Finite Group Databasebeta

← #6565756

Subgroups — #6565756

Order 512.

1927 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×#3361C212yesmaximal, complemented
2#36595C211yesmaximal, characteristic, complemented
3#34317C211yesmaximal, characteristic, complemented
4#29800C212yesmaximal, complemented
5#29794C211yesmaximal, characteristic
6#28751C212yesmaximal, complemented
7#28749C211yesmaximal, characteristic, complemented
8#28207C212yesmaximal, complemented
9#28206C211yesmaximal, characteristic, complemented
10#25280C214yesmaximal, complemented
11#25277C212yesmaximal, complemented
12#24634C214yesmaximal, complemented
13#24631C212yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23676C212yesmaximal, complemented
16#22734C211yesmaximal, characteristic, complemented
17#22543C212yesmaximal, complemented
18C2×#559C2212yes—
19C2×#554C2211yescharacteristic
20#3361—216no—
21C2×#488C2214yes—
22C2×#486C2212yes—
23C2×#477C2212yes—
24C2×#478C2214yes—
25C2×#463C2214yes—
26C2×#459C2212yes—
27C2×C8×Q8C2212yes—
28C2×C4×SD16C2212yes—
29C2×C4×D8C2214yes—
30#2867C2211yescharacteristic
31#2858C2212yes—
32#2548—21no—
33#2548—21no—
34#2548C2212yes—
35#2544C2214yes—
36#2445—22no—
37#2445—22no—
38#2370C2212yes—
39#2369C2211yescharacteristic
40#2366—21no—
41#2366—21no—
42#2362—22no—
43#2362—21no—
44#2357C2212yes—
45#2355C2211yescharacteristic
46#2354C2211yescharacteristic
47#2344—21no—
48#2332—22no—
49#2323—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#127C2212yes—
53C4×#125C2212yes—
54#2293C2211yescharacteristic
55#2293C2212yes—
56#2274—21no—
57C42×Q8C2211yescharacteristic
58#2084C2211yescharacteristic
59#2083C2212yes—
60#2071C2211yescharacteristic
61#2067C2214yes—
62#2066C2212yes—
63#2055C2212yes—
64#2054C2214yes—
65#2043C2212yes—
66#2043C2211yescharacteristic
67#2042C2214yes—
68#2041C2212yes—
69#2041C2212yes—
70#2041C2212yes—
71#2041C2212yes—
72#2013C2211yescharacteristic
73#2012C2214yes—
74#2011C2212yes—
75#2011C2212yes—
76#1983C2214yes—
77#1983C2214yes—
78#1983C2214yes—
79#1982C2212yes—
80#1982C2214yes—
81#1958C2214yes—
82#1957C2212yes—
83#1939C2212yes—
84#1939C2214yes—
85#1928C2214yes—
86#1926C2214yes—
87#1925C2214yes—
88#1831C2211yescharacteristic
89#1831C2212yes—
90#1804C2214yes—
91#1803C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C4×#102C2214yes—
96C22×SD16C2312yes—
97C22×D8C2314yes—
98#559—28no—
99#559—28no—
100#554—24no—
101#554D418yescomplemented
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127—21no—
105C2×#127—21no—
106C2×#127—21no—
107C2×#127—21no—
108C2×#127C2312yes—
109C2×#127C2312yes—
110C2×#127C2312yes—
111C2×#125C2312yes—
112C2×#124—22no—
113C2×#124C2314yes—
114C2×#124C2314yes—
115C2×#123—24no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4C2314yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2312yes—
129C2×#117—21no—
130C2×#117—21no—
131C2×#117—22no—
132C22×C4⋊C4—21no—
133#488—216no—
134#486—28no—
135#478—216no—
136#477—28no—
137#463—216no—
138#459—28no—
139C8×Q8—28no—
140C4×SD16—28no—
141C4×D8—216no—
142C2×#107C2314yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×C4×C8C2312yes—
162C2×C4×C8C2314yes—
163#394—24no—
164#394—24no—
165#394—24no—
166#391—21no—
167#391—21no—
168#391—22no—
169#391C2312yes—
170#390—22no—
171#390—22no—
172#390—22no—
173#390—22no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390C2314yes—
179#390C2312yes—
180#390C2311yescharacteristic
181#390C2311yescharacteristic
182#389—21no—
183#389—22no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389C2311yescharacteristic
188#389C2312yes—
189#389C2311yescharacteristic
190#389C2312yes—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—21no—
195#388—21no—
196#388—22no—
197#388—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#386—24no—
204#386—24no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#382C2312yes—
208#381—22no—
209#380—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D412yes—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2312yes—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—22no—
232C43C2312yes—
233C43C2311yescharacteristic
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2311yescharacteristic
240#336C2314yes—
241#336C2312yes—
242C22×Q8—21no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2412yes—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4C2414yes—
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—24no—
