Finite Group Databasebeta

← #4463780

Subgroups — #4463780

Order 512.

2552 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17216C212yesmaximal, complemented
2#17193C212yesmaximal, complemented
3#17159C212yesmaximal, complemented
4#17107C212yesmaximal, complemented
5#17105C214yesmaximal, complemented
6#17090C214yesmaximal, complemented
7#15200C212yesmaximal, complemented
8#15144C212yesmaximal, complemented
9#15138C212yesmaximal, complemented
10#15127C212yesmaximal, complemented
11#15126C212yesmaximal, complemented
12#15125C211yesmaximal, characteristic, complemented
13#15124C211yesmaximal, characteristic
14#14257C212yesmaximal, complemented
15#13844C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#121C2212yes—
19C22×#117C2211yescharacteristic
20C2×#397C2214yes—
21C2×#394C2214yes—
22C2×#394C2214yes—
23C2×#394C2214yes—
24C2×#393C2212yes—
25C2×#393C2214yes—
26C2×#393C2214yes—
27C2×#392C2212yes—
28C2×#392C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#389C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39#1446C2212yes—
40#1445C2214yes—
41#1444C2211yescharacteristic
42#1444C2211yescharacteristic
43#1442C2211yescharacteristic
44#1442C2212yes—
45#1442C2214yes—
46#1442C2211yescharacteristic
47#1442C2214yes—
48#1442C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2212yes—
56#1439C2212yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2212yes—
62#1438C2214yes—
63#1438C2212yes—
64#1438C2214yes—
65#1438C2214yes—
66#1438C2214yes—
67#1438C2214yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4×#95C2212yes—
75C23×Q8C2312yes—
76C23×D4C2312yes—
77C23×D4C2314yes—
78C23×D4C2312yes—
79C23×D4C2312yes—
80C24×C4C2311yescharacteristic
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—44no—
83C2×#127—42no—
84C2×#127—44no—
85C2×#121—28no—
86C2×#121—44no—
87C2×#121—44no—
88C2×#117—44no—
89C2×#117—42no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
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⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119#397—48no—
120#394—48no—
121#394—48no—
122#394—48no—
123#393—48no—
124#393—44no—
125#393—48no—
126#392—44no—
127#392—44no—
128#390—44no—
129#390—44no—
130#390—44no—
131#390—44no—
132#390—44no—
133#389—44no—
134#380—44no—
135C4×C4⋊C4—44no—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—42no—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2314yes—
162C25C2414yes—
163C25C2411yescharacteristic
164C22×Q8—24no—
165C22×Q8—42no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8—42no—
169C22×D4—28no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—44no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4#4114yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C4⋊Q8—416no—
212#127—416no—
213#121—432no—
214#117—416no—
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—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
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⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297#95—22no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—42no—
306#95—44no—
307#95—41no—
308#95—41no—
309#95—42no—
310#95—42no—
311#95—44no—
312#95—44no—
313#95—42no—
314#95—42no—
315#95—42no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321C24—21no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—22no—
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—
344C24—24no—
345C24—24no—
346C24C2×#4112yes—
347C24C2511yescharacteristic
348C2×Q8—48no—
349C2×Q8—84no—
350C2×Q8—88no—
351C2×Q8—88no—
352C2×Q8—84no—
353C2×D4—416no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—84no—
364C2×D4—88no—
365C2×D4—84no—
366C2×D4—88no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—88no—
370C2×D4—84no—
371C2×D4—88no—
372C2×D4—84no—
373C2×D4—88no—
374C2×D4—88no—
375C2×D4—88no—
376C2×D4—88no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—216no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4H(5,𝔽2)14yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+412yes—
460C22×C42−1+412yes—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—88no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C42—48no—
487C42—44no—
488C42—48no—
489C42—44no—
490C23—24no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
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—44no—
547C23—44no—
548C23C22×#4111yescharacteristic
549C23C2×H(5,𝔽2)12yes—
550C23C2×H(5,𝔽2)12yes—
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×H(5,𝔽2)12yes—
553C23C2×H(5,𝔽2)11yescharacteristic
554C23C2×H(5,𝔽2)11yescharacteristic
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×2−1+411yescharacteristic
557C23C2×2−1+411yescharacteristic
558C23#58512yes—
559Q8—816no—
560D4—816no—
561D4—816no—
562D4—832no—
563D4—816no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—416no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4#53818yes—
640C2×C4#53914yes—
641C2×C4#56014yes—
642C2×C4H(3,𝔽4)14yes—
643C2×C4#56218yes—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—81no—
650C22—82no—
651C22—81no—
652C22—82no—
653C22—81no—
654C22—81no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22C2×#53812yes—
671C22C2×#53911yescharacteristic
672C22#349412yes—
673C22#350811yescharacteristic
674C22#351111yescharacteristic
675C22C2×#56011yescharacteristic
676C22C2×H(3,𝔽4)11yescharacteristic
677C22C2×#56212yes—
678C22#353611yescharacteristic
679C22#353612yes—
680C22#353711yescharacteristic
681C22#354212yes—
682C22#354312yes—
683C22#354312yes—
684C22#354612yes—
685C22#354811yescharacteristic
686C22#354912yes—
687C22#354911yescharacteristic
688C22#355011yescharacteristic
689C22#355111yescharacteristic
690C22#355412yes—
691C22#355412yes—
692C22#355812yes—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—82no—
700C4—84no—
701C4—82no—
702C4—84no—
703C4—82no—
704C4—84no—
705C4#282418yes—
706C2—82no—
707C2—164no—
708C2C2×#282411yescharacteristic
709C2#3728911yescharacteristic
710C2#3731511yescharacteristic
711C2#3733412yes—
712C2#3734612yes—
713C2#5296011yescharacteristic
714C2#5296412yes—
715C2#5296711yescharacteristic
716C2#5296911yescharacteristic
717C2#5301511yescharacteristic
718C2#5310612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15124C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17216C2semidirect
2. #17193C2semidirect
3. #17159C2semidirect
4. #17107C2semidirect
5. #17105C2semidirect
6. #17090C2semidirect
7. #15200C2semidirect
8. #15144C2semidirect
9. #15138C2semidirect
10. #15127C2semidirect
11. #15126C2semidirect
12. #15125C2semidirect
14. #14257C2semidirect
15. #13844C2semidirect

Maximal subgroups

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

subgroupindex
#172162
#171932
#171592
#171072
#171052
#170902
#152002
#151442
#151382
#151272
#151262
#151252
#151242
#142572
#138442