Finite Group Databasebeta

← #6461434

Subgroups — #6461434

Order 512.

1944 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#34250C211yesmaximal, characteristic
3#29344C214yesmaximal, complemented
4#29341C214yesmaximal, complemented
5#29338C212yesmaximal, complemented
6#29335C212yesmaximal, complemented
7#29332C212yesmaximal, complemented
8#29329C212yesmaximal, complemented
9#23747C211yesmaximal, characteristic, complemented
10#23744C211yesmaximal, characteristic, complemented
11#23507C211yesmaximal, characteristic, complemented
12C2×#2043C211yesmaximal, characteristic, complemented
13#21090C212yesmaximal, complemented
14#21088C212yesmaximal, complemented
15#20913C212yesmaximal
16#20913C212yesmaximal
17#20378C211yesmaximal, characteristic
18C2×#536C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#419C2212yes—
24C2×#419C2212yes—
25C22×#103C2212yes—
26C22×#103C2212yes—
27C2×#408C2212yes—
28C2×#405C2212yes—
29#2857C2212yes—
30#2857C2212yes—
31#2621C2214yes—
32#2547—21no—
33#2547—21no—
34#2546—21no—
35#2546—21no—
36#2486—22no—
37#2486—22no—
38#2351—21no—
39Q8×C4⋊C4—21no—
40#2349—22no—
41#2349—21no—
42D4×C4⋊C4—22no—
43D4×C4⋊C4—21no—
44#2317C2211yescharacteristic
45#2317C2212yes—
46#2316C2211yescharacteristic
47#2316C2212yes—
48#2310C2212yes—
49#2296C2212yes—
50#2296C2211yescharacteristic
51C2×#389C2212yes—
52C2×#389C2211yescharacteristic
53#2285C2211yescharacteristic
54#2101C2212yes—
55#2101C2214yes—
56#2101C2212yes—
57#2100C2211yescharacteristic
58#2100C2212yes—
59#2100C2211yescharacteristic
60#2099C2211yescharacteristic
61#2099C2212yes—
62#2099C2211yescharacteristic
63#2043C2212yes—
64#2043C2212yes—
65#2043C2212yes—
66#2043C2212yes—
67#2043C2212yes—
68#2043C2214yes—
69#2043C2212yes—
70#1935C2214yes—
71#1935C2214yes—
72#1935C2214yes—
73#1935C2214yes—
74#1933C2214yes—
75#1933C2214yes—
76#1933C2214yes—
77#1933C2214yes—
78#1873C2214yes—
79#1873C2214yes—
80#1873C2214yes—
81#1873C2214yes—
82#1872C2214yes—
83#1872C2214yes—
84#1872C2214yes—
85#1872C2214yes—
86#1750C2214yes—
87#1749C2212yes—
88C4×C22⋊C8C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92#1735C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8C2312yes—
97C22×M16C2312yes—
98C23×C8C2312yes—
99#536—28no—
100#536—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#124—22no—
108C2×#123—24no—
109C2×C22⋊Q8C2314yes—
110C2×C22⋊Q8C2314yes—
111#520—24no—
112#520—24no—
113#520—44no—
114C4×#41—44no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4D412yes—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D412yes—
126C2×C4×D4D411yescharacteristic
127C2×#117C2312yes—
128C2×#117C2311yescharacteristic
129C2×#117C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2312yes—
132C2×#117C2311yescharacteristic
133C2×#117C2312yes—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—22no—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2312yes—
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144#419—28no—
145#419—44no—
146C2×#103—28no—
147C2×#103—44no—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156#408—28no—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159#405—44no—
160C2×#97C2314yes—
161C2×C4×C8C2314yes—
162#398—24no—
163#398—24no—
164#397—24no—
165#397—24no—
166#391—22no—
167#391—22no—
168#391C2314yes—
169#391C2314yes—
170#389—21no—
171#389—22no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—22no—
180#389—21no—
181#389—22no—
182#389C2311yescharacteristic
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2311yescharacteristic
188#389C2312yes—
189#389C2312yes—
190#389C2311yescharacteristic
191#389C2311yescharacteristic
192#385—21no—
193#385—22no—
194#385—21no—
195#385—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#384—21no—
200#384—22no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#382—21no—
204#382—22no—
205#382C2312yes—
206#381—24no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380—21no—
212#380—21no—
213#380—22no—
214#380—22no—
215#380—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C2×#95—21no—
223C2×#95—22no—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#336C2311yescharacteristic
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2312yes—
240#336C2312yes—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—24no—
244C2×#41#4118yes—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8—24no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8—24no—
252C22×Q8C2×D411yescharacteristic
253C22×Q8C2414yes—
254C22×D4C2×D412yes—
255C22×D4C2×D411yescharacteristic
