Finite Group Databasebeta

← #6471407

Subgroups — #6471407

Order 512.

2504 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic
2C2×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
3#32954C211yesmaximal, characteristic, complemented
4#32903C211yesmaximal, characteristic, complemented
5#32879C212yesmaximal, complemented
6#29951C211yesmaximal, characteristic, complemented
7#29892C212yesmaximal, complemented
8#29713C211yesmaximal, characteristic, complemented
9#27522C214yesmaximal, complemented
10#27226C214yesmaximal, complemented
11#25153C214yesmaximal, complemented
12#24898C214yesmaximal, complemented
13#22041C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic, complemented
15C4×#448C212yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C2×#538—21no—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#475C2212yescomplemented
21C2×#460C2212yescomplemented
22C2×#448C2212yes—
23C2×#448C2212yes—
24C2×#448C2214yes—
25C2×#448C2214yes—
26C2×C4×SD16C2214yescomplemented
27#2857C2211yescharacteristic
28#2856C2211yescharacteristic
29#2847C2212yes—
30#2639—21no—
31#2617—21no—
32#2559—22no—
33#2557—22no—
34#2496C2212yes—
35#2491C2212yes—
36#2428C2212yes—
37#2422C2212yes—
38Q8×C22⋊C4—22no—
39Q8×C22⋊C4—22no—
40Q8×C22⋊C4C2214yes—
41Q8×C22⋊C4C2214yes—
42#2325—21no—
43C4×#121C2212yes—
44C2×#391—21no—
45C2×#391—21no—
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#380—21no—
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4—21no—
51C2×C4×C22⋊C4C2211yescharacteristic
52#2096C2212yescomplemented
53#2077C2214yes—
54#2077C2214yes—
55#2072C2212yescomplemented
56#2066C2214yescomplemented
57#2055C2214yescomplemented
58#2034C2214yes—
59#2034C2214yes—
60#2024C2212yescomplemented
61#2014C2212yescomplemented
62#1968C2214yescomplemented
63#1933C2214yescomplemented
64#1927C2214yescomplemented
65#1923C2214yes—
66#1923C2214yes—
67#1880C2214yescomplemented
68#1880C2214yescomplemented
69#1877C2214yescomplemented
70#1874C2212yes—
71#1874C2212yes—
72#1865C2214yescomplemented
73#1865C2214yescomplemented
74#1853C2212yes—
75#1853C2212yes—
76#1803C2212yescomplemented
77#1803C2212yescomplemented
78#1797C2211yescharacteristic
79#1797C2211yescharacteristic
80#1787C2212yes—
81#1787C2214yes—
82#1787C2214yes—
83#1787C2212yes—
84#1779C2212yes—
85C4×#103C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C23×Q8—24no—
88C23×Q8—24no—
89C23×Q8—22no—
90C23×Q8—21no—
91C23×Q8—44no—
92C23×Q8—44no—
93C23×Q8C2311yescharacteristic
94C23×Q8C2314yes—
95C23×Q8C2311yescharacteristic
96C24×C4—21no—
97C24×C4C2311yescharacteristic
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100#538—48no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#126—22no—
104C2×#124—22no—
105C2×#123—22no—
106C2×C22⋊Q8—21no—
107C2×#121—21no—
108C2×#121—21no—
109C2×#121—22no—
110C2×#121C2312yes—
111C2×#121C2312yes—
112C2×#121C2312yes—
113C2×#120—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—44no—
118C2×C4×Q8—44no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2314yes—
122C2×C4×D4—22no—
123C2×C4×D4C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4D411yescharacteristic
130C22×C22⋊C4D411yescharacteristic
131C22×C42—21no—
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#475—44no—
136#460—44no—
137#448—28no—
138#448—28no—
139#448—48no—
140#448—48no—
141C4×SD16—48no—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C4×C8C2314yes—
159#400—22no—
160#397—22no—
161#397—22no—
162#394—22no—
163#394—22no—
164#394—22no—
165#393—22no—
166#392—22no—
167#391—22no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—24no—
172#391—24no—
173#391—24no—
174#391—24no—
175#391C2312yes—
176#391C2312yes—
177#391C2314yes—
178#390—21no—
179#390—21no—
180#390C2312yes—
181#390C2312yes—
