Finite Group Databasebeta

← #6491905

Subgroups — #6491905

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic, complemented
2#34129C211yesmaximal, characteristic
3#33198C212yesmaximal, complemented
4#33073C212yesmaximal, complemented
5#29995C212yesmaximal, complemented
6#29980C212yesmaximal, complemented
7#26108C212yesmaximal, complemented
8#26086C212yesmaximal, complemented
9#26046C212yesmaximal, complemented
10#26024C212yesmaximal, complemented
11#26011C214yesmaximal, complemented
12#25975C214yesmaximal
13#21982C211yesmaximal, characteristic, complemented
14#21974C211yesmaximal, characteristic, complemented
15#21964C211yesmaximal, characteristic, complemented
16#20737C211yesmaximal, characteristic
17#20648C211yesmaximal, characteristic
18C2×#544—21no—
19C2×C4×#41C2211yescharacteristic
20C2×#477C2214yes—
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#417C2212yes—
24#2868C2212yes—
25#2859C2212yes—
26#2660—22no—
27#2552—21no—
28#2552C2212yes—
29#2551—21no—
30#2551C2212yes—
31#2440—22no—
32#2440C2214yes—
33#2354—21no—
34#2354—21no—
35#2354C2211yescharacteristic
36#2354C2211yescharacteristic
37Q8×C4⋊C4C2211yescharacteristic
38Q8×C4⋊C4C2211yescharacteristic
39#2346—22no—
40#2346C2212yes—
41#2341C2212yes—
42#2317—21no—
43#2293—22no—
44#2293C2212yes—
45#2285—21no—
46#2285C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C42×Q8C2212yes—
50C4×#117—21no—
51C4×#117C2211yescharacteristic
52#2099C2214yes—
53#2093C2214yes—
54#2088C2214yes—
55#2087C2214yes—
56#2067C2214yes—
57#2065C2214yes—
58#2057C2214yes—
59#2023C2214yes—
60#1987C2214yes—
61#1948C2212yes—
62#1948C2212yes—
63#1947C2212yes—
64#1947C2212yes—
65#1947C2212yes—
66#1947C2214yes—
67#1947C2212yes—
68#1939C2212yes—
69#1939C2212yes—
70#1920C2214yes—
71#1918C2214yes—
72#1865C2214yes—
73#1863C2214yes—
74#1852C2214yes—
75#1851C2214yes—
76#1845C2212yes—
77#1842C2212yes—
78#1834C2212yes—
79#1831C2212yes—
80#1825C2212yes—
81#1809C2214yes—
82#1803C2211yescharacteristic
83#1803C2211yescharacteristic
84#1803C2212yes—
85#1803C2214yes—
86#1803C2212yes—
87#1802C2211yescharacteristic
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1788C2212yes—
92C4×#103C2212yes—
93C4×#102C2212yes—
94#1736C2212yes—
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97#544—48no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8C2314yes—
102C2×#125—22no—
103C2×#125—22no—
104C2×C22⋊Q8—24no—
105C4×#41—24no—
106C4×#41Q818yescomplemented
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×#117—21no—
116C2×#117—21no—
117C2×#117—21no—
118C2×#117—21no—
119C2×#117—21no—
120C2×#117—22no—
121C2×#117D411yescharacteristic
122C2×#117D411yescharacteristic
123C2×#117C2311yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2312yes—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129#477—48no—
130#431—24no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135#423—24no—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142#417—44no—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C4×C8C2312yes—
151#398—24no—
152#398—24no—
153#395—24no—
154#395—24no—
155#390—22no—
156#390—22no—
157#390C2312yes—
158#390C2312yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2314yes—
167#389C2314yes—
168#389C2312yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#385—24no—
176#384—21no—
177#384—21no—
178#384—22no—
179#384—22no—
180#384—21no—
181#384—21no—
182#384C2312yes—
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2311yescharacteristic
186#384C2314yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382—22no—
190#382—21no—
191#382—21no—
192#382—22no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—22no—
198#380C2312yes—
199#380C2312yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376—22no—
220#376D412yes—
221C43—22no—
222C43C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#336C2314yes—
234#336C2314yes—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—41no—
239C2×#41—41no—
240C2×#41C2×Q814yes—
241C22×Q8—24no—
242C22×Q8C2411yescharacteristic
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×M16—24no—
253C2×M16—42no—
254C2×M16—42no—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8—48no—
261C4⋊Q8—48no—
262C4⋊Q8—84no—
