Finite Group Databasebeta

← #6477614

Subgroups — #6477614

Order 512.

2158 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
1C2×#3092C212yesmaximal, complemented
2#35032C211yesmaximal, characteristic
3C2×#2344C211yesmaximal, characteristic, complemented
4#33111C212yesmaximal, complemented
5#33108C211yesmaximal, characteristic, complemented
6#33106C211yesmaximal, characteristic, complemented
7#29760C212yesmaximal, complemented
8#29756C211yesmaximal, characteristic, complemented
9#29754C211yesmaximal, characteristic, complemented
10#24589C214yesmaximal, complemented
11#24588C214yesmaximal, complemented
12#24489C214yesmaximal, complemented
13#24488C214yesmaximal, complemented
14#21934C211yesmaximal, characteristic, complemented
15#21769C212yesmaximal, complemented
16C2×#525C2212yes—
17C22×#117—21no—
18C23×C4⋊C4C2211yescharacteristic
19#3092—216no—
20C2×#484C2214yes—
21C2×#481C2214yes—
22C2×#483C2212yes—
23C2×#483C2212yes—
24C2×#480C2212yes—
25C2×#480C2212yes—
26C2×#428C2214yes—
27C2×#417C2214yes—
28C2×#407C2212yes—
29#2857C2211yescharacteristic
30#2856C2211yescharacteristic
31#2850C2212yes—
32#2617—21no—
33#2617—21no—
34#2602—22no—
35#2602—22no—
36#2500C2212yes—
37#2499C2212yes—
38#2432C2212yes—
39#2431C2212yes—
40#2344—22no—
41#2344—22no—
42#2344C2214yes—
43#2344C2214yes—
44#2325—21no—
45#2325—21no—
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#387—21no—
49C2×#387—21no—
50C2×#381—21no—
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55#2085C2214yescomplemented
56#2084C2212yescomplemented
57#2083C2212yescomplemented
58#2068C2214yescomplemented
59#2067C2214yescomplemented
60#2066C2214yescomplemented
61#2065C2214yescomplemented
62#2043C2212yescomplemented
63#2042C2214yescomplemented
64#2041C2212yescomplemented
65#1983C2214yescomplemented
66#1982C2212yescomplemented
67#1981C2212yescomplemented
68#1949C2214yescomplemented
69#1948C2212yescomplemented
70#1947C2212yescomplemented
71#1928C2214yescomplemented
72#1927C2214yescomplemented
73#1926C2214yescomplemented
74#1925C2214yescomplemented
75#1895C2214yes—
76#1894C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1872C2212yes—
80#1872C2212yes—
81#1851C2214yes—
82#1851C2214yes—
83#1825C2214yes—
84#1797C2211yescharacteristic
85#1797C2211yescharacteristic
86#1797C2212yes—
87#1788C2214yes—
88C22×#41C2312yes—
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×M16C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#124C2312yes—
95#525—28no—
96#525—28no—
97C2×#123—22no—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121C2312yes—
103C2×#121C2312yes—
104C2×#120—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×#117—24no—
108C2×#117—24no—
109C2×#117—48no—
110C2×#117C2312yes—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—24no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D418yescomplemented
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4D411yescharacteristic
124C22×C22⋊C4D411yescharacteristic
125C22×C42—21no—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#484—216no—
130#483—28no—
131#483—28no—
132#481—216no—
133#480—28no—
134#480—28no—
135#428—216no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143#417—216no—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152#407—28no—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2314yes—
156#397—22no—
157#397—22no—
158#394—22no—
159#394—22no—
160#394—22no—
161#394—22no—
162#392—22no—
163#392—22no—
164#391C2312yes—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389C2314yes—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#387—24no—
185#387—24no—
186#387—24no—
187#387—24no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#385—22no—
193#385—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#382C2312yes—
197#381—24no—
198#381—24no—
199#380—21no—
200#380—21no—
201#380—22no—
202#380C2314yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4C2312yes—
212C4×C22⋊C4C2312yes—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95—21no—
217C2×#95—21no—
218C2×#95D411yescharacteristic
219C2×#95D411yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#336C2312yes—
225#336C2312yes—
226#336C2314yes—
227C25C2×D411yescharacteristic
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—44no—
231C2×#41—44no—
232C22×Q8C2412yes—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2412yes—
239C23×C4—28no—
240C23×C4—21no—
241C23×C4—24no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D414yes—
256C23×C4C2×D411yescharacteristic
257C23×C4#4114yes—
258C23×C4C2411yescharacteristic
259C23×C4C2412yes—
260C2×M16—28no—
