Finite Group Databasebeta

← #556046

Subgroups — #556046

Order 512.

2862 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#14685C211yesmaximal, characteristic, complemented
3#14682C211yesmaximal, characteristic, complemented
4#14452C212yesmaximal, complemented
5#14446C212yesmaximal, complemented
6#14440C212yesmaximal, complemented
7#14433C214yesmaximal, complemented
8#14417C212yesmaximal
9#14373C214yesmaximal, complemented
10#14352C212yesmaximal, complemented
11#13899C212yesmaximal, complemented
12#13896C212yesmaximal, complemented
13#13890C212yesmaximal, complemented
14#13888C212yesmaximal, complemented
15#13784C212yesmaximal, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2212yes—
19C23×C42C2211yescharacteristic
20C4×#121C4116yescomplemented
21C2×#390C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388C2212yes—
24C2×#387C2214yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2212yes—
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#381C2212yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C22×#95C2212yes—
41#1445C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61#1434C2212yes—
62#1434C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65#1434C2212yes—
66#1434C2212yes—
67#1434C2214yes—
68#1434C2212yes—
69C22⋊#95C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2314yes—
83C2×#121—28no—
84C2×#121C2×C418yes—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C2×C4×D4—48no—
88C2×C4×D4C2×C418yes—
89C2×C4×D4C2×C4116yescomplemented
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C42—42no—
111C22×C42—42no—
112C22×C42C2×C418yescomplemented
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119#390C2×C418yes—
120#389C2×C418yescomplemented
121#388C2×C4116yescomplemented
122#387—48no—
123#386—216no—
124#386—48no—
125#386C2×C4116yescomplemented
126#385—48no—
127#385—48no—
128#382—42no—
129#382—42no—
130#381—44no—
131#380—42no—
132#380—42no—
133#380—44no—
134#380C2×C418yescomplemented
135C4×C4⋊C4C2×C418yescomplemented
136C4×C22⋊C4—44no—
137C4×C22⋊C4C2×C4116yescomplemented
138C2×#95—44no—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C25C2412yes—
166C25C2411yescharacteristic
167C22×D4—28no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—44no—
173C22×D4—42no—
174C22×D4C22×C418yes—
175C22×D4C22×C414yes—
176C22×D4#4118yes—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—28no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—41no—
191C23×C4—41no—
192C23×C4—41no—
193C23×C4—41no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C22×C414yes—
203C23×C4C22×C414yes—
204C23×C4C22×C418yes—
205C23×C4C22×C414yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219#121—232no—
220C4×D4—416no—
221C4×D4—432no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C22×C418yes—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C22×C414yes—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
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⋊C4C22×C418yes—
272C2×C22⋊C4C22×C418yes—
273C2×C22⋊C4C22×C418yes—
274C2×C22⋊C4C22×C418yes—
275C2×C22⋊C4C22×C418yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—416no—
281C2×C42—44no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42—41no—
285C2×C42—41no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42C22×C414yes—
290C2×C42C22×C414yes—
291C2×C42C22×C418yes—
292C2×C42C22×C414yes—
293C2×C42C22×C414yes—
294#95—28no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95C22×C418yes—
318#95C22×C418yes—
319#95C22×C418yes—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—21no—
324C24—22no—
325C24—21no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24C23×C414yes—
338C24C23×C412yes—
339C24C2×#4118yes—
340C24H(5,𝔽2)12yes—
341C242−1+412yes—
342C24C2511yescharacteristic
343C2×D4—216no—
344C2×D4—216no—
345C2×D4—44no—
346C2×D4—48no—
347C2×D4—44no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—416no—
351C2×D4—416no—
352C2×D4#117116yes—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—216no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—416no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4C23×C412yes—
445C22×C4C23×C412yes—
446C22×C4C23×C414yes—
447C22×C4C23×C414yes—
448C22×C4C23×C414yes—
449C22×C4C23×C412yes—
450C22×C4C23×C412yes—
451C22×C4C23×C412yes—
452C22×C4C23×C412yes—
453C22×C4C2×#4114yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C4⋊C4—28no—
460C4⋊C4—216no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4—88no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C22⋊C4—88no—
479C42—28no—
480C42—28no—
481C42—44no—
482C42—44no—
483C42—48no—
484C42—88no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—24no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—24no—
503C23—24no—
504C23—28no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23C2×#11718yes—
530C23#52014yes—
531C23C24×C411yescharacteristic
532C23C22×#4112yes—
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×H(5,𝔽2)12yes—
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×2−1+411yescharacteristic
539C23C2×2−1+411yescharacteristic
540C23#58512yes—
541C23#58512yes—
542D4—416no—
543D4—432no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—416no—
578C2×C4—416no—
579C2×C4—416no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
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—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—816no—
621C2×C4—832no—
622C2×C4C2×#11714yes—
623C2×C4#51814yes—
624C2×C4#51814yes—
625C2×C4#52014yes—
626C22—28no—
627C22—24no—
628C22—24no—
629C22—28no—
630C22—24no—
631C22—28no—
632C22—42no—
633C22—48no—
634C22—44no—
635C22—48no—
636C22—44no—
637C22—48no—
638C22—44no—
639C22—48no—
640C22—44no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22C22×#11711yescharacteristic
649C22C2×#51811yescharacteristic
650C22C2×#51811yescharacteristic
651C22C2×#52011yescharacteristic
652C22C2×#52011yescharacteristic
653C22#342811yescharacteristic
654C22#342811yescharacteristic
655C22#345912yes—
656C22#345912yes—
657C22#351412yes—
658C22#351512yes—
659C22#351912yes—
660C22#352012yes—
661C22#353412yes—
662C22#353412yes—
663C22#353611yescharacteristic
664C22#353611yescharacteristic
665C22#353711yescharacteristic
666C22#353811yescharacteristic
667C22#354512yes—
668C22#354612yes—
669C22#355312yes—
670C22#355412yes—
671C4—24no—
672C4—24no—
673C4—24no—
674C4—24no—
675C4—42no—
676C4—44no—
677C4—42no—
678C4—44no—
679C4—44no—
680C4—84no—
681C4—84no—
682C4—84no—
683C4—84no—
684C2—48no—
685C2—44no—
686C2#3405711yescharacteristic
687C2#3419111yescharacteristic
688C2#3419411yescharacteristic
689C2#3435811yescharacteristic
690C2#3436011yescharacteristic
691C2#3439011yescharacteristic
692C2#3439311yescharacteristic
693C2#3715412yes—
694C2#3715512yes—
695C2#3716512yes—
696C2#3716612yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14685C2semidirect
3. #14682C2semidirect
4. #14452C2semidirect
5. #14446C2semidirect
6. #14440C2semidirect
7. #14433C2semidirect
9. #14373C2semidirect
10. #14352C2semidirect
11. #13899C2semidirect
12. #13896C2semidirect
13. #13890C2semidirect
14. #13888C2semidirect
15. #13784C2semidirect
20. C4×#121C4semidirect
89. C2×C4×D4C2×C4semidirect
112. C22×C42C2×C4semidirect
120. #389C2×C4semidirect
121. #388C2×C4semidirect
125. #386C2×C4semidirect
134. #380C2×C4semidirect
135. C4×C4⋊C4C2×C4semidirect
137. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#146852
#146822
#144522
#144462
#144402
#144332
#144172
#143732
#143522
#138992
#138962
#138902
#138882
#137842