263#125—48no—
264#125—48no—
265C22⋊Q8—416no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—42no—
271C4×Q8—42no—
272C4×Q8C2×Q814yes—
273C4×Q8C2×Q814yes—
274C4×D4—28no—
275C4×D4—28no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4C2×Q814yes—
279C4×D4C2×Q814yes—
280#117—22no—
281#117—24no—
282#117—24no—
283#117—28no—
284#117—44no—
285#117—44no—
286#117—44no—
287#117—84no—
288#117—82no—
289#117C2×Q814yes—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—28no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×Q814yes—
357C2×C42#4114yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—44no—
363#107—44no—
364#107—44no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—22no—
369C4⋊C8—22no—
370C4⋊C8—44no—
371C4⋊C8—44no—
372#103—28no—
373#103—24no—
374#103—24no—
375#102—42no—
376#102—42no—
377#102—44no—
378C22⋊C8—24no—
379C4×C8—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—21no—
395C24C22×D411yescharacteristic
396#41—28no—
397#41—44no—
398#41—44no—
399#41—44no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—48no—
404C2×Q8—84no—
405C2×Q8—84no—
406C2×Q8C22×Q812yes—
407C2×Q8H(5,𝔽2)12yes—
408C2×Q82−1+412yes—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4C22×Q812yes—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4C22⋊Q814yes—
447C22×C4C22⋊Q814yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×Q812yes—
458C22×C4C2×#4112yes—
459C22×C4C2511yescharacteristic
460M16—84no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4AGL(1,ℤ/8)14yes—
509C4⋊C4#13714yes—
510C4⋊C4C22×Q812yes—
511C4⋊C4C22×Q812yes—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C42—24no—
524C42—24no—
525C42—24no—
526C42—24no—
527C42—24no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—44no—
535C42—44no—
536C42—44no—
537C42—44no—
538C42—44no—
539C42—44no—
540C42C22⋊Q814yes—
541C42C22⋊Q814yes—
542C42C22×Q812yes—
543C42C22×Q812yes—
544C42C2×#4112yes—
545C42C2×#4112yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×C22⋊Q812yes—
560C23C23×D411yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—22no—
564Q8—168no—
565Q8#54414yescomplemented
566D4—22no—
567D4—22no—
568D4—48no—
569D4—41no—
570D4—41no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—48no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—164no—
621C2×C4C2×C22⋊Q812yes—
622C2×C4C2×C22⋊Q812yes—
623C2×C4C2×C22⋊Q812yes—
624C2×C4C2×C22⋊Q812yes—
625C2×C4C2×C22⋊Q812yes—
626C2×C4C2×AGL(1,ℤ/8)14yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4C2×#13714yes—
629C2×C4C2×#13712yes—
630C2×C4#57612yes—
631C2×C4#57812yes—
632C2×C4C23×Q811yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4#58511yescharacteristic
637C2×C4#58511yescharacteristic
638C8—42no—
639C8—44no—
640C8—42no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22C22×C22⋊Q811yescharacteristic
656C22#348811yescharacteristic
657C22#348911yescharacteristic
658C22C22×AGL(1,ℤ/8)11yescharacteristic
659C22C22×#13711yescharacteristic
660C22C2×#57611yescharacteristic
661C22C2×#57811yescharacteristic
662C4—24no—
663C4—22no—
664C4—22no—
665C4—24no—
666C4—22no—
667C4—24no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—82no—
673C4—84no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4#307914yes—
678C4#308314yes—
679C4C2×#54411yescharacteristic
680C4#345811yescharacteristic
681C4#349411yescharacteristic
682C4#349511yescharacteristic
683C2—42no—
684C2—82no—
685C2#3525311yescharacteristic
686C2C2×#307911yescharacteristic
687C2C2×#308311yescharacteristic
688C2#6198411yescharacteristic
689C2#6200111yescharacteristic
690C2#6213211yescharacteristic
691C2#6214911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35275C2semidirect
3. #33198C2semidirect
4. #33073C2semidirect
5. #29995C2semidirect
6. #29980C2semidirect
7. #26108C2semidirect
8. #26086C2semidirect
9. #26046C2semidirect
10. #26024C2semidirect
11. #26011C2semidirect
13. #21982C2semidirect
14. #21974C2semidirect
15. #21964C2semidirect
106. C4×#41Q8semidirect
565. Q8#544semidirect

Maximal subgroups

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

subgroupindex
#352752
#341292
#331982
#330732
#299952
#299802
#261082
#260862
#260462
#260242
#260112
#259752
#219822
#219742
#219642
#207372
#206482