Finite Group Databasebeta

← #7510499

Subgroups — #7510499

Order 512.

1868 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35515C211yesmaximal, characteristic, complemented
2#35338C211yesmaximal, characteristic, complemented
3#32620C214yesmaximal, complemented
4#32614C214yesmaximal, complemented
5#32605C214yesmaximal, complemented
6#32599C214yesmaximal, complemented
7#23736C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic, complemented
9#23614C211yesmaximal, characteristic, complemented
10#23541C211yesmaximal, characteristic, complemented
11#23506C211yesmaximal, characteristic, complemented
12C2×#2012C211yesmaximal, characteristic
13#21304C211yesmaximal, characteristic, complemented
14#21233C211yesmaximal, characteristic
15#21088C211yesmaximal, characteristic, complemented
16#21060C211yesmaximal, characteristic, complemented
17#21032C211yesmaximal, characteristic, complemented
18#20934C211yesmaximal, characteristic
19#20933C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C23×D8C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2861—22no—
38#2640—22no—
39#2592C2214yescomplemented
40#2592C2214yescomplemented
41#2580C2214yescomplemented
42#2561—24no—
43#2538C2214yescomplemented
44#2454—22no—
45#2452—22no—
46#2437—24no—
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2310C2211yescharacteristic
50#2310C2211yescharacteristic
51#2309C2211yescharacteristic
52#2309C2211yescharacteristic
53#2309C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2097C2212yescomplemented
58#2097C2212yescomplemented
59#2096C2212yescomplemented
60#2095C2212yescomplemented
61#2073C2214yescomplemented
62#2071C2214yescomplemented
63#2067C2214yescomplemented
64#2065C2214yescomplemented
65#2055C2214yescomplemented
66#2055C2214yescomplemented
67#2042C2214yescomplemented
68#2042C2214yescomplemented
69#2012C2214yes—
70#2012C2214yes—
71#1985C2214yescomplemented
72#1984C2214yescomplemented
73#1972C2212yes—
74#1972C2212yes—
75#1970C2212yes—
76#1970C2212yes—
77#1935C2214yescomplemented
78#1933C2214yescomplemented
79#1926C2214yescomplemented
80#1925C2214yescomplemented
81#1921C2212yescomplemented
82#1920C2212yescomplemented
83#1919C2212yescomplemented
84#1918C2212yescomplemented
85#1879C2214yes—
86#1879C2214yes—
87#1877C2214yescomplemented
88#1877C2214yescomplemented
89#1736C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—42no—
102C22×SD16—42no—
103C22×SD16C2314yes—
104C22×D8—24no—
105C22×D8—24no—
106C22×D8—42no—
107C22×D8C2312yes—
108C22×D8C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#533—42no—
114#533—42no—
115C2×C4⋊Q8D412yes—
116C2×#125D412yes—
117C2×#124—21no—
118C2×#124—21no—
119C2×#124—42no—
120C2×#124—42no—
121C2×#124C2314yes—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8D414yes—
124#518—42no—
125#518—42no—
126C2×C4×Q8C2314yes—
127C2×C4×D4—22no—
128C2×C4×D4C2314yes—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C22⋊C4—22no—
139C22×C42C2311yescharacteristic
140#428—42no—
141#427—24no—
142C2×#107C2311yescharacteristic
143C2×#107C2311yescharacteristic
144C2×#107C2312yes—
145C2×#106C2311yescharacteristic
146C2×#106C2311yescharacteristic
147C2×#106C2312yes—
148#424—42no—
149C2×C4⋊C8C2314yes—
150#419—42no—
151#419—42no—
152#418—24no—
153#418—42no—
154#418—42no—
155#418—42no—
156C2×#103—24no—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2314yes—
162C2×#102—42no—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170#405—42no—
171C2×#97C2312yes—
172C2×C4×C8C2312yes—
173#399—24no—
174#397—24no—
175#397—24no—
176#396—24no—
177#396—24no—
178#393—24no—
179#393—24no—
180#393—24no—
181#391—22no—
182#391—22no—
183#391C2314yes—
184#391C2312yes—
185#391C2312yes—
186#390—22no—
187#390—22no—
188#390—22no—
189#390—21no—
190#390—21no—
191#390C2312yes—
192#390C2312yes—
193#390C2312yes—
194#390C2312yes—
195#390C2314yes—
196#390C2314yes—
197#389—22no—
198#389—22no—
199#389C2314yes—
200#389C2312yes—
201#389C2312yes—
202#388—24no—
203#388—24no—
204#388—24no—
205#388—24no—
206#385—22no—
207#385—24no—
208#385—24no—
209#385—22no—
210#384C2312yes—
211#382C2312yes—
212#381—22no—
213#381—22no—
214#380—24no—
215#380—22no—
216#379—22no—
217#379—22no—
218#379—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—22no—
224#376—21no—
225C43—21no—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239C25—21no—
240C25—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8C2×D414yes—
249C22×Q8C2412yes—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—22no—
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#4114yes—
264C22×D4C2412yes—
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—24no—
