Finite Group Databasebeta

← #949913

Subgroups — #949913

Order 512.

2388 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17153C214yesmaximal, complemented
2#17080C212yesmaximal, complemented
3#17012C212yesmaximal, complemented
4#16823C212yesmaximal, complemented
5#16339C212yesmaximal, complemented
6#15331C214yesmaximal
7#15146C212yesmaximal, complemented
8#15125C212yesmaximal, complemented
9#15125C212yesmaximal, complemented
10#14264C212yesmaximal, complemented
11#14130C212yesmaximal, complemented
12#14053C212yesmaximal, complemented
13#13906C211yesmaximal, characteristic
14#13872C211yesmaximal, characteristic
15#13839C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#400C2214yes—
21C2×#393C2214yes—
22C2×#393C2214yes—
23C2×#391C2214yes—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#386C2212yes—
28C2×#385C2214yes—
29C2×#384C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37#1446C2214yes—
38#1446C2214yes—
39#1446C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2211yescharacteristic
50#1441C2211yescharacteristic
51#1441C2214yes—
52#1441C2212yes—
53#1440C2212yes—
54#1439C2212yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2211yescharacteristic
60#1439C2211yescharacteristic
61#1439C2214yes—
62#1439C2212yes—
63#1438C2212yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C4×#95C2212yes—
75C23×Q8C2314yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—44no—
82C2×C4⋊Q8—44no—
83C2×C4×D4—42no—
84C2×C4×D4—42no—
85C2×C4×D4Q818yescomplemented
86C2×#117—44no—
87C2×#117—42no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117#400—48no—
118#393—48no—
119#393—48no—
120#391—48no—
121#390—44no—
122#389—44no—
123#389—44no—
124#386—44no—
125#385—48no—
126#384—42no—
127#380—28no—
128#380—28no—
129#380—44no—
130#380—42no—
131C4×C4⋊C4—42no—
132C4×C22⋊C4—44no—
133C4×C22⋊C4—42no—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2312yes—
161C25C2411yescharacteristic
162C25C2411yescharacteristic
163C22×Q8—28no—
164C22×Q8—28no—
165C22×Q8—44no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8—44no—
169C22×Q8—44no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4C2×Q814yes—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—44no—
187C23×C4—42no—
188C23×C4—44no—
189C23×C4—42no—
190C23×C4C2×Q814yes—
191C23×C4C2×Q814yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C4⋊Q8—432no—
206C4×D4—416no—
207#117—416no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—416no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×Q814yes—
243C2×C4⋊C4#4118yes—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4C2×Q814yes—
273C2×C22⋊C4C2×Q814yes—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42C2×Q814yes—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—41no—
297#95—41no—
298#95—42no—
299#95—42no—
300#95—42no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24C22×Q812yes—
332C24C22×Q812yes—
333C24H(5,𝔽2)12yes—
334C242−1+412yes—
335C24C2511yescharacteristic
336C2×Q8—416no—
337C2×Q8—416no—
338C2×Q8—88no—
339C2×Q8—88no—
340C2×Q8—88no—
341C2×Q8—88no—
342C2×Q8—88no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—88no—
431C22×C4—84no—
432C22×C4—88no—
433C22×C4—84no—
434C22×C4C22×Q812yes—
435C22×C4C22×Q812yes—
436C22×C4C22×Q812yes—
437C22×C4C22×Q812yes—
438C22×C4C22×Q812yes—
439C22×C4C2×#4118yes—
440C22×C4C2×#4114yes—
441C22×C4H(5,𝔽2)14yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C4H(5,𝔽2)14yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C42−1+414yes—
447C22×C42−1+412yes—
448C4⋊C4—216no—
449C4⋊C4—44no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—416no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—88no—
469C22⋊C4—84no—
470C22⋊C4—88no—
471C42—28no—
472C42—48no—
473C42—48no—
474C42—84no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—42no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—41no—
520C23#54414yes—
521C23C23×Q811yescharacteristic
522C23C22×#4112yes—
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)12yes—
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×H(5,𝔽2)11yescharacteristic
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)12yes—
529C23C2×2−1+411yescharacteristic
530C23C2×2−1+411yescharacteristic
531C23C2×2−1+412yes—
532Q8—832no—
533D4—416no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—816no—
614C2×C4#53818yes—
615C2×C4#54314yes—
616C2×C4#54414yes—
617C2×C4#55418yes—
618C2×C4#55914yes—
619C2×C4H(3,𝔽4)14yes—
620C22—24no—
621C22—28no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—81no—
637C22—81no—
638C22—82no—
639C22—82no—
640C22—81no—
641C22—81no—
642C22C2×#53812yes—
643C22C2×#54311yescharacteristic
644C22C2×#54411yescharacteristic
645C22C2×#54411yescharacteristic
646C22C2×#55412yes—
647C22C2×#55911yescharacteristic
648C22#350312yes—
649C22#351912yes—
650C22#352012yes—
651C22#352212yes—
652C22#352312yes—
653C22C2×H(3,𝔽4)11yescharacteristic
654C22#353611yescharacteristic
655C22#353612yes—
656C22#353611yescharacteristic
657C22#353612yes—
658C22#353612yes—
659C22#353711yescharacteristic
660C22#353811yescharacteristic
661C22#354912yes—
662C22#354911yescharacteristic
663C22#355011yescharacteristic
664C22#355012yes—
665C4—24no—
666C4—42no—
667C4—42no—
668C4—48no—
669C4—44no—
670C4—44no—
671C4—42no—
672C4—82no—
673C4—84no—
674C4—84no—
675C4—82no—
676C4—84no—
677C4#280418yes—
678C2—44no—
679C2—82no—
680C2#3730611yescharacteristic
681C2#3747711yescharacteristic
682C2#3809912yes—
683C2#3896412yes—
684C2#4260612yes—
685C2#4286212yes—
686C2#4504711yescharacteristic
687C2#4534511yescharacteristic
688C2C2×#280411yescharacteristic
689C2#5295811yescharacteristic
690C2#5301511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17153C2semidirect
2. #17080C2semidirect
3. #17012C2semidirect
4. #16823C2semidirect
5. #16339C2semidirect
7. #15146C2semidirect
8. #15125C2semidirect
9. #15125C2semidirect
10. #14264C2semidirect
11. #14130C2semidirect
12. #14053C2semidirect
85. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#171532
#170802
#170122
#168232
#163392
#153312
#151462
#151252
#142642
#141302
#140532
#139062
#138722
#138392