Finite Group Databasebeta

← #6433532

Subgroups — #6433532

Order 512.

1840 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic, complemented
2#34269C211yesmaximal, characteristic
3#30180C214yesmaximal, complemented
4#30178C214yesmaximal, complemented
5#30173C212yesmaximal, complemented
6#30172C212yesmaximal, complemented
7#30166C212yesmaximal, complemented
8#30165C212yesmaximal, complemented
9#23658C211yesmaximal, characteristic, complemented
10#23641C212yesmaximal, complemented
11C2×#2071C211yesmaximal, characteristic, complemented
12#23551C211yesmaximal, characteristic, complemented
13#23483C212yesmaximal
14#23454C211yesmaximal, characteristic, complemented
15#21036C212yesmaximal, complemented
16#20879C212yesmaximal, complemented
17#20684C211yesmaximal, characteristic, complemented
18C2×#535C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×#518C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2212yes—
23C2×#427C2212yes—
24C2×#418C2212yes—
25C22×#102C2212yes—
26C2×#408C2212yes—
27#2857C2212yescomplemented
28#2857C2212yescomplemented
29#2629C2214yescomplemented
30#2582—21no—
31#2582—21no—
32#2538—21no—
33#2538—21no—
34#2417—22no—
35#2417—22no—
36#2367—21no—
37#2367—21no—
38#2366—21no—
39#2366—21no—
40#2358—22no—
41#2358—22no—
42#2324C2212yes—
43#2324C2212yes—
44#2316C2212yes—
45#2311C2211yescharacteristic
46#2311C2212yes—
47C4×#121C2212yes—
48C4×#121C2212yes—
49#2296C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2274C2211yescharacteristic
53#2084C2212yescomplemented
54#2084C2212yescomplemented
55#2083C2212yescomplemented
56#2083C2212yescomplemented
57#2077C2214yescomplemented
58#2077C2214yescomplemented
59#2076C2214yescomplemented
60#2075C2214yescomplemented
61#2071C2212yes—
62#2071C2212yes—
63#2071C2214yes—
64#2059C2212yescomplemented
65#2059C2212yescomplemented
66#2058C2211yescharacteristic, complemented
67#2058C2211yescharacteristic, complemented
68#2057C2211yescharacteristic, complemented
69#2057C2211yescharacteristic, complemented
70#2034C2214yes—
71#2034C2214yes—
72#2033C2214yes—
73#2033C2214yes—
74#2025C2214yes—
75#2023C2212yes—
76#2023C2212yes—
77#1921C2214yescomplemented
78#1920C2214yescomplemented
79#1919C2214yescomplemented
80#1918C2214yescomplemented
81#1864C2214yes—
82#1864C2214yes—
83#1862C2214yes—
84#1862C2214yes—
85#1818C2214yescomplemented
86#1817C2212yescomplemented
87#1817C2212yescomplemented
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×D4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#535—24no—
100#535—24no—
101#533—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127—21no—
105C2×#125C2312yes—
106C2×#124—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×#121C2312yes—
112C2×#121C2312yes—
113C2×#120—22no—
114#518—44no—
115C4×#41—44no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4D412yes—
125C2×C4×D4D412yes—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×#117C2312yes—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2312yes—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—22no—
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144#428—28no—
145#427—44no—
146C2×#107C2314yes—
147C2×#107C2312yes—
148C2×#107C2312yes—
149C2×#106C2314yes—
150C2×#106C2312yes—
151C2×#106C2312yes—
152#418—44no—
153C2×#102—28no—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158#408—44no—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161#398—24no—
162#398—24no—
163#396—24no—
164#396—24no—
165#391—22no—
166#391—21no—
167#391—22no—
168#391—22no—
169#390—21no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390—21no—
174#390—21no—
175#390C2312yes—
176#390C2312yes—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389C2311yescharacteristic
182#389C2311yescharacteristic
183#389C2311yescharacteristic
184#389C2311yescharacteristic
185#389C2314yes—
186#389C2314yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#386—22no—
192#386—24no—
193#386—22no—
194#386—22no—
195#386—22no—
196#386—22no—
197#385—21no—
198#385—21no—
199#385—22no—
200#385—22no—
201#384—21no—
202#384—21no—
203#384C2312yes—
204#384C2312yes—
205#380—22no—
206#380—22no—
207#380—24no—
208#380C2312yes—
209#380C2312yes—
210#380C2314yes—
211#380C2314yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C25—21no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—24no—
242C2×#41—24no—
243C22×Q8—21no—
244C22×Q8—21no—
245C22×Q8C2×D411yescharacteristic
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—24no—
251C22×D4—24no—
252C22×D4C2×D412yes—
253C22×D4C2×D411yescharacteristic
254C22×D4#4114yes—
255C22×D4C2412yes—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2412yes—
265C23×C4C2411yescharacteristic
266C2×M16—24no—
267C2×M16—24no—
268C2×M16—24no—
269C22×C8—24no—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2414yes—
275C4⋊Q8—42no—
276C4⋊Q8—42no—
277#127—28no—
278#125—42no—
279#125—42no—
280#124—28no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—84no—
