Finite Group Databasebeta

← #582880

Subgroups — #582880

Order 512.

2490 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
1C2×#2487C211yesmaximal, characteristic
2#14669C212yesmaximal, complemented
3#14648C214yesmaximal, complemented
4#14636C212yesmaximal, complemented
5#14551C212yesmaximal, complemented
6#14402C212yesmaximal, complemented
7#14391C214yesmaximal
8#14266C212yesmaximal, complemented
9#14130C212yesmaximal, complemented
10#14046C212yesmaximal, complemented
11#13993C212yesmaximal, complemented
12#13974C212yesmaximal, complemented
13#13907C212yesmaximal, complemented
14#13872C211yesmaximal, characteristic
15#13847C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2487C4116yescomplemented
20C2×#400C2212yes—
21C2×#399C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#385C2214yes—
31C2×#385C2212yes—
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×#380C2214yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×#376C2211yescharacteristic
42#1445C2212yes—
43#1445C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2212yes—
55#1441C2214yes—
56#1440C2214yes—
57#1439C2211yescharacteristic
58#1439C2211yescharacteristic
59#1439C2211yescharacteristic
60#1439C2211yescharacteristic
61#1439C2214yes—
62#1439C2212yes—
63#1438C2212yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C2×C4×D4—42no—
82C2×C4×D4—42no—
83C2×C4×D4C2×C418yes—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4C2×C418yescomplemented
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2314yes—
116#400C2×C4116yescomplemented
117#399C2×C4116yes—
118#389—48no—
119#389—44no—
120#389—44no—
121#389—44no—
122#389C2×C4116yescomplemented
123#388—48no—
124#386—44no—
125#386C2×C4116yes—
126#385—48no—
127#385C2×C4116yescomplemented
128#380—48no—
129#380—44no—
130#380—42no—
131#380C2×C4116yes—
132#380C2×C418yescomplemented
133C4×C4⋊C4—42no—
134C4×C4⋊C4—44no—
135C4×C22⋊C4—44no—
136C4×C22⋊C4—44no—
137#376—42no—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C25C2411yescharacteristic
166C25C2411yescharacteristic
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4C22×C414yes—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—44no—
186C23×C4—42no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—42no—
191C23×C4C22×C414yes—
192C23×C4C22×C414yes—
193C23×C4C22×C418yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C4×D4—416no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
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—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—416no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—416no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C22×C418yes—
242C2×C4⋊C4C22×C418yes—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C22×C4116yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4C22×C418yes—
273C2×C22⋊C4C22×C414yes—
274C2×C22⋊C4C22×C414yes—
275C2×C22⋊C4C22×C4116yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42C22×C414yes—
288#95—28no—
289#95—44no—
290#95—44no—
291#95—41no—
292#95—41no—
293#95—42no—
294#95—42no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—42no—
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#95C22×C4116yes—
313#95C22×C418yes—
314#95C22×C4116yes—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—22no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—22no—
325C24—22no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24C23×C412yes—
332C24C23×C412yes—
333C24H(5,𝔽2)12yes—
334C242−1+412yes—
335C242−1+412yes—
336C242−1+412yes—
337C24C2511yescharacteristic
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—88no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4—84no—
436C22×C4C23×C412yes—
437C22×C4C23×C412yes—
438C22×C4C23×C418yes—
439C22×C4C23×C412yes—
440C22×C4C23×C418yes—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C42−1+414yes—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—48no—
451C4⋊C4—416no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—88no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C22⋊C4—216no—
462C22⋊C4—416no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—84no—
470C22⋊C4—88no—
471C42—28no—
472C42—48no—
473C42—48no—
474C42—88no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23#51914yes—
520C23#52014yes—
521C23C24×C411yescharacteristic
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)12yes—
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×H(5,𝔽2)12yes—
527C23C2×2−1+411yescharacteristic
528C23C2×2−1+411yescharacteristic
529C23C2×2−1+412yes—
530C23C2×2−1+411yescharacteristic
531C23#58512yes—
532D4—416no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—816no—
614C2×C4—816no—
615C2×C4#51814yes—
616C2×C4#52018yes—
617C2×C4#56014yes—
618C2×C4#56218yes—
619C2×C4#56314yes—
620C22—24no—
621C22—24no—
622C22—28no—
623C22—28no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22C2×#51811yescharacteristic
643C22C2×#51911yescharacteristic
644C22C2×#52012yes—
645C22C2×#52011yescharacteristic
646C22C4×H(5,𝔽2)12yes—
647C22C2×#56011yescharacteristic
648C22C2×#56212yes—
649C22C2×#56311yescharacteristic
650C22#353611yescharacteristic
651C22#353611yescharacteristic
652C22#353612yes—
653C22#353711yescharacteristic
654C22#353712yes—
655C22#353811yescharacteristic
656C22#353811yescharacteristic
657C22#354011yescharacteristic
658C22#354112yes—
659C22#354112yes—
660C22#354612yes—
661C22#355212yes—
662C22#355212yes—
663C22#355912yes—
664C22#356312yes—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—48no—
670C4—48no—
671C4—44no—
672C4—82no—
673C4—82no—
674C4—84no—
675C4—82no—
676C4—84no—
677C4#237618yes—
678C2—44no—
679C2—44no—
680C2C2×#237611yescharacteristic
681C2#3438011yescharacteristic
682C2#3441111yescharacteristic
683C2#3446112yes—
684C2#3446412yes—
685C2#5296312yes—
686C2#5298212yes—
687C2#5298711yescharacteristic
688C2#5298811yescharacteristic
689C2#5299111yescharacteristic
690C2#5313711yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14669C2semidirect
3. #14648C2semidirect
4. #14636C2semidirect
5. #14551C2semidirect
6. #14402C2semidirect
8. #14266C2semidirect
9. #14130C2semidirect
10. #14046C2semidirect
11. #13993C2semidirect
12. #13974C2semidirect
13. #13907C2semidirect
19. #2487C4semidirect
88. C22×C4⋊C4C2×C4semidirect
116. #400C2×C4semidirect
122. #389C2×C4semidirect
127. #385C2×C4semidirect
132. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
#146692
#146482
#146362
#145512
#144022
#143912
#142662
#141302
#140462
#139932
#139742
#139072
#138722
#138472