Finite Group Databasebeta

← #7506752

Subgroups — #7506752

Order 512.

1844 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35673C211yesmaximal, characteristic, complemented
2#35531C211yesmaximal, characteristic
3#31960C214yesmaximal, complemented
4#31959C214yesmaximal, complemented
5#31948C214yesmaximal, complemented
6#31947C214yesmaximal, complemented
7#23732C211yesmaximal, characteristic, complemented
8#23660C211yesmaximal, characteristic, complemented
9#23628C211yesmaximal, characteristic, complemented
10#23614C211yesmaximal, characteristic, complemented
11#23483C211yesmaximal, characteristic
12#23388C211yesmaximal, characteristic
13C2×#1973C211yesmaximal, characteristic, complemented
14#21047C211yesmaximal, characteristic, complemented
15#20936C211yesmaximal, characteristic, complemented
16#20913C211yesmaximal, characteristic, complemented
17#20847C211yesmaximal, characteristic, complemented
18#20762C211yesmaximal, characteristic, complemented
19#20651C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#535C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×#408C2211yescharacteristic
37C22×#97C2211yescharacteristic
38#2504C2212yes—
39#2501C2212yes—
40#2500C2212yes—
41#2500C2212yes—
42#2498—24no—
43#2448—22no—
44#2448—22no—
45#2446—22no—
46#2445—22no—
47#2423C2214yescomplemented
48#2421C2214yescomplemented
49#2418—24no—
50#2317C2211yescharacteristic
51#2316C2211yescharacteristic
52#2311C2211yescharacteristic
53#2309C2211yescharacteristic
54#2297C2211yescharacteristic
55C2×#391C2211yescharacteristic
56C2×#390C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2097C2212yescomplemented
60#2097C2212yescomplemented
61#2096C2212yescomplemented
62#2095C2212yescomplemented
63#2085C2214yescomplemented
64#2083C2214yescomplemented
65#2073C2214yes—
66#2071C2214yes—
67#2067C2214yes—
68#2065C2214yes—
69#2034C2214yes—
70#2033C2214yes—
71#2000C2214yes—
72#2000C2214yes—
73#1973C2214yes—
74#1973C2214yes—
75#1924C2214yescomplemented
76#1924C2214yescomplemented
77#1880C2214yes—
78#1880C2214yes—
79#1873C2214yescomplemented
80#1872C2214yescomplemented
81#1854C2214yescomplemented
82#1852C2214yescomplemented
83#1836C2212yescomplemented
84#1836C2212yescomplemented
85#1836C2212yescomplemented
86#1836C2212yescomplemented
87#1805C2214yescomplemented
88#1803C2214yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—24no—
102C22×SD16—42no—
103C22×SD16C2314yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#536—24no—
109#536—42no—
110#535—42no—
111#535—42no—
112C2×C4⋊Q8—21no—
113C2×#127—21no—
114C2×#124—21no—
115C2×#124—21no—
116C2×#123—22no—
117C2×#123—24no—
118C2×C22⋊Q8C2314yes—
119C2×#121C2312yes—
120C2×#121C2312yes—
121C2×#120—22no—
122#520—42no—
123#520—42no—
124#519—24no—
125#519—42no—
126C2×C4×Q8—21no—
127C2×C4×Q8—22no—
128C2×C4×Q8C2314yes—
129C2×C4×D4—22no—
130C2×C4×D4—21no—
131C2×#117—22no—
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×C42C2311yescharacteristic
143#428—42no—
144#428—42no—
145C2×#107—42no—
146C2×#107C2312yes—
147C2×#107C2312yes—
148C2×#107C2314yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151#424—42no—
152C2×C4⋊C8C2314yes—
153#419—42no—
154#419—42no—
155#418—42no—
156#418—42no—
157C2×#103—42no—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#102—42no—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167#408—24no—
168#408—42no—
169C2×C22⋊C8C2314yes—
170C2×C22⋊C8C2314yes—
171C2×#97—24no—
172C2×#97C2312yes—
173C2×#97C2312yes—
174#400—24no—
175#399—24no—
176#399—24no—
177#398—24no—
178#397—24no—
179#397—24no—
180#394—24no—
181#394—24no—
182#391—21no—
183#391—22no—
184#391D412yes—
185#391C2312yes—
186#391C2312yes—
187#390—21no—
188#390—22no—
189#390—22no—
190#390D412yes—
191#390C2312yes—
192#390C2312yes—
193#390C2312yes—
194#390C2312yes—
195#389—22no—
196#389—22no—
197#389C2314yes—
198#388—24no—
199#388—24no—
200#388—24no—
201#388—24no—
202#387—24no—
203#387—24no—
204#386—22no—
205#386—24no—
206#386D414yes—
207#385—22no—
208#384C2312yes—
209#384C2312yes—
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#382C2312yes—
213#382C2312yes—
214#382C2311yescharacteristic
215#382C2311yescharacteristic
216#380—22no—
217#380—22no—
218#380—22no—
219#380C2314yes—
220#379—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C22⋊C4C2312yes—
224C4×C22⋊C4C2312yes—
225C2×#95—22no—
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—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8C2×D411yescharacteristic
253C22×Q8#4114yes—
254C22×Q8C2412yes—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—21no—
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×D4C2412yes—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—21no—
271C23×C4C2×D412yes—
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C2×SD16—28no—
275C2×SD16—28no—
276C2×SD16—44no—
277C2×SD16—416no—
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8#4114yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C4⋊Q8—44no—
295#127—44no—
296#124—44no—
297#124—44no—
298#123—48no—
299#123—88no—
300C22⋊Q8—28no—
301C22⋊Q8—44no—
302#121—42no—
303#121—42no—
304#121—42no—
305#121—42no—
306#120—48no—
307C4×Q8—28no—
308C4×Q8—44no—
309C4×Q8—48no—
310C4×Q8—44no—
311C4×D4—48no—
312C4×D4—44no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—48no—
