Finite Group Databasebeta

← #6470506

Subgroups — #6470506

Order 512.

2420 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37306C211yesmaximal, characteristic, complemented
2#34258C211yesmaximal, characteristic
3#32963C212yesmaximal, complemented
4#32958C212yesmaximal, complemented
5#29959C212yesmaximal, complemented
6#29950C212yesmaximal, complemented
7#25975C214yesmaximal, complemented
8#25974C214yesmaximal, complemented
9#25959C214yesmaximal, complemented
10#25950C214yesmaximal, complemented
11#21947C212yesmaximal, complemented
12#21936C211yesmaximal, characteristic, complemented
13#20602C212yesmaximal
14C22×H(5,𝔽2)C2211yescharacteristic
15C2×#520C2211yescharacteristic
16C2×C4×#41C2211yescharacteristic
17C2×#477C2212yescomplemented
18C2×#477C2212yescomplemented
19C2×#476C2212yescomplemented
20C2×#474C2212yescomplemented
21C2×#417C2212yes—
22C2×#418C2212yes—
23C2×#416C2212yes—
24C22×#102C2212yes—
25C2×#408C2212yes—
26#2857C2212yescomplemented
27#2811—21no—
28#2809—21no—
29#2804—21no—
30#2692—24no—
31#2683—21no—
32#2628C2212yescomplemented
33#2440C2214yescomplemented
34#2439C2214yescomplemented
35#2341C2214yes—
36D4×C22⋊C4—24no—
37D4×C22⋊C4C2214yes—
38#2311—21no—
39#2311C2212yes—
40#2311C2211yescharacteristic
41#2309—21no—
42#2309C2211yescharacteristic
43#2297—21no—
44#2297C2211yescharacteristic
45C2×#390—21no—
46C2×#390C2211yescharacteristic
47#2093C2214yescomplemented
48#2092C2214yescomplemented
49#2067C2214yescomplemented
50#2067C2214yescomplemented
51#2067C2214yescomplemented
52#2065C2214yescomplemented
53#2023C2214yescomplemented
54#2023C2214yescomplemented
55#1865C2214yescomplemented
56#1863C2214yescomplemented
57#1862C2214yescomplemented
58#1862C2214yescomplemented
59#1854C2214yescomplemented
60#1854C2214yescomplemented
61#1852C2214yescomplemented
62#1852C2214yescomplemented
63#1851C2214yescomplemented
64#1850C2214yescomplemented
65#1809C2214yescomplemented
66#1808C2214yescomplemented
67#1805C2212yescomplemented
68#1804C2212yescomplemented
69#1803C2212yescomplemented
70#1802C2212yescomplemented
71#1798C2212yescomplemented
72#1797C2212yescomplemented
73#1788C2214yes—
74#1788C2214yes—
75#1786C2214yes—
76#1786C2214yes—
77C2×H(5,𝔽2)—24no—
78C2×H(5,𝔽2)—42no—
79C2×H(5,𝔽2)—42no—
80C2×H(5,𝔽2)—42no—
81C2×H(5,𝔽2)—42no—
82C2×H(5,𝔽2)Q818yescomplemented
83C22×#41—21no—
84C22×#41C2311yescharacteristic
85C22×#41C2312yes—
86C22×#41C2311yescharacteristic
87C23×D4—21no—
88C23×D4—22no—
89C23×D4C2312yes—
90C23×D4C2311yescharacteristic
91C22×M16C2312yes—
92C23×C8C2312yes—
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2312yes—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—24no—
97#520—42no—
98#520—42no—
99C4×#41—42no—
100C4×#41—42no—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2312yes—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C2×#117—21no—
108C2×#117—21no—
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4—22no—
116C22×C22⋊C4—24no—
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119#477—44no—
120#477—44no—
121#476—44no—
122#474—44no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131#418—44no—
132#417—44no—
133#416—44no—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102—44no—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143#408—28no—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146#400—22no—
147#399—22no—
148#398—24no—
149#396—24no—
150#395—22no—
151#395—24no—
152#393—22no—
153#393—24no—
154#391—21no—
155#391—22no—
156#391—22no—
157#391—24no—
158#391D411yescharacteristic
159#391D411yescharacteristic
160#390—21no—
161#390—22no—
162#390—21no—
163#390—21no—
164#390—24no—
165#390—24no—
166#390—21no—
167#390—21no—
168#390D411yescharacteristic
169#390D411yescharacteristic
170#390C2314yes—
171#390C2314yes—
172#389—22no—
173#389C2314yes—
174#389C2314yes—
175#387—24no—
176#386—22no—
177#386—22no—
178#386—22no—
179#386—24no—
180#386—24no—