182#389—21no—
183#389C2312yes—
184#389C2312yes—
185#388—22no—
186#388—22no—
187#387—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#380—21no—
197#380—22no—
198#380—24no—
199#380—24no—
200#380C2312yes—
201#380C2312yes—
202#380C2314yes—
203#380C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—24no—
206C4×C22⋊C4—24no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4D412yes—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4C2312yes—
211C4×C22⋊C4C2312yes—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95—21no—
217C2×#95D411yescharacteristic
218C2×#95D411yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#336C2312yes—
224#336C2312yes—
225#336C2314yes—
226C25C2×D411yescharacteristic
227C22×Q8—24no—
228C22×Q8—24no—
229C22×Q8—28no—
230C22×Q8—216no—
231C22×Q8—28no—
232C22×Q8—28no—
233C22×Q8—21no—
234C22×Q8—24no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—28no—
240C22×Q8—28no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8—48no—
248C22×Q8—48no—
249C22×Q8—432no—
250C22×Q8—432no—
251C22×Q8#4114yes—
252C22×Q8#4114yes—
253C22×Q8C2412yes—
254C22×Q8C2412yes—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4C2412yes—
261C23×C4—21no—
262C23×C4—28no—
263C23×C4—21no—
264C23×C4—24no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—42no—
271C23×C4—42no—
272C23×C4—44no—
273C23×C4—44no—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D412yes—
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2412yes—
281C23×C4C2411yescharacteristic
282C2×SD16—28no—
283C2×SD16—28no—
284C2×SD16—28no—
285C2×SD16—28no—
286C2×SD16—44no—
287C2×SD16—44no—
288C22×C8C2414yes—
289C22×C8C2414yes—
290C4⋊Q8—42no—
291#127—42no—
292#126—48no—
293#124—48no—
294#123—48no—
295C22⋊Q8—44no—
296#121—24no—
297#121—24no—
298#121—44no—
299#121—44no—
300#121—48no—
301#121—44no—
302#121—44no—
303#120—48no—
304C4×Q8—28no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—44no—
308C4×Q8—42no—
309C4×Q8—42no—
310C4×Q8—48no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×D4—44no—
314C4×D4—48no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—42no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4SD1618yescomplemented
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—22no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—42no—
360C2×C42—42no—
361C2×C42—44no—
362C2×C42—44no—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#106—44no—
370#106—44no—
371C4⋊C8—42no—
372C4⋊C8—42no—
373#103—28no—
374#103—28no—
375#103—28no—
376#103—28no—
377#103—44no—
378#103—44no—
379#102—44no—
380#102—44no—
381C22⋊C8—24no—
382C22⋊C8—24no—
383C22⋊C8—44no—
384C22⋊C8—44no—
385C4×C8—44no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—41no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24C2×SD1614yes—
411C24C22×D412yes—
412C24C22×D411yescharacteristic
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—216no—
416C2×Q8—216no—
417C2×Q8—216no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—48no—
427C2×Q8—48no—
428C2×Q8—48no—
429C2×Q8—432no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—48no—
433C2×Q8—416no—
434C2×Q8—416no—
435C2×Q8—464no—
436C2×Q8—416no—
437C2×Q8—416no—
438C2×Q8—42no—
439C2×Q8—81no—
440C2×Q8—81no—
441C2×Q8C2×#4114yes—
442C2×Q8C2×#4114yes—
443C2×Q8H(5,𝔽2)14yes—
444C2×Q8H(5,𝔽2)14yes—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—42no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—42no—
456C2×D4—81no—
457C2×D4—81no—
458C2×D4—81no—
459C2×D4—81no—
460C2×D4—82no—
461C2×D4—82no—
462C2×D4—82no—
463C2×D4—82no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—216no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—432no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—44no—
512C22×C4—82no—
