Finite Group Databasebeta

← #6471408

Subgroups — #6471408

Order 512.

2338 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic
2C2×Q8×C22⋊C4C211yesmaximal, characteristic
3#32957C211yesmaximal, characteristic, complemented
4#32906C211yesmaximal, characteristic, complemented
5#32882C212yesmaximal
6#29953C211yesmaximal, characteristic, complemented
7#29895C212yesmaximal
8#29715C211yesmaximal, characteristic, complemented
9#27524C214yesmaximal, complemented
10#27228C214yesmaximal, complemented
11#25161C214yesmaximal, complemented
12#24906C214yesmaximal, complemented
13#22044C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic
15C4×#451C212yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#477C2212yescomplemented
20C2×#462C2212yescomplemented
21C2×#451C2212yes—
22C2×#451C2212yes—
23C2×#451C2214yes—
24C2×#451C2214yes—
25C2×C4×Q16C2214yescomplemented
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2848C2212yes—
29#2646—21no—
30#2626—21no—
31#2589—22no—
32#2587—22no—
33#2497C2212yes—
34#2492C2212yes—
35#2429C2212yes—
36#2423C2212yes—
37Q8×C22⋊C4—22no—
38Q8×C22⋊C4—22no—
39Q8×C22⋊C4C2214yes—
40Q8×C22⋊C4C2214yes—
41#2329—21no—
42#2312—21no—
43C4×C22⋊Q8C2212yes—
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#2095C2212yescomplemented
53#2076C2214yes—
54#2076C2214yes—
55#2072C2212yescomplemented
56#2068C2214yescomplemented
57#2056C2214yescomplemented
58#2036C2214yes—
59#2036C2214yes—
60#2024C2212yescomplemented
61#2016C2212yescomplemented
62#1969C2214yescomplemented
63#1933C2214yescomplemented
64#1928C2214yescomplemented
65#1923C2214yes—
66#1923C2214yes—
67#1878C2214yescomplemented
68#1878C2214yescomplemented
69#1878C2214yescomplemented
70#1872C2212yes—
71#1872C2212yes—
72#1863C2214yescomplemented
73#1863C2214yescomplemented
74#1852C2212yes—
75#1852C2212yes—
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×Q16C2314yes—
99C22×Q16C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—44no—
110C2×C4×Q8—44no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2314yes—
114C2×C4×Q8C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4D411yescharacteristic
121C22×C22⋊C4D411yescharacteristic
122C22×C42—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#477—44no—
127#462—44no—
128#451—28no—
129#451—28no—
130#451—48no—
131#451—48no—
132C4×Q16—48no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2314yes—
150#400—22no—
151#400—22no—
152#399—22no—
153#398—22no—
154#398—22no—
155#396—22no—
156#396—22no—
157#393—22no—
158#391—21no—
159#391—22no—
160#391—21no—
161#391—21no—
162#391—22no—
163#391—24no—
164#391—24no—
165#391—24no—
166#391—24no—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#391C2314yes—
171#391C2312yes—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#380—21no—
192#380—21no—
193#380—21no—
194#380—22no—
195#380—24no—
196#380—24no—
197#380C2312yes—
198#380C2312yes—
199#380C2314yes—
200#380C2312yes—
201#379—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—24no—
206C4×C22⋊C4—24no—
207C4×C22⋊C4D412yes—
208C4×C22⋊C4D412yes—
209C4×C22⋊C4C2312yes—
210C4×C22⋊C4C2312yes—
211C2×#95—21no—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95D411yescharacteristic
217C2×#95D411yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#336C2312yes—
223#336C2312yes—
224#336C2314yes—
225C25C2×D411yescharacteristic
226C22×Q8—24no—
227C22×Q8—24no—
228C22×Q8—28no—
229C22×Q8—216no—
230C22×Q8—28no—
231C22×Q8—28no—
232C22×Q8—21no—
233C22×Q8—24no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—28no—
239C22×Q8—28no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—48no—
247C22×Q8—48no—
248C22×Q8—432no—
249C22×Q8—432no—
250C22×Q8#4114yes—
251C22×Q8#4114yes—
252C22×Q8C2412yes—
253C22×Q8C2412yes—
254C22×Q8C2412yes—
255C23×C4—21no—
256C23×C4—28no—
257C23×C4—21no—
258C23×C4—24no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—42no—
264C23×C4—42no—
265C23×C4—44no—
266C23×C4—44no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D412yes—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2412yes—
274C23×C4C2411yescharacteristic
275C2×Q16—28no—
276C2×Q16—28no—
277C2×Q16—28no—
278C2×Q16—28no—
279C2×Q16—44no—
280C2×Q16—44no—
281C22×C8C2414yes—
282C22×C8C2414yes—
283C4⋊Q8—42no—
284C4⋊Q8—42no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C4×Q8—28no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—44no—
294C4×Q8—42no—
295C4×Q8—42no—
296C4×Q8—44no—
297C4×Q8—48no—
298C4×Q8—44no—
299C4×Q8—44no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