181#386D414yes—
182#385—24no—
183#384C2312yes—
184#382C2312yes—
185#380—22no—
186#380—24no—
187#380C2314yes—
188#380C2314yes—
189#380C2314yes—
190#379—24no—
191#379—22no—
192C4×C22⋊C4—24no—
193C4×C22⋊C4C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—22no—
201C25—21no—
202H(5,𝔽2)—416no—
203H(5,𝔽2)—416no—
204H(5,𝔽2)—416no—
205H(5,𝔽2)—416no—
206C2×#41—28no—
207C2×#41—22no—
208C2×#41—22no—
209C2×#41—22no—
210C2×#41—22no—
211C2×#41—22no—
212C2×#41—24no—
213C2×#41—24no—
214C2×#41—24no—
215C2×#41—41no—
216C2×#41—41no—
217C2×#41—41no—
218C2×#41—41no—
219C2×#41—42no—
220C2×#41—42no—
221C2×#41—44no—
222C2×#41—44no—
223C2×#41C2×Q814yes—
224C2×#41C2×Q818yes—
225C2×#41C2×Q814yes—
226C2×#41#4114yes—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8C2×D411yescharacteristic
230C22×Q8C2411yescharacteristic
231C22×D4—21no—
232C22×D4—24no—
233C22×D4—24no—
234C22×D4—22no—
235C22×D4—28no—
236C22×D4—216no—
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—24no—
241C22×D4—41no—
242C22×D4—41no—
243C22×D4—41no—
244C22×D4—41no—
245C22×D4—42no—
246C22×D4—42no—
247C22×D4—44no—
248C22×D4—44no—
249C22×D4—48no—
250C22×D4—48no—
251C22×D4C2×D412yes—
252C22×D4C2×D411yescharacteristic
253C22×D4C2×Q818yes—
254C22×D4C2×Q814yes—
255C22×D4C2412yes—
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—22no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4C2×D412yes—
264C23×C4C2411yescharacteristic
265C23×C4C2412yes—
266C2×M16—24no—
267C2×M16—42no—
268C2×M16—42no—
269C22×C8—24no—
270C22×C8—42no—
271C22×C8—42no—
272C22×C8C2414yes—
273C22×C8C2414yes—
274C4⋊Q8—42no—
275C4⋊Q8—42no—
276C4⋊Q8—42no—
277C4⋊Q8—42no—
278C22⋊Q8—88no—
279C22⋊Q8—84no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—42no—
287C4×D4—42no—
288C4×D4—48no—
289C4×D4—48no—
290#117—41no—
291#117—41no—
292#117—41no—
293#117—41no—
294#117—44no—
295#117—44no—
296#117—42no—
297#117—42no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—41no—
309C2×C4⋊C4—41no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—88no—
331C2×C22⋊C4—84no—
332C2×C22⋊C4C2×D414yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D414yes—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—41no—
340C2×C42—41no—
341C2×C42—44no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2412yes—
346#107—44no—
347#107—44no—
348#107—44no—
349#106—44no—
350C4⋊C8—42no—
351C4⋊C8—42no—
352C4⋊C8—42no—
353C4⋊C8—42no—
354#103—28no—
355#103—28no—
356#102—28no—
357#102—28no—
358#102—44no—
359#102—44no—
360#102—44no—
361#102—44no—
362C22⋊C8—28no—
363C22⋊C8—28no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95C2×D414yes—
372#95C2×D414yes—
373C24—21no—
374C24—22no—
375C24—28no—
376C24—24no—
377C24—44no—
378C24—44no—
379C24—42no—
380C24—42no—
381C24—84no—
382C24—82no—
383C24—84no—
384C24—82no—
385C24C22×D412yes—
386C24C22×D411yescharacteristic
387#41—28no—
388#41—28no—
389#41—28no—
390#41—28no—
391#41—216no—
392#41—44no—
393#41—44no—
394#41—44no—
395#41—44no—
396#41—48no—
397#41—48no—
398#41—48no—
399#41—432no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—41no—
404C2×Q8—41no—
405C2×Q8—44no—
406C2×Q8—84no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8C22⋊Q814yes—
411C2×Q8C22×Q812yes—
412C2×Q8H(5,𝔽2)12yes—
413C2×Q8H(5,𝔽2)12yes—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—216no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—24no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—48no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—48no—
433C2×D4—48no—
434C2×D4—41no—
435C2×D4—42no—
436C2×D4—41no—
437C2×D4—42no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—432no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—464no—
448C2×D4C22⋊Q818yes—
449C2×D4C22⋊Q814yes—
450C2×D4C22×Q814yes—
451C2×D4C22×Q812yes—
452C2×D4H(5,𝔽2)12yes—
453C2×D4H(5,𝔽2)12yes—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—28no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—82no—
