Finite Group Databasebeta

← #6434218

Subgroups — #6434218

Order 512.

1946 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2644C211yesmaximal, characteristic, complemented
2#34070C211yesmaximal, characteristic, complemented
3#30142C214yesmaximal, complemented
4#30141C214yesmaximal, complemented
5#30139C214yesmaximal, complemented
6#30137C214yesmaximal, complemented
7#23747C211yesmaximal, characteristic, complemented
8#23731C211yesmaximal, characteristic, complemented
9#23617C212yesmaximal, complemented
10#23546C212yesmaximal, complemented
11#23508C211yesmaximal, characteristic, complemented
12#23428C211yesmaximal, characteristic
13#20928C212yesmaximal, complemented
14#20875C212yesmaximal, complemented
15#20753C211yesmaximal, characteristic, complemented
16C22×AGL(1,ℤ/8)C2212yes—
17C2×#532C2211yescharacteristic
18C2×C4×#41C2212yes—
19C2×#428C2212yes—
20C2×#423C2212yes—
21C2×#417C2212yes—
22C2×#417C2212yes—
23C2×#417C2212yes—
24C2×#418C2212yes—
25#2644—22no—
26#2644—22no—
27#2644C2214yes—
28#2644C2214yes—
29C2×#397—21no—
30C2×#397—21no—
31C2×#397—21no—
32C2×#396—21no—
33#2328C2212yescomplemented
34#2327C2212yescomplemented
35#2326C2212yescomplemented
36#2326C2212yescomplemented
37#2325—21no—
38#2325—21no—
39#2325—21no—
40#2325—21no—
41#2297C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48#2290—22no—
49#2290—22no—
50#2285C2211yescharacteristic
51C2×#376C2211yescharacteristic
52#2101C2214yescomplemented
53#2100C2212yescomplemented
54#2099C2212yescomplemented
55#2094C2212yes—
56#2093C2212yes—
57#2092C2214yes—
58#2068C2214yescomplemented
59#2067C2214yescomplemented
60#2066C2214yescomplemented
61#2065C2214yescomplemented
62#2055C2214yes—
63#2055C2214yes—
64#2054C2214yes—
65#2054C2214yes—
66#2043C2212yescomplemented
67#2042C2214yescomplemented
68#2041C2212yescomplemented
69#2013C2212yes—
70#2012C2214yes—
71#2011C2212yes—
72#1880C2214yes—
73#1880C2214yes—
74#1879C2214yes—
75#1879C2214yes—
76#1865C2214yescomplemented
77#1864C2214yescomplemented
78#1863C2214yescomplemented
79#1862C2214yescomplemented
80#1832C2214yescomplemented
81#1832C2214yescomplemented
82#1736C2212yes—
83#1736C2212yes—
84#1728C2211yescharacteristic
85#1728C2212yes—
86#1728C2211yescharacteristic
87C22×#41C2312yes—
88C23×D4C2311yescharacteristic
89C2×AGL(1,ℤ/8)—28no—
90C2×AGL(1,ℤ/8)—44no—
91C2×AGL(1,ℤ/8)—44no—
92C22×SD16C2314yes—
93C22×D8C2314yes—
94C22×M16C2312yes—
95C22×M16C2312yes—
96#532—28no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127C2311yescharacteristic
99C2×#124C2312yes—
100C2×#121—21no—
101C2×#121—21no—
102C4×#41—44no—
103C4×#41—44no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C2×#117C2312yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C42—22no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#428—44no—
131C2×#107C2314yes—
132C2×#106C2314yes—
133#423—44no—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136#418—44no—
137#417—28no—
138#417—44no—
139#417—44no—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148#397—24no—
149#397—24no—
150#397—24no—
151#397—24no—
152#397—24no—
153#397—24no—
154#396—24no—
155#396—24no—
156#391C2312yes—
157#391C2312yes—
158#390—22no—
159#390—21no—
160#390—21no—
161#390—22no—
162#390C2314yes—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#390C2314yes—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389C2314yes—
173#389C2312yes—
174#389C2312yes—
175#389C2314yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#385—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#382C2312yes—
197#380—21no—
198#380—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203#376D412yes—
204#376D412yes—
205#376C2312yes—
206#376C2312yes—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2312yes—
221#336C2312yes—
222#336C2314yes—
223#336C2314yes—
224C25—21no—
225C2×#41—24no—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—24no—
229C2×#41—24no—
230C2×#41#4118yes—
231C22×Q8C2412yes—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4—24no—
237C22×D4—22no—
238C22×D4—24no—
239C22×D4—22no—
240C22×D4C2412yes—
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255AGL(1,ℤ/8)—432no—
256C2×SD16—28no—
257C2×SD16—44no—
258C2×SD16—44no—
259C2×D8—28no—
260C2×D8—44no—
261C2×D8—44no—
262C2×M16—24no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—24no—
271#127—24no—
272#124—24no—
273#124—24no—
274#121—28no—
275#121—28no—
276C4×Q8—42no—
277C4×Q8—42no—
278C4×Q8—42no—
