Finite Group Databasebeta

← #6433944

Subgroups — #6433944

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34634C211yesmaximal, characteristic, complemented
2C2×#2624C211yesmaximal, characteristic
3#30373C214yesmaximal, complemented
4#30372C214yesmaximal, complemented
5#30369C214yesmaximal, complemented
6#30368C214yesmaximal, complemented
7#23747C211yesmaximal, characteristic, complemented
8#23731C211yesmaximal, characteristic, complemented
9#23642C212yesmaximal, complemented
10#23554C211yesmaximal, characteristic, complemented
11#23480C212yesmaximal
12#23454C211yesmaximal, characteristic, complemented
13#21159C212yesmaximal, complemented
14#21095C212yesmaximal, complemented
15#20377C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#418C2212yes—
22C2×#418C2212yes—
23C2×#407C2212yes—
24C2×C4×M16C2212yes—
25#2629C2212yescomplemented
26#2629C2212yescomplemented
27#2628C2212yescomplemented
28#2628C2212yescomplemented
29#2627—21no—
30#2627—21no—
31#2626—21no—
32#2626—21no—
33#2624—22no—
34#2624—22no—
35#2624C2214yes—
36#2624C2214yes—
37#2396—22no—
38#2396—22no—
39C2×#399—21no—
40C2×#397—21no—
41C2×#394—21no—
42C2×#394—21no—
43#2296C2212yes—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48#2285C2211yescharacteristic
49#2280C2212yes—
50C2×#380C2211yescharacteristic
51#2101C2214yescomplemented
52#2100C2212yescomplemented
53#2099C2212yescomplemented
54#2094C2212yescomplemented
55#2093C2212yescomplemented
56#2092C2214yescomplemented
57#2077C2214yescomplemented
58#2077C2214yescomplemented
59#2076C2214yescomplemented
60#2075C2214yescomplemented
61#2059C2214yescomplemented
62#2058C2212yescomplemented
63#2057C2212yescomplemented
64#2034C2214yes—
65#2034C2214yes—
66#2033C2214yes—
67#2033C2214yes—
68#2025C2214yescomplemented
69#2023C2212yescomplemented
70#2023C2212yescomplemented
71#1944C2214yescomplemented
72#1943C2214yescomplemented
73#1942C2214yescomplemented
74#1941C2214yescomplemented
75#1936C2214yescomplemented
76#1936C2214yescomplemented
77#1934C2214yescomplemented
78#1934C2214yescomplemented
79#1750C2214yescomplemented
80#1750C2214yescomplemented
81#1736C2212yes—
82#1736C2212yes—
83#1736C2212yes—
84#1736C2212yes—
85#1728C2211yescharacteristic
86#1728C2211yescharacteristic
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C22×M16C2312yes—
91C22×M16C2312yes—
92#533—44no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×#125C2312yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×#121—22no—
99C2×#121—44no—
100C2×#121C2314yes—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×#117—44no—
104C2×#117D412yes—
105C2×#117D412yes—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4D411yescharacteristic
124C22×C22⋊C4D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#428—28no—
128#428—44no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133#418—28no—
134#418—44no—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139#407—28no—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C4×M16—44no—
143C2×#97C2314yes—
144C2×C4×C8C2314yes—
145#400—22no—
146#400—22no—
147#399—24no—
148#399—24no—
149#399—22no—
150#399—22no—
151#398—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#397—24no—
156#397—24no—
157#396—22no—
158#396—22no—
159#396—22no—
160#396—22no—
161#395—22no—
162#395—22no—
163#394—24no—
164#394—24no—
165#394—24no—
166#394—24no—
167#393—22no—
168#393—22no—
169#390—21no—
170#390—21no—
171#390—22no—
172#390C2312yes—
173#390C2312yes—
174#390C2314yes—
175#389—21no—
176#389—21no—
177#389—22no—
178#389C2312yes—
179#389C2312yes—
180#389C2314yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#385—22no—
186#385—22no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382C2312yes—
190#381—22no—
191#380—22no—
192#380—21no—
193#380—21no—
194#380C2312yes—
195#380C2314yes—
196#380C2312yes—
197#380C2312yes—
198#380C2312yes—
199#379—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2312yes—
220#336C2312yes—
221C25—21no—
222C25C2×D411yescharacteristic
223C22×Q8—21no—
224C22×Q8—21no—
225C22×D4—24no—
226C22×D4—21no—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—44no—
232C22×D4—82no—
233C22×D4—82no—
234C22×D4C2412yes—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C2×M16—24no—
251C2×M16—24no—
252C2×M16—24no—
253C2×M16—24no—
254C2×M16—24no—
255C2×M16#4118yes—
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—42no—
259C4⋊Q8—42no—
260#125—42no—
261#125—42no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264#121—44no—
265#121—44no—
266#121—48no—
267C4×Q8—44no—
268C4×Q8—44no—
269#117—24no—
