Finite Group Databasebeta

← #7506750

Subgroups — #7506750

Order 512.

1956 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35541C211yesmaximal, characteristic, complemented
2#35531C211yesmaximal, characteristic, complemented
3#31961C214yesmaximal, complemented
4#31958C214yesmaximal, complemented
5#31949C214yesmaximal, complemented
6#31946C214yesmaximal, complemented
7#23731C211yesmaximal, characteristic, complemented
8#23661C211yesmaximal, characteristic, complemented
9#23628C211yesmaximal, characteristic, complemented
10#23614C211yesmaximal, characteristic, complemented
11#23483C211yesmaximal, characteristic, complemented
12#23387C211yesmaximal, characteristic
13C2×#1972C211yesmaximal, characteristic, complemented
14#21047C211yesmaximal, characteristic, complemented
15#20934C211yesmaximal, characteristic, complemented
16#20914C211yesmaximal, characteristic, complemented
17#20848C211yesmaximal, characteristic, complemented
18#20762C211yesmaximal, characteristic, complemented
19#20650C211yesmaximal, characteristic, complemented
20C23×D8C2211yescharacteristic
21C2×#535C2211yescharacteristic
22C2×#535C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×#408C2211yescharacteristic
37C22×#97C2211yescharacteristic
38#2504C2212yescomplemented
39#2501C2212yescomplemented
40#2500C2212yescomplemented
41#2500C2212yescomplemented
42#2448—22no—
43#2447—22no—
44#2445—22no—
45#2445—22no—
46#2422C2214yescomplemented
47#2420C2214yescomplemented
48#2418—24no—
49#2412—24no—
50#2317C2211yescharacteristic
51#2316C2211yescharacteristic
52#2311C2211yescharacteristic
53#2309C2211yescharacteristic
54#2297C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#390C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2094C2212yescomplemented
60#2093C2212yescomplemented
61#2092C2212yescomplemented
62#2092C2212yescomplemented
63#2085C2214yescomplemented
64#2084C2214yescomplemented
65#2073C2214yescomplemented
66#2071C2214yescomplemented
67#2067C2214yescomplemented
68#2065C2214yescomplemented
69#2034C2214yescomplemented
70#2033C2214yescomplemented
71#1999C2214yes—
72#1999C2214yes—
73#1972C2214yes—
74#1972C2214yes—
75#1924C2214yescomplemented
76#1924C2214yescomplemented
77#1879C2214yes—
78#1879C2214yes—
79#1875C2214yescomplemented
80#1874C2214yescomplemented
81#1854C2214yescomplemented
82#1853C2214yescomplemented
83#1836C2212yescomplemented
84#1836C2212yescomplemented
85#1836C2212yescomplemented
86#1836C2212yescomplemented
87#1804C2214yescomplemented
88#1802C2214yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×D8—24no—
101C22×D8—24no—
102C22×D8—42no—
103C22×D8C2314yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#535—24no—
109#535—42no—
110#535—42no—
111#535—42no—
112C2×#127—21no—
113C2×#127—21no—
114C2×#124—21no—
115C2×#124—21no—
116C2×#123—24no—
117C2×#123—24no—
118C2×#121C2314yes—
119C2×#121C2312yes—
120C2×#121C2312yes—
121C2×#120—22no—
122C2×#120—22no—
123#520—42no—
124#520—42no—
125#518—24no—
126#518—42no—
127C2×C4×Q8—21no—
128C2×C4×D4—22no—
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131C2×C4×D4C2314yes—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2312yes—
137C2×#117C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C22⋊C4—22no—
142C22×C22⋊C4—22no—
143C22×C22⋊C4—22no—
144C22×C42C2311yescharacteristic
145#428—42no—
146#428—42no—
147C2×#107C2312yes—
148C2×#107C2312yes—
149C2×#106—42no—
150C2×#106C2312yes—
151C2×#106C2312yes—
152C2×#106C2314yes—
153#424—42no—
154C2×C4⋊C8C2314yes—
155#418—42no—
156#418—42no—
157#418—42no—
158#418—42no—
159C2×#102—42no—
160C2×#102—42no—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169#408—24no—
170#408—42no—
171C2×C22⋊C8C2314yes—
172C2×C22⋊C8C2314yes—
173C2×#97—24no—
174C2×#97C2312yes—
175C2×#97C2312yes—
176#399—24no—
177#397—24no—
178#397—24no—
179#397—24no—
180#394—24no—