279C4×Q8—42no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—42no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—42no—
286C4×D4—48no—
287C4×D4—48no—
288#117—24no—
289#117—24no—
290#117—48no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4—82no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4—82no—
340C2×C22⋊C4—82no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—48no—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357#107—44no—
358#106—44no—
359C4⋊C8—44no—
360C4⋊C8—44no—
361#103—28no—
362#103—44no—
363#103—44no—
364#102—28no—
365#102—44no—
366#102—44no—
367#102—44no—
368#102—44no—
369#95—24no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—22no—
388C24—22no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C22×D411yescharacteristic
397#41—216no—
398#41—216no—
399#41—48no—
400#41—48no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8C2×#4114yes—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—81no—
427C2×D4—81no—
428C2×D4—81no—
429C2×D4—81no—
430C2×D4C2×#4114yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—28no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—41no—
445C22×C4—41no—
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—41no—
455C22×C4—41no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C2×#4114yes—
483C22×C4H(5,𝔽2)14yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+412yes—
486C22×C4C2511yescharacteristic
487SD16—416no—
488D8—416no—
489M16—48no—
490M16—48no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4—84no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4—82no—
520C22⋊C4—28no—
521C22⋊C4—28no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C42—22no—
538C42—22no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—44no—
548C42—44no—
549C42—44no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—24no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23#52512yes—
576C23#53512yes—
577C23#53712yes—
578C23C23×D411yescharacteristic
579Q8—24no—
580Q8—24no—
581Q8—42no—
582Q8—42no—
583D4—24no—
584D4—24no—
585D4—416no—
586D4—42no—
587D4—44no—
588D4—42no—
589D4—44no—
590D4—84no—
591D4—84no—
592D4—84no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—416no—
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—82no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—88no—
643C2×C4—88no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4#52512yes—
648C2×C4#53512yes—
649C2×C4#53712yes—
650C2×C4#53818yes—
651C2×C4#53918yes—
652C2×C4C22×#4112yes—
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×H(5,𝔽2)12yes—
655C2×C4C2×2−1+411yescharacteristic
656C8—44no—
657C8—44no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—168no—
679C22#263614yes—
680C22C2×#52511yescharacteristic
681C22C2×#53511yescharacteristic
682C22C2×#53711yescharacteristic
683C22#358511yescharacteristic
684C22#358511yescharacteristic
685C22#358511yescharacteristic
686C22#358611yescharacteristic
687C4—44no—
688C4—42no—
689C4—42no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—42no—
695C4—42no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4#263614yes—
702C4C2×#53812yes—
703C4C2×#53912yes—
704C2—24no—
705C2—84no—
706C2—162no—
707C2C2×#263611yescharacteristic
708C2#6056711yescharacteristic
709C2#6056811yescharacteristic
710C2#6080811yescharacteristic
711C2#6081011yescharacteristic
712C2#6091911yescharacteristic
713C2#6092011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23428C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2644C2semidirect
2. #34070C2semidirect
3. #30142C2semidirect
4. #30141C2semidirect
5. #30139C2semidirect
6. #30137C2semidirect
7. #23747C2semidirect
8. #23731C2semidirect
9. #23617C2semidirect
10. #23546C2semidirect
11. #23508C2semidirect
13. #20928C2semidirect
14. #20875C2semidirect
15. #20753C2semidirect
33. #2328C22semidirect
34. #2327C22semidirect
35. #2326C22semidirect
36. #2326C22semidirect
52. #2101C22semidirect
53. #2100C22semidirect
54. #2099C22semidirect
58. #2068C22semidirect
59. #2067C22semidirect
60. #2066C22semidirect
61. #2065C22semidirect
66. #2043C22semidirect
67. #2042C22semidirect
68. #2041C22semidirect
76. #1865C22semidirect
77. #1864C22semidirect
78. #1863C22semidirect
79. #1862C22semidirect
80. #1832C22semidirect
81. #1832C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26442
#340702
#301422
#301412
#301392
#301372
#237472
#237312
#236172
#235462
#235082
#234282
#209282
#208752
#207532