Finite Group Databasebeta

← #6471080

Subgroups — #6471080

Order 512.

1902 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic
2C2×#2362C211yesmaximal, characteristic, complemented
3#33146C211yesmaximal, characteristic, complemented
4#33106C211yesmaximal, characteristic, complemented
5#32981C212yesmaximal, complemented
6#29951C211yesmaximal, characteristic, complemented
7#29713C211yesmaximal, characteristic, complemented
8#29595C212yesmaximal, complemented
9#27058C214yesmaximal, complemented
10#26748C214yesmaximal, complemented
11#26647C214yesmaximal, complemented
12#26569C214yesmaximal, complemented
13#22081C212yesmaximal, complemented
14#21906C211yesmaximal, characteristic, complemented
15#21454C212yesmaximal, complemented
16C2×#538—21no—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C2×#482C2214yes—
21C2×#480C2212yes—
22C2×#448C2212yes—
23C2×#440C2214yescomplemented
24#2857C2211yescharacteristic
25#2856C2211yescharacteristic
26#2847C2212yes—
27#2639—21no—
28#2617—21no—
29#2559—22no—
30#2557—22no—
31#2496C2212yes—
32#2491C2212yes—
33#2428C2212yes—
34#2422C2212yes—
35#2362—22no—
36#2362—22no—
37#2362C2214yes—
38#2362C2214yes—
39#2325—21no—
40C4×#121C2212yes—
41C2×#391C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#388—21no—
45C2×#387—21no—
46C2×C4×C4⋊C4—21no—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2096C2212yescomplemented
50#2072C2212yescomplemented
51#2066C2214yescomplemented
52#2055C2214yescomplemented
53#2036C2214yes—
54#2036C2214yes—
55#2024C2212yescomplemented
56#2014C2212yescomplemented
57#2000C2214yes—
58#2000C2214yes—
59#1988C2212yescomplemented
60#1982C2212yescomplemented
61#1947C2212yescomplemented
62#1947C2212yescomplemented
63#1943C2214yescomplemented
64#1936C2214yescomplemented
65#1925C2214yescomplemented
66#1924C2214yes—
67#1924C2214yes—
68#1921C2214yescomplemented
69#1918C2214yescomplemented
70#1880C2214yescomplemented
71#1875C2214yes—
72#1874C2212yes—
73#1874C2212yes—
74#1864C2214yescomplemented
75#1863C2214yescomplemented
76#1854C2214yes—
77#1853C2212yes—
78#1852C2212yes—
79#1825C2214yes—
80#1823C2212yes—
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1787C2212yes—
84#1779C2212yes—
85#1763C2214yescomplemented
86#1749C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×SD16C2314yes—
90#538—48no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#126—22no—
94C2×#124—22no—
95C2×#123—22no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8C2314yes—
101C2×#121—21no—
102C2×#121—21no—
103C2×#121—22no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#121C2312yes—
107C2×#120—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—44no—
110C2×C4×Q8C2314yes—
111C2×C4×D4—22no—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C2×#117—44no—
115C2×#117C2314yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42—21no—
127C22×C42—21no—
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#482—48no—
132#480—28no—
133#448—28no—
134#440—48no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2314yes—
150C2×#97C2314yes—
151#400—22no—
152#397—22no—
153#397—22no—
154#394—22no—
155#394—22no—
156#394—22no—
157#393—22no—
158#392—22no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391C2312yes—
163#391C2312yes—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#390C2312yes—
168#389—21no—
169#389—22no—
170#389C2312yes—
171#389C2314yes—
172#389C2312yes—
173#388—24no—
174#388—24no—
175#388—24no—
176#388—24no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—24no—
181#387—24no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#385—22no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#380—21no—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—24no—
197C4×C4⋊C4—24no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—22no—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4C2312yes—
205C4×C22⋊C4C2312yes—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—42no—
226C22×Q8—42no—
227C22×Q8#4114yes—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C2×SD16—28no—