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—24no—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—48no—
281#124—48no—
282#123—416no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—28no—
286C22⋊Q8—28no—
287C4×Q8—28no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×Q8—44no—
291C4×D4—216no—
292C4×D4—28no—
293C4×D4—42no—
294C4×D4—42no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—48no—
298C4×D4—44no—
299#117—22no—
300#117—24no—
301#117—24no—
302#117—24no—
303#117—24no—
304#117—24no—
305#117—24no—
306#117—44no—
307#117—44no—
308#117#4114yes—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4—44no—
340C2×C4⋊C4—82no—
341C2×C4⋊C4—84no—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4#4114yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2411yescharacteristic
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—24no—
368C2×C42—22no—
369C2×C42—21no—
370C2×C42—22no—
371C2×C42—44no—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D412yes—
375C2×C42C2×D412yes—
376C2×C42C2411yescharacteristic
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379#103—28no—
380#103—28no—
381#103—28no—
382#103—28no—
383#103—44no—
384#103—44no—
385#103—44no—
386#103—44no—
387C22⋊C8—28no—
388C22⋊C8—28no—
389#97—44no—
390C4×C8—44no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—22no—
398#95—21no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407C24C22×D412yes—
408C24C22×D411yescharacteristic
409#41—216no—
410#41—216no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—48no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×Q8—48no—
426C2×Q8#12414yes—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4#12418yes—
432C2×D4#12414yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—28no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—81no—
465C22×C4—82no—
466C22×C4—81no—
467C22×C4—82no—
468C22×C4—81no—
469C22×C4—82no—
470C22×C4—81no—
471C22×C4—82no—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4118yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4H(5,𝔽2)14yes—
485C22×C4C2511yescharacteristic
486M16—48no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—48no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—81no—
529C4⋊C4—82no—
530C4⋊C4—81no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4C2×#4112yes—
536C4⋊C4C2×#4114yes—
537C4⋊C4C2×#4112yes—
538C22⋊C4—28no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—42no—
548C42—28no—
549C42—24no—
550C42—22no—
551C42—22no—
552C42—24no—
553C42—22no—
554C42—22no—
555C42—42no—
556C42—42no—
557C42—44no—
558C42—44no—
559C42—42no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23#52512yes—
571C23C2×#12414yes—
572C23C2×#12412yes—
573C23#53412yes—
574C23C23×D411yescharacteristic
575Q8—28no—
576Q8—44no—
577Q8—44no—
578Q8—44no—
579Q8—44no—
580Q8—88no—
581D4—216no—
582D4—28no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—88no—
636C2×C4—88no—
637C2×C4—162no—
638C2×C4—164no—
639C2×C4#52512yes—
640C2×C4C2×#12414yes—
641C2×C4C2×#12412yes—
642C2×C4#53412yes—
643C2×C4#55318yes—
644C2×C4#55318yes—
645C2×C4#57712yes—
646C2×C4#57812yes—
647C2×C4C22×#4111yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4112yes—
650C2×C4C2×H(5,𝔽2)12yes—
651C8—42no—
652C8—42no—
653C8—44no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22#259114yes—
668C22C2×#52511yescharacteristic
669C22C22×#12411yescharacteristic
670C22C2×#53411yescharacteristic
671C22C2×#57711yescharacteristic
672C22C2×#57811yescharacteristic
673C22#358611yescharacteristic
674C22#358611yescharacteristic
675C4—24no—
676C4—22no—
677C4—22no—
678C4—41no—
679C4—42no—
680C4—41no—
681C4—42no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4—84no—
686C4—82no—
687C4—81no—
688C4—81no—
689C4—82no—
690C4—82no—
691C4#259114yes—
692C4C2×#55312yes—
693C4C2×#55312yes—
694C2—24no—
695C2—44no—
696C2C2×#259111yescharacteristic
697C2#6059111yescharacteristic
698C2#6059411yescharacteristic
699C2#6062311yescharacteristic
700C2#6062711yescharacteristic
701C2#6075811yescharacteristic
702C2#6075911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29344C2semidirect
4. #29341C2semidirect
5. #29338C2semidirect
6. #29335C2semidirect
7. #29332C2semidirect
8. #29329C2semidirect
9. #23747C2semidirect
10. #23744C2semidirect
11. #23507C2semidirect
12. C2×#2043C2semidirect
13. #21090C2semidirect
14. #21088C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#342502
#293442
#293412
#293382
#293352
#293322
#293292
#237472
#237442
#235072
C2×#20432
#210902
#210882
#209132
#203782