181#394—24no—
182#394—24no—
183#394—24no—
184#391—21no—
185#391D412yes—
186#390—21no—
187#390—22no—
188#390—22no—
189#390—22no—
190#390D412yes—
191#390C2312yes—
192#390C2312yes—
193#390C2312yes—
194#390C2312yes—
195#390C2312yes—
196#390C2312yes—
197#389—22no—
198#389—22no—
199#389C2314yes—
200#388—24no—
201#388—24no—
202#386—22no—
203#386—24no—
204#386—24no—
205#386D414yes—
206#385—22no—
207#385—24no—
208#384C2312yes—
209#384C2312yes—
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#382C2312yes—
213#382C2312yes—
214#382C2311yescharacteristic
215#382C2311yescharacteristic
216#381—24no—
217#380—22no—
218#380—22no—
219#380—22no—
220#380C2314yes—
221#379—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C22⋊C4C2312yes—
225C4×C22⋊C4C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239C25—21no—
240C25—21no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C22×Q8C2×D411yescharacteristic
248C22×D4—22no—
249C22×D4—24no—
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4C2×D412yes—
265C22×D4C2×D411yescharacteristic
266C22×D4#4114yes—
267C22×D4C2412yes—
268C22×D4C2412yes—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—22no—
272C23×C4—24no—
273C23×C4C2×D412yes—
274C23×C4C2412yes—
275C23×C4C2411yescharacteristic
276C2×D8—28no—
277C2×D8—28no—
278C2×D8—44no—
279C2×D8—416no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8#4114yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296#127—44no—
297#127—44no—
298#124—44no—
299#124—44no—
300#123—88no—
301#123—88no—
302#121—28no—
303#121—44no—
304#121—42no—
305#121—42no—
306#121—42no—
307#121—42no—
308#120—48no—
309#120—48no—
310C4×Q8—44no—
311C4×D4—28no—
312C4×D4—44no—
313C4×D4—48no—
314C4×D4—48no—
315C4×D4—44no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—42no—
327#117—42no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—42no—
342C2×C4⋊C4—42no—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2411yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—48no—
369C2×C22⋊C4—84no—
370C2×C22⋊C4—84no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D414yes—
373C2×C22⋊C4C2×D414yes—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42C2×D412yes—
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—42no—
386#107—42no—
387#106—44no—
388#106—42no—
389#106—42no—
390C4⋊C8—44no—
391#102—44no—
392#102—44no—
393#102—42no—
394#102—42no—
395#102—44no—
396#102—44no—
397#102—42no—
398#102—42no—
399C22⋊C8—28no—
400C22⋊C8—44no—
401#97—24no—
402#97—24no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95C2×D414yes—
411#95C2×D414yes—
412C24—22no—
413C24—24no—
414C24—22no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—82no—
421C24—82no—
422C24—82no—
423C24—82no—
424C24C22×D412yes—
425C24C22×D411yescharacteristic
426#41—44no—
427#41—44no—
428#41—44no—
429#41—44no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—22no—
437C2×D4—28no—
438C2×D4—24no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—28no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—48no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—84no—
468C2×D4—84no—
469C2×D4—84no—
470C2×D4—84no—
471C2×D4C2×#4114yes—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—48no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—84no—
494C22×C4—84no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—84no—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D414yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C4H(5,𝔽2)12yes—
507C22×C4H(5,𝔽2)12yes—
508C22×C42−1+412yes—
509C22×C4C2511yescharacteristic
510D8—416no—
511M16—44no—
512M16—44no—
513C2×C8—28no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—44no—
526C2×C8C2×#4114yes—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—84no—
552C4⋊C4—84no—
553C22⋊C4—24no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—48no—
557C22⋊C4—48no—