261C2×M16—44no—
262C2×M16—44no—
263C22×C8C2414yes—
264C22×C8C2414yes—
265C4⋊Q8—42no—
266#127—42no—
267#124—44no—
268#123—48no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273#121—24no—
274#121—24no—
275#121—24no—
276#121—24no—
277#121—44no—
278#121—44no—
279#120—48no—
280C4×D4—48no—
281C4×D4—48no—
282#117—24no—
283#117—24no—
284#117—416no—
285#117—416no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—216no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D4116yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42—44no—
346C2×C42—44no—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—28no—
354#107—28no—
355#106—28no—
356#106—28no—
357C4⋊C8—44no—
358C4⋊C8—42no—
359C4⋊C8—42no—
360#103—28no—
361#103—28no—
362#103—44no—
363#103—44no—
364#102—28no—
365#102—28no—
366#102—44no—
367#102—44no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C22⋊C8—28no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—41no—
392C24—41no—
393C24—41no—
394C24—41no—
395C24#12114yes—
396C24C22×D412yes—
397C24C22×D411yescharacteristic
398#41—416no—
399#41—88no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—81no—
416C2×D4—81no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—216no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—48no—
455C22×C4—432no—
456C22×C4—48no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4#12114yes—
471C22×C4#12118yes—
472C22×C4AGL(1,ℤ/8)14yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D414yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4C2511yescharacteristic
486M16—416no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C4⋊C4—216no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4—81no—
518C4⋊C4C22×D414yes—
519C4⋊C4C22×D414yes—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—41no—
540C42—41no—
541C42—42no—
542C42—48no—
543C42—44no—
544C42—44no—
545C42—44no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—28no—
555C23—22no—
556C23—22no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23C2×#12114yes—
571C23C2×#12112yes—
572C23C2×AGL(1,ℤ/8)12yes—
573C23#57512yes—
574C23C23×D411yescharacteristic
575Q8—44no—
576Q8—44no—
577D4—44no—
578D4—44no—
579D4—88no—
580D4—88no—
581D4—88no—
582D4—162no—
583D4—162no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—432no—
615C2×C4—48no—
616C2×C4—48no—
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—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—88no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4C2×#12114yes—
642C2×C4C2×#12112yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C23×D411yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4#58512yes—
648C2×C4#58512yes—
649C8—44no—
650C8—44no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—416no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22#303614yes—
673C22C22×#12111yescharacteristic
674C22#348511yescharacteristic
675C22#348511yescharacteristic
676C22C22×AGL(1,ℤ/8)11yescharacteristic
677C22C2×#57511yescharacteristic
678C22#358611yescharacteristic
679C22#358611yescharacteristic
680C4—22no—
681C4—22no—
682C4—24no—
683C4—22no—
684C4—42no—
685C4—41no—
686C4—41no—
687C4—42no—
688C4—44no—
689C4—44no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4#344912yes—
696C4#349112yes—
697C2—24no—
698C2—44no—
699C2—162no—
700C2#3508911yescharacteristic
701C2C2×#303611yescharacteristic
702C2#6042411yescharacteristic
703C2#6170611yescharacteristic
704C2#6171111yescharacteristic
705C2#6171611yescharacteristic
706C2#6171711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35032C2
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
1. C2×#3092C2semidirect
3. C2×#2344C2semidirect
4. #33111C2semidirect
5. #33108C2semidirect
6. #33106C2semidirect
7. #29760C2semidirect
8. #29756C2semidirect
9. #29754C2semidirect
10. #24589C2semidirect
11. #24588C2semidirect
12. #24489C2semidirect
13. #24488C2semidirect
14. #21934C2semidirect
15. #21769C2semidirect
55. #2085C22semidirect
56. #2084C22semidirect
57. #2083C22semidirect
58. #2068C22semidirect
59. #2067C22semidirect
60. #2066C22semidirect
61. #2065C22semidirect
62. #2043C22semidirect
63. #2042C22semidirect
64. #2041C22semidirect
65. #1983C22semidirect
66. #1982C22semidirect
67. #1981C22semidirect
68. #1949C22semidirect
69. #1948C22semidirect
70. #1947C22semidirect
71. #1928C22semidirect
72. #1927C22semidirect
73. #1926C22semidirect
74. #1925C22semidirect
116. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30922
#350322
C2×#23442
#331112
#331082
#331062
#297602
#297562
#297542
#245892
#245882
#244892
#244882
#219342
#217692