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—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4Q1618yescomplemented
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—42no—
347C2×C42—42no—
348C2×C42—44no—
349C2×C42—44no—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—44no—
357#107—44no—
358C4⋊C8—42no—
359C4⋊C8—42no—
360#103—28no—
361#103—28no—
362#103—28no—
363#103—28no—
364#103—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C22⋊C8—44no—
371C22⋊C8—44no—
372C4×C8—44no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—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—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—22no—
395C24—22no—
396C24—41no—
397C24—41no—
398C24—41no—
399C24—41no—
400C24C2×Q1614yes—
401C24C22×D412yes—
402C24C22×D411yescharacteristic
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—216no—
406C2×Q8—216no—
407C2×Q8—216no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—48no—
421C2×Q8—48no—
422C2×Q8—48no—
423C2×Q8—432no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—48no—
427C2×Q8—42no—
428C2×Q8—416no—
429C2×Q8—416no—
430C2×Q8—464no—
431C2×Q8—416no—
432C2×Q8—416no—
433C2×Q8—81no—
434C2×Q8—81no—
435C2×Q8C2×#4114yes—
436C2×Q8C2×#4114yes—
437C2×Q8H(5,𝔽2)14yes—
438C2×Q8H(5,𝔽2)14yes—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—216no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—432no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4C2×Q1614yes—
488C22×C4C2×Q1614yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2511yescharacteristic
500Q16—416no—
501Q16—88no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4—81no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—81no—
531C4⋊C4—81no—
532C4⋊C4—81no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—48no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4C2×Q1618yes—
543C22⋊C4C2×Q1618yes—
544C42—24no—
545C42—22no—
546C42—22no—
547C42—42no—
548C42—41no—
549C42—41no—
550C42—48no—
551C42—44no—
552C42—44no—
553C42—82no—
554C42—82no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—22no—
561C23—28no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23#52512yes—
573C23C22×Q1614yes—
574C23C22×Q1612yes—
575C23#57512yes—
576C23C23×D411yescharacteristic
577Q8—216no—
578Q8—24no—
579Q8—24no—
580Q8—24no—
581Q8—28no—
582Q8—432no—
583Q8—48no—
584Q8—44no—
585Q8—44no—
586Q8—44no—
587Q8—162no—
588Q8#53818yes—
589C2×C4—216no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—432no—
619C2×C4—48no—
620C2×C4—44no—
621C2×C4—48no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4#52512yes—
649C2×C4C22×Q1614yes—
650C2×C4C22×Q1612yes—
651C2×C4C22×Q1612yes—
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C2×H(5,𝔽2)12yes—
655C2×C4C2×H(5,𝔽2)12yes—
656C8—44no—
657C8—44no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—416no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22#310414yes—
673C22C2×#52511yescharacteristic
674C22#348411yescharacteristic
675C22#348411yescharacteristic
676C22C23×Q1611yescharacteristic
677C22C2×#57511yescharacteristic
678C22#358611yescharacteristic
679C22#358611yescharacteristic
680C4—24no—
681C4—22no—
682C4—22no—
683C4—28no—
684C4—22no—
685C4—42no—
686C4—41no—
687C4—41no—
688C4—42no—
689C4—44no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4C2×#53812yes—
697C4#349012yes—
698C2—24no—
699C2—44no—
700C2#3498811yescharacteristic
701C2C2×#310411yescharacteristic
702C2#6056611yescharacteristic
703C2#6156311yescharacteristic
704C2#6156311yescharacteristic
705C2#6157811yescharacteristic
706C2#6158311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #32957C2semidirect
4. #32906C2semidirect
6. #29953C2semidirect
8. #29715C2semidirect
9. #27524C2semidirect
10. #27228C2semidirect
11. #25161C2semidirect
12. #24906C2semidirect
13. #22044C2semidirect
15. C4×#451C2semidirect
19. C2×#477C22semidirect
20. C2×#462C22semidirect
25. C2×C4×Q16C22semidirect
52. #2095C22semidirect
55. #2072C22semidirect
56. #2068C22semidirect
57. #2056C22semidirect
60. #2024C22semidirect
61. #2016C22semidirect
62. #1969C22semidirect
63. #1933C22semidirect
64. #1928C22semidirect
67. #1878C22semidirect
68. #1878C22semidirect
69. #1878C22semidirect
72. #1863C22semidirect
73. #1863C22semidirect
76. #1803C22semidirect
77. #1803C22semidirect
85. C4×#103C22semidirect
336. C2×C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×Q8×C22⋊C42
#329572
#329062
#328822
#299532
#298952
#297152
#275242
#272282
#251612
#249062
#220442
#219022
C4×#4512