481C22×C4—84no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4—84no—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C22×Q814yes—
492C22×C4C22×Q812yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4114yes—
495C22×C4C2511yescharacteristic
496M16—84no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—84no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—82no—
520C4⋊C4—84no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C22⋊C4—28no—
525C22⋊C4—44no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—48no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4—82no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—41no—
543C42—41no—
544C42—44no—
545C42—44no—
546C42—42no—
547C42—82no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—24no—
558C23—22no—
559C23—24no—
560C23—416no—
561C23—48no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—816no—
569C23—88no—
570C23—168no—
571C23—164no—
572C23—164no—
573C23—164no—
574C23—162no—
575C23—162no—
576C23C2×C22⋊Q812yes—
577C23C2×C22⋊Q814yes—
578C23C23×D411yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—44no—
583Q8—164no—
584Q8—164no—
585Q8#54314yes—
586D4—216no—
587D4—22no—
588D4—24no—
589D4—24no—
590D4—22no—
591D4—22no—
592D4—44no—
593D4—48no—
594D4—416no—
595D4—416no—
596D4—48no—
597D4—42no—
598D4—42no—
599D4#54314yes—
600C2×C4—216no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—416no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—84no—
638C2×C4—88no—
639C2×C4—164no—
640C2×C4—162no—
641C2×C4—164no—
642C2×C4C2×C22⋊Q812yes—
643C2×C4C2×C22⋊Q814yes—
644C2×C4#57614yes—
645C2×C4C23×Q811yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C8—44no—
652C8—44no—
653C22—28no—
654C22—24no—
655C22—22no—
656C22—24no—
657C22—24no—
658C22—44no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—41no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—41no—
667C22—42no—
668C22—48no—
669C22—44no—
670C22—84no—
671C22—82no—
672C22—82no—
673C22—1616no—
674C22—168no—
675C22C22×C22⋊Q811yescharacteristic
676C22#348411yescharacteristic
677C22#348411yescharacteristic
678C22C2×#57612yes—
679C22#358612yes—
680C4—22no—
681C4—24no—
682C4—24no—
683C4—24no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—82no—
688C4—84no—
689C4—82no—
690C4—84no—
691C4C2×#54311yescharacteristic
692C4C2×#54311yescharacteristic
693C4#349011yescharacteristic
694C4#349011yescharacteristic
695C2—42no—
696C2—44no—
697C2—44no—
698C2—84no—
699C2#3498211yescharacteristic
700C2#6049412yes—
701C2#6162412yes—
702C2#6166111yescharacteristic
703C2#6167411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37306C2semidirect
3. #32963C2semidirect
4. #32958C2semidirect
5. #29959C2semidirect
6. #29950C2semidirect
7. #25975C2semidirect
8. #25974C2semidirect
9. #25959C2semidirect
10. #25950C2semidirect
11. #21947C2semidirect
12. #21936C2semidirect
17. C2×#477C22semidirect
18. C2×#477C22semidirect
19. C2×#476C22semidirect
20. C2×#474C22semidirect
26. #2857C22semidirect
32. #2628C22semidirect
33. #2440C22semidirect
34. #2439C22semidirect
47. #2093C22semidirect
48. #2092C22semidirect
49. #2067C22semidirect
50. #2067C22semidirect
51. #2067C22semidirect
52. #2065C22semidirect
53. #2023C22semidirect
54. #2023C22semidirect
55. #1865C22semidirect
56. #1863C22semidirect
57. #1862C22semidirect
58. #1862C22semidirect
59. #1854C22semidirect
60. #1854C22semidirect
61. #1852C22semidirect
62. #1852C22semidirect
63. #1851C22semidirect
64. #1850C22semidirect
65. #1809C22semidirect
66. #1808C22semidirect
67. #1805C22semidirect
68. #1804C22semidirect
69. #1803C22semidirect
70. #1802C22semidirect
71. #1798C22semidirect
72. #1797C22semidirect
82. C2×H(5,𝔽2)Q8semidirect

Maximal subgroups

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

subgroupindex
#373062
#342582
#329632
#329582
#299592
#299502
#259752
#259742
#259592
#259502
#219472
#219362
#206022