253C2×SD16—44no—
254C2×SD16—44no—
255C22×C8C2414yes—
256C22×C8C2414yes—
257C4⋊Q8—42no—
258#127—42no—
259#126—48no—
260#124—48no—
261#123—48no—
262C22⋊Q8—216no—
263C22⋊Q8—28no—
264C22⋊Q8—28no—
265C22⋊Q8—44no—
266#121—24no—
267#121—24no—
268#121—48no—
269#121—44no—
270#121—44no—
271#121—44no—
272#121—44no—
273#120—48no—
274C4×Q8—28no—
275C4×Q8—44no—
276C4×Q8—48no—
277C4×D4—44no—
278C4×D4—48no—
279#117—28no—
280#117—44no—
281#117—48no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4#4114yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—28no—
346#107—28no—
347#106—44no—
348C4⋊C8—42no—
349C4⋊C8—42no—
350#103—28no—
351#103—44no—
352#103—44no—
353#102—28no—
354#102—28no—
355#102—44no—
356C22⋊C8—24no—
357C22⋊C8—24no—
358C22⋊C8—44no—
359C22⋊C8—44no—
360#97—44no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—41no—
386C24C22×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—48no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—48no—
396C2×Q8—42no—
397C2×Q8—81no—
398C2×Q8—81no—
399C2×Q8C2×#4114yes—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—81no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4—82no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4AGL(1,ℤ/8)14yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2511yescharacteristic
478SD16—416no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C4⋊C4—24no—
486C4⋊C4—216no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4C2×#4114yes—
506C4⋊C4H(5,𝔽2)14yes—
507C22⋊C4—216no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—41no—
528C42—41no—
529C42—42no—
530C42—48no—
531C42—48no—
532C42—44no—
533C42—44no—
534C42—44no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—82no—
557C23—82no—
558C23—82no—
559C23—82no—
560C23#52512yes—
561C23C2×AGL(1,ℤ/8)12yes—
562C23#57512yes—
563C23C23×D411yescharacteristic
564Q8—28no—
565Q8—48no—
566Q8—44no—
567Q8—162no—
568D4—44no—
569D4—44no—
570D4—88no—
571D4—162no—
572D4—162no—
573D4—164no—
574D4—164no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—84no—
631C2×C4—164no—
632C2×C4#52512yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C22×#4111yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×H(5,𝔽2)12yes—
638C2×C4#58512yes—
639C8—44no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—88no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22#311514yes—
663C22C2×#52511yescharacteristic
664C22#348411yescharacteristic
665C22#348511yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22C2×#57511yescharacteristic
668C22#358611yescharacteristic
669C22#358611yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—41no—
674C4—41no—
675C4—44no—
676C4—42no—
677C4—42no—
678C4—44no—
679C4—42no—
680C4—44no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#345912yes—
686C4#350612yes—
687C2—24no—
688C2—82no—
689C2—162no—
690C2#3498611yescharacteristic
691C2C2×#311511yescharacteristic
692C2#6056611yescharacteristic
693C2#6157811yescharacteristic
694C2#6158511yescharacteristic
695C2#6170411yescharacteristic
696C2#6171211yescharacteristic
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×#2362C2semidirect
3. #33146C2semidirect
4. #33106C2semidirect
5. #32981C2semidirect
6. #29951C2semidirect
7. #29713C2semidirect
8. #29595C2semidirect
9. #27058C2semidirect
10. #26748C2semidirect
11. #26647C2semidirect
12. #26569C2semidirect
13. #22081C2semidirect
14. #21906C2semidirect
15. #21454C2semidirect
23. C2×#440C22semidirect
49. #2096C22semidirect
50. #2072C22semidirect
51. #2066C22semidirect
52. #2055C22semidirect
55. #2024C22semidirect
56. #2014C22semidirect
59. #1988C22semidirect
60. #1982C22semidirect
61. #1947C22semidirect
62. #1947C22semidirect
63. #1943C22semidirect
64. #1936C22semidirect
65. #1925C22semidirect
68. #1921C22semidirect
69. #1918C22semidirect
70. #1880C22semidirect
74. #1864C22semidirect
75. #1863C22semidirect
85. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#23622
#331462
#331062
#329812
#299512
#297132
#295952
#270582
#267482
#266472
#265692
#220812
#219062
#214542