324#117—42no—
325#117—42no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
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⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2412yes—
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—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—84no—
366C2×C22⋊C4C2×D412yes—
367C2×C22⋊C4C2×D414yes—
368C2×C22⋊C4C2×D414yes—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—24no—
372C2×C42—21no—
373C2×C42—22no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42C2×D412yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—44no—
382#107—42no—
383#107—42no—
384#106—42no—
385#106—42no—
386C4⋊C8—44no—
387#103—44no—
388#103—42no—
389#103—42no—
390#103—42no—
391#103—42no—
392#102—44no—
393#102—44no—
394#102—44no—
395C22⋊C8—28no—
396C22⋊C8—44no—
397#97—24no—
398#97—24no—
399#95—24no—
400#95—24no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95C2×D414yes—
409#95C2×D414yes—
410C24—22no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—82no—
415C24—82no—
416C24C22×D412yes—
417C24C22×D411yescharacteristic
418#41—44no—
419#41—44no—
420#41—44no—
421#41—44no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—28no—
428C2×Q8—28no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—42no—
434C2×Q8—44no—
435C2×Q8—42no—
436C2×Q8—44no—
437C2×Q8C2×#4114yes—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
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—44no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—84no—
460C2×D4—84no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—82no—
486C22×C4—84no—
487C22×C4—82no—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D414yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C42−1+412yes—
499C22×C4C2511yescharacteristic
500SD16—416no—
501M16—44no—
502M16—44no—
503C2×C8—28no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—44no—
516C2×C8C2×#4114yes—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—84no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—84no—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—84no—
558C22⋊C4—84no—
559C22⋊C4—84no—
560C22⋊C4—84no—
561C42—24no—
562C42—21no—
563C42—21no—
564C42—22no—
565C42—22no—
566C42—21no—
567C42—21no—
568C42—48no—
569C42—44no—
570C42—42no—
571C42—42no—
572C42—42no—
573C42—42no—
574C23—22no—
575C23—22no—
576C23—24no—
577C23—24no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—44no—
582C23—44no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—88no—
587C23—84no—
588C23—164no—
589C23—162no—
590C23—162no—
591C23#53512yes—
592C23#54612yes—
593C23#54712yes—
594C23C23×D411yescharacteristic
595Q8—24no—
596Q8—24no—
597Q8—42no—
598Q8—48no—
599Q8—44no—
600Q8—42no—
601D4—48no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—44no—
606D4—42no—
607D4—84no—
608D4—164no—
609D4—164no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—48no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—84no—
649C2×C4—162no—
650C2×C4—162no—
651C2×C4—164no—
652C2×C4#53512yes—
653C2×C4#53914yes—
654C2×C4#54014yes—
655C2×C4#54612yes—
656C2×C4#54712yes—
657C2×C4#55114yes—
658C2×C4#56214yes—
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×2−1+411yescharacteristic
665C8—24no—
666C8—42no—
667C8—42no—
668C8—42no—
669C22—24no—
670C22—24no—
671C22—22no—
672C22—22no—
673C22—44no—
674C22—44no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—84no—
679C22—84no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—168no—
684C22#270614yes—
685C22C2×#53511yescharacteristic
686C22C2×#54611yescharacteristic
687C22C2×#54711yescharacteristic
688C22#358512yes—
689C22#358612yes—
690C4—44no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—82no—
696C4—82no—
697C4—81no—
698C4—81no—
699C4—82no—
700C4—82no—
701C4—81no—
702C4—81no—
703C4#270614yes—
704C4C2×#53911yescharacteristic
705C4C2×#54011yescharacteristic
706C4C2×#55111yescharacteristic
707C4C2×#56211yescharacteristic
708C2—24no—
709C2—82no—
710C2—162no—
711C2C2×#270611yescharacteristic
712C2#6082611yescharacteristic
713C2#6082711yescharacteristic
714C2#6122012yes—
715C2#6126812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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. #35673C2semidirect
3. #31960C2semidirect
4. #31959C2semidirect
5. #31948C2semidirect
6. #31947C2semidirect
7. #23732C2semidirect
8. #23660C2semidirect
9. #23628C2semidirect
10. #23614C2semidirect
13. C2×#1973C2semidirect
14. #21047C2semidirect
15. #20936C2semidirect
16. #20913C2semidirect
17. #20847C2semidirect
18. #20762C2semidirect
19. #20651C2semidirect
47. #2423C22semidirect
48. #2421C22semidirect
59. #2097C22semidirect
60. #2097C22semidirect
61. #2096C22semidirect
62. #2095C22semidirect
63. #2085C22semidirect
64. #2083C22semidirect
75. #1924C22semidirect
76. #1924C22semidirect
79. #1873C22semidirect
80. #1872C22semidirect
81. #1854C22semidirect
82. #1852C22semidirect
83. #1836C22semidirect
84. #1836C22semidirect
85. #1836C22semidirect
86. #1836C22semidirect
87. #1805C22semidirect
88. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#356732
#355312
#319602
#319592
#319482
#319472
#237322
#236602
#236282
#236142
#234832
#233882
C2×#19732
#210472
#209362
#209132
#208472
#207622
#206512