284C22⋊Q8—84no—
285#121—24no—
286#121—24no—
287#121—24no—
288#121—24no—
289#120—216no—
290C4×Q8—42no—
291C4×Q8—42no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×D4—42no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—84no—
303C4×D4—84no—
304#117—22no—
305#117—24no—
306#117—24no—
307#117—24no—
308#117—24no—
309#117—44no—
310#117#4114yes—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4—82no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—48no—
360C2×C22⋊C4—82no—
361C2×C22⋊C4—82no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C42—21no—
366C2×C42—24no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—44no—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D412yes—
375C2×C42C2×D412yes—
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—28no—
380#107—42no—
381#107—42no—
382#106—28no—
383#106—42no—
384#106—42no—
385#102—28no—
386#102—28no—
387#102—44no—
388#102—44no—
389C22⋊C8—44no—
390C22⋊C8—44no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—21no—
398#95—21no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403C24—22no—
404C24—24no—
405C24—42no—
406C24—42no—
407C24—81no—
408C24—81no—
409C24—81no—
410C24—81no—
411C24C22×D412yes—
412C24C22×D411yescharacteristic
413#41—48no—
414#41—48no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—28no—
428C2×D4—28no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—216no—
434C2×D4—22no—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—48no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4C2×#4114yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)14yes—
492C22×C42−1+414yes—
493C22×C4C2511yescharacteristic
494M16—48no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—48no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—81no—
531C4⋊C4—81no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4C2×#4112yes—
537C4⋊C4C2×#4112yes—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—24no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—24no—
562C42—42no—
563C42—42no—
564C42—42no—
565C42—44no—
566C42—82no—
567C42—82no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—48no—
587C23—84no—
588C23—84no—
589C23—162no—
590C23—162no—
591C23—162no—
592C23—162no—
593C23#52512yes—
594C23#53412yes—
595C23#53612yes—
596C23C23×D411yescharacteristic
597Q8—44no—
598Q8—84no—
599Q8—84no—
600D4—28no—
601D4—48no—
602D4—48no—
603D4—44no—
604D4—44no—
605D4—88no—
606D4—88no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—164no—
657C2×C4—164no—
658C2×C4—164no—
659C2×C4—162no—
660C2×C4#52512yes—
661C2×C4#53412yes—
662C2×C4#53612yes—
663C2×C4#53918yes—
664C2×C4#54218yes—
665C2×C4#57612yes—
666C2×C4#57712yes—
667C2×C4C22×#4111yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C2×H(5,𝔽2)12yes—
670C2×C4C2×2−1+412yes—
671C8—42no—
672C8—42no—
673C8—44no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—42no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—44no—
686C22—42no—
687C22—44no—
688C22—42no—
689C22—42no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—168no—
695C22#263814yes—
696C22C2×#52511yescharacteristic
697C22C2×#53411yescharacteristic
698C22C2×#53611yescharacteristic
699C22C2×#57611yescharacteristic
700C22C2×#57711yescharacteristic
701C22#358511yescharacteristic
702C22#358611yescharacteristic
703C4—22no—
704C4—22no—
705C4—42no—
706C4—41no—
707C4—42no—
708C4—41no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—81no—
713C4—82no—
714C4—81no—
715C4—82no—
716C4—84no—
717C4—82no—
718C4#263814yes—
719C4C2×#53912yes—
720C4C2×#54212yes—
721C2—24no—
722C2—82no—
723C2—84no—
724C2C2×#263811yescharacteristic
725C2#6059011yescharacteristic
726C2#6059111yescharacteristic
727C2#6076211yescharacteristic
728C2#6076511yescharacteristic
729C2#6087711yescharacteristic
730C2#6088511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35280C2semidirect
3. #30180C2semidirect
4. #30178C2semidirect
5. #30173C2semidirect
6. #30172C2semidirect
7. #30166C2semidirect
8. #30165C2semidirect
9. #23658C2semidirect
10. #23641C2semidirect
11. C2×#2071C2semidirect
12. #23551C2semidirect
14. #23454C2semidirect
15. #21036C2semidirect
16. #20879C2semidirect
17. #20684C2semidirect
27. #2857C22semidirect
28. #2857C22semidirect
29. #2629C22semidirect
53. #2084C22semidirect
54. #2084C22semidirect
55. #2083C22semidirect
56. #2083C22semidirect
57. #2077C22semidirect
58. #2077C22semidirect
59. #2076C22semidirect
60. #2075C22semidirect
64. #2059C22semidirect
65. #2059C22semidirect
66. #2058C22semidirect
67. #2058C22semidirect
68. #2057C22semidirect
69. #2057C22semidirect
77. #1921C22semidirect
78. #1920C22semidirect
79. #1919C22semidirect
80. #1918C22semidirect
85. #1818C22semidirect
86. #1817C22semidirect
87. #1817C22semidirect

Maximal subgroups

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

subgroupindex
#352802
#342692
#301802
#301782
#301732
#301722
#301662
#301652
#236582
#236412
C2×#20712
#235512
#234832
#234542
#210362
#208792
#206842