558C22⋊C4—48no—
559C22⋊C4—48no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C22⋊C4—84no—
568C22⋊C4—84no—
569C22⋊C4—84no—
570C22⋊C4—84no—
571C22⋊C4—84no—
572C42—24no—
573C42—21no—
574C42—21no—
575C42—22no—
576C42—22no—
577C42—21no—
578C42—21no—
579C42—44no—
580C42—42no—
581C42—42no—
582C42—42no—
583C42—42no—
584C23—22no—
585C23—22no—
586C23—24no—
587C23—24no—
588C23—22no—
589C23—24no—
590C23—22no—
591C23—22no—
592C23—44no—
593C23—44no—
594C23—42no—
595C23—42no—
596C23—48no—
597C23—44no—
598C23—44no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—88no—
603C23—88no—
604C23—84no—
605C23—164no—
606C23—164no—
607C23—162no—
608C23—162no—
609C23—162no—
610C23—162no—
611C23#53512yes—
612C23#54612yes—
613C23#54712yes—
614C23C23×D411yescharacteristic
615Q8—44no—
616D4—24no—
617D4—24no—
618D4—48no—
619D4—42no—
620D4—42no—
621D4—42no—
622D4—42no—
623D4—48no—
624D4—44no—
625D4—42no—
626D4—42no—
627D4—84no—
628D4—168no—
629D4—164no—
630D4—164no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—84no—
666C2×C4—168no—
667C2×C4—162no—
668C2×C4—162no—
669C2×C4—164no—
670C2×C4#53512yes—
671C2×C4#53914yes—
672C2×C4#54014yes—
673C2×C4#54612yes—
674C2×C4#54712yes—
675C2×C4#55114yes—
676C2×C4#56214yes—
677C2×C4C22×#4111yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C2×C4C2×H(5,𝔽2)11yescharacteristic
680C2×C4C2×H(5,𝔽2)11yescharacteristic
681C2×C4C2×H(5,𝔽2)11yescharacteristic
682C2×C4C2×2−1+411yescharacteristic
683C8—24no—
684C8—42no—
685C8—42no—
686C8—42no—
687C22—24no—
688C22—24no—
689C22—22no—
690C22—22no—
691C22—48no—
692C22—44no—
693C22—44no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—44no—
698C22—84no—
699C22—82no—
700C22—82no—
701C22—84no—
702C22—84no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—168no—
707C22—168no—
708C22#270614yes—
709C22C2×#53511yescharacteristic
710C22C2×#54611yescharacteristic
711C22C2×#54711yescharacteristic
712C22#358512yes—
713C22#358512yes—
714C4—44no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—82no—
719C4—82no—
720C4—81no—
721C4—82no—
722C4—81no—
723C4—82no—
724C4—81no—
725C4—81no—
726C4#270614yes—
727C4C2×#53911yescharacteristic
728C4C2×#54011yescharacteristic
729C4C2×#55111yescharacteristic
730C4C2×#56211yescharacteristic
731C2—24no—
732C2—82no—
733C2—84no—
734C2—162no—
735C2C2×#270611yescharacteristic
736C2#6082511yescharacteristic
737C2#6082711yescharacteristic
738C2#6121812yes—
739C2#6126712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23387C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35541C2semidirect
2. #35531C2semidirect
3. #31961C2semidirect
4. #31958C2semidirect
5. #31949C2semidirect
6. #31946C2semidirect
7. #23731C2semidirect
8. #23661C2semidirect
9. #23628C2semidirect
10. #23614C2semidirect
11. #23483C2semidirect
13. C2×#1972C2semidirect
14. #21047C2semidirect
15. #20934C2semidirect
16. #20914C2semidirect
17. #20848C2semidirect
18. #20762C2semidirect
19. #20650C2semidirect
38. #2504C22semidirect
39. #2501C22semidirect
40. #2500C22semidirect
41. #2500C22semidirect
46. #2422C22semidirect
47. #2420C22semidirect
59. #2094C22semidirect
60. #2093C22semidirect
61. #2092C22semidirect
62. #2092C22semidirect
63. #2085C22semidirect
64. #2084C22semidirect
65. #2073C22semidirect
66. #2071C22semidirect
67. #2067C22semidirect
68. #2065C22semidirect
69. #2034C22semidirect
70. #2033C22semidirect
75. #1924C22semidirect
76. #1924C22semidirect
79. #1875C22semidirect
80. #1874C22semidirect
81. #1854C22semidirect
82. #1853C22semidirect
83. #1836C22semidirect
84. #1836C22semidirect
85. #1836C22semidirect
86. #1836C22semidirect
87. #1804C22semidirect
88. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#355412
#355312
#319612
#319582
#319492
#319462
#237312
#236612
#236282
#236142
#234832
#233872
C2×#19722
#210472
#209342
#209142
#208482
#207622
#206502