513C22×C4—82no—
514C22×C4C2×SD1614yes—
515C22×C4C2×SD1614yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D412yes—
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C2×#4112yes—
525C22×C4C2×#4112yes—
526C22×C4C2511yescharacteristic
527SD16—416no—
528SD16—88no—
529C2×C8—24no—
530C2×C8—24no—
531C2×C8—24no—
532C2×C8—24no—
533C2×C8—24no—
534C2×C8—24no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—22no—
542C4⋊C4—48no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—81no—
553C4⋊C4—81no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—44no—
557C22⋊C4—48no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4C2×SD1618yes—
569C22⋊C4C2×SD1618yes—
570C42—24no—
571C42—22no—
572C42—22no—
573C42—42no—
574C42—41no—
575C42—41no—
576C42—48no—
577C42—44no—
578C42—44no—
579C42—44no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—22no—
585C23—22no—
586C23—28no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23#52512yes—
605C23C22×SD1614yes—
606C23C22×SD1612yes—
607C23#57512yes—
608C23C23×D411yescharacteristic
609Q8—216no—
610Q8—24no—
611Q8—24no—
612Q8—24no—
613Q8—28no—
614Q8—432no—
615Q8—48no—
616Q8—44no—
617Q8—162no—
618Q8#53818yescomplemented
619D4—44no—
620D4—44no—
621D4—88no—
622D4—162no—
623D4—162no—
624D4—164no—
625D4—164no—
626C2×C4—216no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—22no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—432no—
656C2×C4—48no—
657C2×C4—44no—
658C2×C4—48no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—82no—
679C2×C4—82no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—164no—
683C2×C4#52512yes—
684C2×C4C22×SD1614yes—
685C2×C4C22×SD1612yes—
686C2×C4C22×SD1612yes—
687C2×C4C22×#4111yescharacteristic
688C2×C4C22×#4111yescharacteristic
689C2×C4C2×H(5,𝔽2)12yes—
690C2×C4C2×H(5,𝔽2)12yes—
691C8—44no—
692C8—44no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22—416no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22#310314yes—
715C22C2×#52511yescharacteristic
716C22#348411yescharacteristic
717C22#348411yescharacteristic
718C22C23×SD1611yescharacteristic
719C22C2×#57511yescharacteristic
720C22#358611yescharacteristic
721C22#358611yescharacteristic
722C4—24no—
723C4—22no—
724C4—22no—
725C4—28no—
726C4—22no—
727C4—42no—
728C4—41no—
729C4—41no—
730C4—42no—
731C4—44no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4C2×#53812yes—
738C4#349012yes—
739C2—24no—
740C2—44no—
741C2—162no—
742C2#3498811yescharacteristic
743C2C2×#310311yescharacteristic
744C2#6056611yescharacteristic
745C2#6156211yescharacteristic
746C2#6156211yescharacteristic
747C2#6157811yescharacteristic
748C2#6158311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35005C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C22⋊C4C2semidirect
3. #32954C2semidirect
4. #32903C2semidirect
5. #32879C2semidirect
6. #29951C2semidirect
7. #29892C2semidirect
8. #29713C2semidirect
9. #27522C2semidirect
10. #27226C2semidirect
11. #25153C2semidirect
12. #24898C2semidirect
13. #22041C2semidirect
14. #21902C2semidirect
15. C4×#448C2semidirect
20. C2×#475C22semidirect
21. C2×#460C22semidirect
26. C2×C4×SD16C22semidirect
52. #2096C22semidirect
55. #2072C22semidirect
56. #2066C22semidirect
57. #2055C22semidirect
60. #2024C22semidirect
61. #2014C22semidirect
62. #1968C22semidirect
63. #1933C22semidirect
64. #1927C22semidirect
67. #1880C22semidirect
68. #1880C22semidirect
69. #1877C22semidirect
72. #1865C22semidirect
73. #1865C22semidirect
76. #1803C22semidirect
77. #1803C22semidirect
85. C4×#103C22semidirect
350. C2×C22⋊C4SD16semidirect
618. Q8#538semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×Q8×C22⋊C42
#329542
#329032
#328792
#299512
#298922
#297132
#275222
#272262
#251532
#248982
#220412
#219022
C4×#4482