Finite Group Databasebeta

← #6461335

Subgroups — #6461335

Order 512.

1868 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#34250C211yesmaximal, characteristic
3#29311C214yesmaximal, complemented
4#29308C214yesmaximal, complemented
5#29304C212yesmaximal, complemented
6#29301C212yesmaximal, complemented
7#29299C212yesmaximal, complemented
8#29297C212yesmaximal, complemented
9#23657C211yesmaximal, characteristic, complemented
10#23656C211yesmaximal, characteristic, complemented
11#23552C211yesmaximal, characteristic, complemented
12C2×#2058C211yesmaximal, characteristic, complemented
13#21160C212yesmaximal, complemented
14#21155C212yesmaximal, complemented
15#20849C212yesmaximal
16#20848C212yesmaximal
17#20378C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#520C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2212yes—
23C2×#428C2212yes—
24C2×#427C2212yes—
25C22×#106C2212yes—
26C2×#408C2212yes—
27C2×#405C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2628C2214yes—
31#2568—21no—
32#2567—21no—
33#2563—21no—
34#2542—21no—
35#2487—22no—
36#2409—22no—
37#2351—21no—
38Q8×C4⋊C4—21no—
39#2349—22no—
40#2349—21no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4—22no—
43#2317C2212yes—
44#2317C2211yescharacteristic
45#2317C2212yes—
46#2316C2212yes—
47#2316C2211yescharacteristic
48#2311C2212yes—
49#2296C2211yescharacteristic
50#2296C2212yes—
51C2×#389C2211yescharacteristic
52C2×#389C2212yes—
53#2285C2211yescharacteristic
54#2085C2212yes—
55#2085C2212yes—
56#2085C2214yes—
57#2084C2211yescharacteristic
58#2084C2211yescharacteristic
59#2084C2212yes—
60#2083C2211yescharacteristic
61#2083C2211yescharacteristic
62#2083C2212yes—
63#2058C2212yes—
64#2058C2212yes—
65#2058C2212yes—
66#2058C2212yes—
67#2058C2214yes—
68#2057C2212yes—
69#2057C2212yes—
70#1944C2214yes—
71#1943C2214yes—
72#1943C2214yes—
73#1943C2214yes—
74#1942C2214yes—
75#1941C2214yes—
76#1941C2214yes—
77#1941C2214yes—
78#1854C2214yes—
79#1854C2214yes—
80#1854C2214yes—
81#1854C2214yes—
82#1853C2214yes—
83#1853C2214yes—
84#1853C2214yes—
85#1852C2214yes—
86#1750C2214yes—
87#1749C2212yes—
88C4×C22⋊C8C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106#520—24no—
107#520—24no—
108#520—44no—
109C4×#41—44no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D412yes—
120C2×C4×D4D412yes—
121C2×C4×D4D411yescharacteristic
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C2×#117C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2312yes—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#428—28no—
140#428—44no—
141#427—28no—
142C2×#107C2312yes—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×#107C2314yes—
146C2×#106—44no—
147C2×#106C2312yes—
148C2×#106C2314yes—
149C2×#106C2312yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153#408Q8116yescomplemented
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156#405—44no—
157C2×#97C2314yes—
158C2×C4×C8C2314yes—
159#400—24no—
160#399—24no—
161#395—24no—
162#393—24no—
163#391—22no—
164#391—22no—
165#391—22no—
166#390—22no—
167#389—21no—
168#389—22no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—21no—
174#389—22no—
175#389—21no—
176#389—22no—
177#389C2311yescharacteristic
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2311yescharacteristic
181#389C2311yescharacteristic
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2314yes—
188#389C2314yes—
189#386—24no—
190#385—21no—
191#385—22no—
192#385—21no—
193#385—22no—
194#385—22no—
195#385—22no—
196#385—22no—
197#385—24no—
198#384—22no—
199#384—21no—
200#384—22no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#382—22no—
204#382—21no—
205#382—22no—
206#382C2312yes—
207#380—22no—
208#380—22no—
209#380—21no—
210#380—21no—
211#380—22no—
212#380—22no—
213#380—22no—
214#380C2314yes—
215#380C2314yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C2×#95—21no—
223C2×#95—22no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2311yescharacteristic
237#336C2311yescharacteristic
238#336C2312yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—24no—
242C2×#41C2×Q818yes—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—44no—
246C22×Q8—82no—
247C22×Q8—82no—