270#117—24no—
271#117—24no—
272#117—24no—
273#117—44no—
274#117—44no—
275#117—42no—
276#117—42no—
277#117—42no—
278#117—42no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
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—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—42no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—82no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4—82no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352#107—28no—
353#107—44no—
354#106—28no—
355#106—44no—
356#102—28no—
357#102—28no—
358#102—44no—
359#102—44no—
360C22⋊C8—28no—
361C22⋊C8—28no—
362#97—44no—
363C4×C8—44no—
364#95—24no—
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#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—41no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24C22×D412yes—
394C24C22×D411yescharacteristic
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—81no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×Q8—81no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—48no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—84no—
412C2×D4—84no—
413C2×D4—816no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
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—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2×#4114yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C42−1+412yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469M16—216no—
470M16—48no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8C2×#4118yes—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4—82no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—84no—
519C22⋊C4—82no—
520C22⋊C4—84no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—84no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—44no—
537C42—44no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—82no—
560C23—82no—
561C23—82no—
562C23—82no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23#53212yes—
571C23#53412yes—
572C23#53612yes—
573C23C23×D411yescharacteristic
574Q8—84no—
575Q8—84no—
576D4—44no—
577D4—44no—
578D4—816no—
579D4—84no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
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—82no—
631C2×C4—82no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4#53212yes—
636C2×C4#53412yes—
637C2×C4#53612yes—
638C2×C4#53918yes—
639C2×C4#54218yes—
640C2×C4C22×#4112yes—
641C2×C4C2×H(5,𝔽2)12yes—
642C2×C4C2×2−1+411yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C8—28no—
645C8—44no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—88no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—168no—
668C22#266314yes—
669C22C2×#53211yescharacteristic
670C22C2×#53411yescharacteristic
671C22C2×#53611yescharacteristic
672C22#358511yescharacteristic
673C22#358511yescharacteristic
674C22#358511yescharacteristic
675C22#358611yescharacteristic
676C4—42no—
677C4—41no—
678C4—42no—
679C4—41no—
680C4—41no—
681C4—42no—
682C4—41no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4#266314yes—
691C4C2×#53912yes—
692C4C2×#54212yes—
693C2—24no—
694C2—82no—
695C2—162no—
696C2C2×#266311yescharacteristic
697C2#6070311yescharacteristic
698C2#6070411yescharacteristic
699C2#6075311yescharacteristic
700C2#6075511yescharacteristic
701C2#6086811yescharacteristic
702C2#6087011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34634C2semidirect
3. #30373C2semidirect
4. #30372C2semidirect
5. #30369C2semidirect
6. #30368C2semidirect
7. #23747C2semidirect
8. #23731C2semidirect
9. #23642C2semidirect
10. #23554C2semidirect
12. #23454C2semidirect
13. #21159C2semidirect
14. #21095C2semidirect
15. #20377C2semidirect
25. #2629C22semidirect
26. #2629C22semidirect
27. #2628C22semidirect
28. #2628C22semidirect
51. #2101C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
54. #2094C22semidirect
55. #2093C22semidirect
56. #2092C22semidirect
57. #2077C22semidirect
58. #2077C22semidirect
59. #2076C22semidirect
60. #2075C22semidirect
61. #2059C22semidirect
62. #2058C22semidirect
63. #2057C22semidirect
68. #2025C22semidirect
69. #2023C22semidirect
70. #2023C22semidirect
71. #1944C22semidirect
72. #1943C22semidirect
73. #1942C22semidirect
74. #1941C22semidirect
75. #1936C22semidirect
76. #1936C22semidirect
77. #1934C22semidirect
78. #1934C22semidirect
79. #1750C22semidirect
80. #1750C22semidirect

Maximal subgroups

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

subgroupindex
#346342
C2×#26242
#303732
#303722
#303692
#303682
#237472
#237312
#236422
#235542
#234802
#234542
#211592
#210952
#203772