Finite Group Databasebeta

← #818994

Subgroups — #818994

Order 512.

2794 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2572C212yesmaximal, complemented
2#16652C212yesmaximal, complemented
3#16051C214yesmaximal, complemented
4#16047C212yesmaximal, complemented
5#15952C212yesmaximal, complemented
6#15144C212yesmaximal, complemented
7#15127C211yesmaximal, characteristic
8#14817C214yesmaximal, complemented
9#14155C211yesmaximal, characteristic, complemented
10#14025C212yesmaximal
11#14022C212yesmaximal, complemented
12#13921C212yesmaximal, complemented
13#13916C212yesmaximal, complemented
14C4×#400C211yesmaximal, characteristic, complemented
15#13820C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×C4×D4C2212yes—
19#2572—216no—
20C2×#400C2211yescharacteristic
21C2×#394C2212yes—
22C2×#394C2212yes—
23C2×#393C2214yes—
24C2×#393C2214yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#388C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#383C2214yes—
36C2×#381C2214yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2214yes—
42#1445C2211yescharacteristic
43#1445C2212yes—
44#1445C2212yes—
45#1443C2214yes—
46#1443C2212yes—
47#1443C2214yes—
48#1442C2211yescharacteristic
49#1442C2212yes—
50#1441C2211yescharacteristic
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2214yes—
58#1438C2212yes—
59#1438C2212yes—
60#1438C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C4×#95C2212yes—
74C4×#95C2211yescharacteristic
75C4×#95C2212yes—
76C23×Q8C2312yes—
77C23×D4C2312yes—
78C23×D4C2312yes—
79C23×D4C2312yes—
80C24×C4C2314yes—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8—42no—
84C2×#127—28no—
85C2×#127—42no—
86C2×C4×D4—28no—
87C2×C4×D4—28no—
88C2×C4×D4—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
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×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2314yes—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115#400D418yescomplemented
116#394—28no—
117#394—44no—
118#393—216no—
119#393—48no—
120#391—28no—
121#391—44no—
122#390—28no—
123#390—28no—
124#390—28no—
125#390—44no—
126#390—44no—
127#388—216no—
128#386—216no—
129#386—48no—
130#383—216no—
131#381—216no—
132C4×C4⋊C4—28no—
133C4×C4⋊C4—42no—
134C4×C4⋊C4—42no—
135C4×C4⋊C4—42no—
136C4×C22⋊C4—216no—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C25C2414yes—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4#4118yes—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C4⋊Q8—416no—
212#127—416no—
213C4×D4—432no—
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—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4#4118yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
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×C42—22no—
271C2×C42—22no—
272C2×C42—28no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—44no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42#4114yes—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—24no—
297#95—28no—
298#95—24no—
299#95—42no—
300#95—42no—
301#95—42no—
302#95—44no—
303#95—41no—
304#95—42no—
305#95—42no—
306#95—41no—
307#95—41no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—42no—
314#95C2×D418yes—
315#95C2×D414yes—
316#95C2×D418yes—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24C2×#4118yes—
332C24C2511yescharacteristic
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—84no—
337C2×Q8—84no—
338C2×Q8—84no—
339C2×Q8—84no—
340C2×D4—216no—
341C2×D4—216no—
342C2×D4—216no—
343C2×D4—416no—
344C2×D4—416no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—88no—
354C2×D4—88no—
355C2×D4—84no—
356C2×D4—84no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C22×D412yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C2×#4118yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4H(5,𝔽2)12yes—
450C4⋊C4—216no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C22⋊C4—416no—
459C22⋊C4—416no—
460C22⋊C4—416no—
461C22⋊C4—416no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C42—28no—
470C42—416no—
471C42—44no—
472C42—48no—
473C42—44no—
474C42—48no—
475C42—84no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23C23×D411yescharacteristic
512C23C22×#4111yescharacteristic
513C23C22×#4112yes—
514C23C22×#4112yes—
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)11yescharacteristic
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×H(5,𝔽2)11yescharacteristic
519C23#58512yes—
520C23#58512yes—
521C23#58511yescharacteristic
522Q8—816no—
523D4—432no—
524D4—816no—
525D4—816no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—832no—
607C2×C4#52918yes—
608C2×C4#53814yes—
609C2×C4#55014yes—
610C2×C4#55114yes—
611C2×C4#55118yes—
612C2×C4#55514yes—
613C22—48no—
614C22—48no—
615C22—48no—
616C22—48no—
617C22—48no—
618C22—48no—
619C22—48no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22C2×#52912yes—
629C22C2×#53811yescharacteristic
630C22C2×#55011yescharacteristic
631C22D4×#4112yes—
632C22D4×#4112yes—
633C22C2×#55112yes—
634C22C2×#55111yescharacteristic
635C22C2×#55511yescharacteristic
636C22#349611yescharacteristic
637C22#349811yescharacteristic
638C22#350611yescharacteristic
639C22#351012yes—
640C22#351312yes—
641C22#351412yes—
642C22#351412yes—
643C22#351712yes—
644C22#352112yes—
645C22#352412yes—
646C22#353711yescharacteristic
647C22#354011yescharacteristic
648C22#355111yescharacteristic
649C22#355811yescharacteristic
650C22#356212yes—
651C4—24no—
652C4—48no—
653C4—44no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—84no—
660C4—82no—
661C4—82no—
662C4—81no—
663C4—84no—
664C4#257218yescomplemented
665C2—88no—
666C2C2×#257211yescharacteristic
667C2#3597012yes—
668C2#3611512yes—
669C2#3938311yescharacteristic
670C2#4296111yescharacteristic
671C2#4520912yes—
672C2#4521411yescharacteristic
673C2#4540811yescharacteristic
674C2#5937511yescharacteristic
675C2#5947612yes—
676C2#5984111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2572C2semidirect
2. #16652C2semidirect
3. #16051C2semidirect
4. #16047C2semidirect
5. #15952C2semidirect
6. #15144C2semidirect
8. #14817C2semidirect
9. #14155C2semidirect
11. #14022C2semidirect
12. #13921C2semidirect
13. #13916C2semidirect
14. C4×#400C2semidirect
15. #13820C2semidirect
115. #400D4semidirect
664. C4#2572semidirect

Maximal subgroups

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

subgroupindex
C2×#25722
#166522
#160512
#160472
#159522
#151442
#151272
#148172
#141552
#140252
#140222
#139212
#139162
C4×#4002
#138202