248C22×Q8C2×D411yescharacteristic
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C2×M16—24no—
263C2×M16—24no—
264C2×M16C2×Q818yes—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8C2×Q818yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C22⋊Q8—84no—
276C22⋊Q8—84no—
277C4×Q8—28no—
278C4×Q8—42no—
279C4×Q8—42no—
280C4×Q8—44no—
281C4×D4—216no—
282C4×D4—28no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—44no—
289#117—22no—
290#117—24no—
291#117—24no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—44no—
297#117—44no—
298#117#4114yes—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4—84no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4#4114yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—44no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370#107—24no—
371#107—24no—
372#107—28no—
373#107—44no—
374#106—24no—
375#106—24no—
376#106—28no—
377#106—44no—
378#106—44no—
379#106—44no—
380C22⋊C8C2×Q8116yescomplemented
381C22⋊C8C2×Q8116yescomplemented
382#97—44no—
383C4×C8—44no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400C24C22×D412yes—
401C24C22×D411yescharacteristic
402#41—216no—
403#41—216no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×Q8—816no—
413C2×Q8C4⋊Q814yes—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4C4⋊Q818yes—
419C2×D4C4⋊Q814yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—81no—
450C22×C4—82no—
451C22×C4—81no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—81no—
456C22×C4—82no—
457C22×C4—81no—
458C22×C4—82no—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×Q818yes—
468C22×C4C22×Q814yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)14yes—
472C22×C4C2511yescharacteristic
473M16—216no—
474C2×C8—216no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8C22×Q814yes—
482C2×C8C22×Q814yes—
483C2×C8C22×Q818yes—
484C4⋊C4—28no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—82no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4C2×#4112yes—
524C4⋊C4C2×#4114yes—
525C4⋊C4C2×#4112yes—
526C22⋊C4—28no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C42—28no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42—44no—
547C42—42no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23#52512yes—
559C23C2×C4⋊Q814yes—
560C23C2×C4⋊Q812yes—
561C23#53412yes—
562C23C23×D411yescharacteristic
563Q8—28no—
564Q8—88no—
565Q8—88no—
566D4—216no—
567D4—28no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4—88no—
622C2×C4—162no—
623C2×C4—164no—
624C2×C4#52512yes—
625C2×C4C2×C4⋊Q814yes—
626C2×C4C2×C4⋊Q812yes—
627C2×C4#53412yes—
628C2×C4#55418yes—
629C2×C4#55418yes—
630C2×C4#57712yes—
631C2×C4#57712yes—
632C2×C4C23×Q812yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)12yes—
636C8—24no—
637C8—24no—
638C8—28no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22#258314yes—
653C22C2×#52511yescharacteristic
654C22C22×C4⋊Q811yescharacteristic
655C22C2×#53411yescharacteristic
656C22C2×#57711yescharacteristic
657C22C2×#57711yescharacteristic
658C22#358511yescharacteristic
659C22#358611yescharacteristic
660C4—24no—
661C4—22no—
662C4—22no—
663C4—41no—
664C4—42no—
665C4—41no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—81no—
671C4—84no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—82no—
676C4#258314yes—
677C4C2×#55412yes—
678C4C2×#55412yes—
679C2—24no—
680C2—44no—
681C2C2×#258311yescharacteristic
682C2#6059711yescharacteristic
683C2#6059911yescharacteristic
684C2#6066511yescharacteristic
685C2#6066811yescharacteristic
686C2#6075711yescharacteristic
687C2#6075811yescharacteristic
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. #29311C2semidirect
4. #29308C2semidirect
5. #29304C2semidirect
6. #29301C2semidirect
7. #29299C2semidirect
8. #29297C2semidirect
9. #23657C2semidirect
10. #23656C2semidirect
11. #23552C2semidirect
12. C2×#2058C2semidirect
13. #21160C2semidirect
14. #21155C2semidirect
153. #408Q8semidirect
380. C22⋊C8C2×Q8semidirect
381. C22⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#352792
#342502
#293112
#293082
#293042
#293012
#292992
#292972
#236572
#236562
#235522
C2×#20582
#211602
#211552
#208492
#208482
#203782