2×Q8—88no—
350C2×Q8—88no—
351C2×Q8—84no—
352C2×Q8—84no—
353C2×D4—416no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—416no—
361C2×D4—48no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4—88no—
372C2×D4—88no—
373C2×D4—88no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—84no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C2×#4118yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C4H(5,𝔽2)14yes—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4—88no—
468C4⋊C4—84no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C42—216no—
484C42—44no—
485C42—416no—
486C42—48no—
487C42—48no—
488C42—44no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
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—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#53714yes—
547C23C23×D411yescharacteristic
548C23C22×#4111yescharacteristic
549C23C22×#4111yescharacteristic
550C23C22×#4112yes—
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×H(5,𝔽2)11yescharacteristic
554C23C2×H(5,𝔽2)12yes—
555C23C2×H(5,𝔽2)12yes—
556C23C2×2−1+411yescharacteristic
557C23#58512yes—
558Q8—816no—
559D4—816no—
560D4—832no—
561D4—816no—
562D4—816no—
563C2×C4—232no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—432no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4#53214yes—
632C2×C4#53414yes—
633C2×C4#55518yes—
634C22—28no—
635C22—24no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22—44no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22C2×#53211yescharacteristic
661C22C2×#53411yescharacteristic
662C22C2×#53711yescharacteristic
663C22#346012yes—
664C22D4×#4112yes—
665C22C2×#55512yes—
666C22#348311yescharacteristic
667C22#348312yes—
668C22#349012yes—
669C22#349011yescharacteristic
670C22#349011yescharacteristic
671C22#349211yescharacteristic
672C22#349211yescharacteristic
673C22#349412yes—
674C22#349812yes—
675C22#351412yes—
676C22#352012yes—
677C22#352911yescharacteristic
678C22#353412yes—
679C22#353712yes—
680C22#354911yescharacteristic
681C22#354911yescharacteristic
682C22#356112yes—
683C4—28no—
684C4—24no—
685C4—24no—
686C4—48no—
687C4—44no—
688C4—44no—
689C4—44no—
690C4—44no—
691C4—82no—
692C4—84no—
693C4—84no—
694C2—44no—
695C2—164no—
696C2#3505111yescharacteristic
697C2#3627312yes—
698C2#3729711yescharacteristic
699C2#3794912yes—
700C2#3842612yes—
701C2#3882312yes—
702C2#4623211yescharacteristic
703C2#4628711yescharacteristic
704C2#5216711yescharacteristic
705C2#5923311yescharacteristic
706C2#6004811yescharacteristic
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×#2610C2semidirect
2. #17072C2semidirect
3. #17068C2semidirect
4. #16878C2semidirect
5. #15476C2semidirect
6. #15320C2semidirect
7. #14928C2semidirect
8. #14886C2semidirect
9. #14698C2semidirect
10. #14589C2semidirect
11. #14445C2semidirect
12. #14074C2semidirect
13. #13756C2semidirect
86. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26102
#170722
#170682
#168782
#154762
#153202
#149282
#148862
#146982
#145892
#144452
#140742
#137562