256C2×SD16—24no—
257C2×SD16—42no—
258C2×SD16—42no—
259C2×D8—28no—
260C2×D8—44no—
261C2×D8—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—22no—
266C4⋊Q8—42no—
267C4⋊Q8C2×D414yes—
268#127—24no—
269#127—24no—
270#127—44no—
271#127—44no—
272#127—44no—
273#127—44no—
274#127—82no—
275#127—82no—
276#125—24no—
277#125C2×D418yes—
278#124—28no—
279#124—28no—
280#124—44no—
281#124—44no—
282#124—48no—
283#123—416no—
284C22⋊Q8—28no—
285C22⋊Q8—216no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—42no—
290C4×Q8—82no—
291C4×Q8C2×D414yes—
292C4×Q8#4114yes—
293C4×D4—28no—
294C4×D4—24no—
295C4×D4—22no—
296C4×D4—28no—
297C4×D4—48no—
298C4×D4—84no—
299C4×D4—84no—
300C4×D4C2×D418yes—
301C4×D4C2×D414yes—
302#117—44no—
303#117—44no—
304#117—84no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366#107—28no—
367#106—24no—
368C4⋊C8—24no—
369C4⋊C8—28no—
370C4⋊C8—44no—
371C4⋊C8—42no—
372#103—24no—
373#103—28no—
374#103—42no—
375#103—44no—
376#103—42no—
377#102—24no—
378#102—28no—
379#102—28no—
380#102—28no—
381#102—44no—
382#102—44no—
383#102—44no—
384#102—44no—
385C4×C8—28no—
386C4×C8—24no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—21no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403C24—21no—
404C24—24no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—81no—
415C2×Q8—81no—
416C2×Q8#12114yes—
417C2×D4—24no—
418C2×D4—21no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4C22×D412yes—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—48no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C2511yescharacteristic
472SD16—48no—
473D8—416no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—28no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—82no—
507C4⋊C4—81no—
508C4⋊C4#12114yes—
509C4⋊C4#12118yes—
510C4⋊C4C22×D412yes—
511C4⋊C4C22×D414yes—
512C22⋊C4—24no—
513C22⋊C4—28no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—81no—
522C22⋊C4—82no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C42—24no—
528C42—22no—
529C42—24no—
530C42—22no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—42no—
535C42—42no—
536C42—41no—
537C42—44no—
538C42—44no—
539C42—42no—
540C42—82no—
541C42—82no—
542C42—82no—
543C42#13514yes—
544C42C22×D414yes—
545C42C22×D412yes—
546C42C2×#4114yes—
547C42C2×#4112yes—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23C23×D411yescharacteristic
563Q8—28no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—84no—
569D4—24no—
570D4—44no—
571D4—44no—
572D4—42no—
573D4—44no—
574D4—44no—
575D4—88no—
576D4—88no—
577D4—88no—
578D4—164no—
579D4—164no—
580D4—164no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—88no—
625C2×C4—168no—
626C2×C4C2×#12114yes—
627C2×C4C2×#12112yes—
628C2×C4C2×#12114yes—
629C2×C4C2×#12112yes—
630C2×C4C2×#13512yes—
631C2×C4C2×#13512yes—
632C2×C4C2×#13512yes—
633C2×C4#57512yes—
634C2×C4#57612yes—
635C2×C4C23×D411yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4#58512yes—
638C2×C4#58512yes—
639C8—42no—
640C8—44no—
641C8—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22C22×#12111yescharacteristic
657C22#348711yescharacteristic
658C22#350011yescharacteristic
659C22C22×#13511yescharacteristic
660C22C2×#57511yescharacteristic
661C22C2×#57611yescharacteristic
662C22#358511yescharacteristic
663C4—24no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—24no—
669C4—44no—
670C4—42no—
671C4—42no—
672C4—41no—
673C4—42no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4—81no—
678C4—82no—
679C4#303514yes—
680C4#348712yes—
681C4#352412yes—
682C2—82no—
683C2—164no—
684C2#3601811yescharacteristic
685C2C2×#303511yescharacteristic
686C2#6042811yescharacteristic
687C2#6188511yescharacteristic
688C2#6189811yescharacteristic
689C2#6252711yescharacteristic
690C2#6259011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#29794C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3361C2semidirect
2. #36595C2semidirect
3. #34317C2semidirect
4. #29800C2semidirect
6. #28751C2semidirect
7. #28749C2semidirect
8. #28207C2semidirect
9. #28206C2semidirect
10. #25280C2semidirect
11. #25277C2semidirect
12. #24634C2semidirect
13. #24631C2semidirect
14. #23677C2semidirect
15. #23676C2semidirect
16. #22734C2semidirect
17. #22543C2semidirect
101. #554D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33612
#365952
#343172
#298002
#297942
#287512
#287492
#282072
#282062
#252802
#252772
#246342
#246312
#236772
#236762
#227342
#225432