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271C2×SD16—28no—
272C2×SD16—44no—
273C2×SD16—44no—
274C2×SD16—416no—
275C2×D8—24no—
276C2×D8—24no—
277C2×D8—24no—
278C2×D8—24no—
279C2×D8—42no—
280C2×D8—42no—
281C2×D8—416no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8#4114yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2411yescharacteristic
298C22×C8C2411yescharacteristic
299C4⋊Q8—44no—
300#125—44no—
301#124—44no—
302#124—44no—
303#124—44no—
304#124—44no—
305C22⋊Q8—48no—
306C22⋊Q8—88no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×D4—44no—
310C4×D4—44no—
311C4×D4—48no—
312#117—22no—
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—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2×D414yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—48no—
359C2×C22⋊C4—48no—
360C2×C22⋊C4—84no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2412yes—
374C2×C42C2412yes—
375C2×C42C2412yes—
376#107—22no—
377#107—22no—
378#107—42no—
379#106—22no—
380#106—22no—
381#106—42no—
382C4⋊C8—44no—
383#103—24no—
384#103—24no—
385#103—42no—
386#103—42no—
387#103—44no—
388#102—28no—
389#102—44no—
390#102—42no—
391#102—42no—
392#102—42no—
393#102—42no—
394#102—44no—
395#97—42no—
396C4×C8—42no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407C24—22no—
408C24—24no—
409C24—22no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—82no—
416C24—82no—
417C24—82no—
418C24—82no—
419C24C22×D411yescharacteristic
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×D4—28no—
437C2×D4—24no—
438C2×D4—28no—
439C2×D4—22no—
440C2×D4—22no—
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—42no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4C2×#4114yes—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—48no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—82no—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C42−1+412yes—
492C22×C4C2511yescharacteristic
493SD16—416no—
494D8—48no—
495D8—48no—
496M16—44no—
497M16—44no—
498C2×C8—28no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—22no—
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—44no—
513C2×C8C2×#4112yes—
514C2×C8C2×#4112yes—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—82no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—84no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—42no—
565C42—42no—
566C42—42no—
567C42—42no—
568C42—82no—
569C42—82no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23—44no—
574C23—44no—
575C23—42no—
576C23—42no—
577C23—48no—
578C23—44no—
579C23—44no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—88no—
584C23—88no—
585C23—84no—
586C23—164no—
587C23—164no—
588C23—162no—
589C23—162no—
590C23—162no—
591C23—162no—
592C23#53412yes—
593C23#54812yes—
594C23#55012yes—
595C23C23×D411yescharacteristic
596Q8—42no—
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—88no—
601Q8—84no—
602D4—24no—
603D4—24no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—48no—
609D4—42no—
610D4—42no—
611D4—84no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—168no—
646C2×C4—162no—
647C2×C4#53412yes—
648C2×C4#53814yes—
649C2×C4#54214yes—
650C2×C4#54812yes—
651C2×C4#55012yes—
652C2×C4#55114yes—
653C2×C4#56214yes—
654C2×C4C22×#4111yescharacteristic
655C2×C4C22×#4111yescharacteristic
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4C2×2−1+411yescharacteristic
660C8—22no—
661C8—22no—
662C8—42no—
663C8—42no—
664C8—42no—
665C22—24no—
666C22—24no—
667C22—22no—
668C22—22no—
669C22—48no—
670C22—44no—
671C22—84no—
672C22—82no—
673C22—82no—
674C22—84no—
675C22—84no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—168no—
680C22—168no—
681C22#273014yes—
682C22C2×#53411yescharacteristic
683C22C2×#54811yescharacteristic
684C22C2×#55011yescharacteristic
685C22#358412yes—
686C22#358512yes—
687C4—42no—
688C4—42no—
689C4—44no—
690C4—44no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4—82no—
699C4#273014yes—
700C4C2×#53811yescharacteristic
701C4C2×#54211yescharacteristic
702C4C2×#55111yescharacteristic
703C4C2×#56211yescharacteristic
704C2—24no—
705C2—84no—
706C2—162no—
707C2C2×#273011yescharacteristic
708C2#6076012yes—
709C2#6134312yes—
710C2#6141311yescharacteristic
711C2#6141711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35515C2semidirect
2. #35338C2semidirect
3. #32620C2semidirect
4. #32614C2semidirect
5. #32605C2semidirect
6. #32599C2semidirect
7. #23736C2semidirect
8. #23632C2semidirect
9. #23614C2semidirect
10. #23541C2semidirect
11. #23506C2semidirect
13. #21304C2semidirect
15. #21088C2semidirect
16. #21060C2semidirect
17. #21032C2semidirect
19. #20933C2semidirect
39. #2592C22semidirect
40. #2592C22semidirect
41. #2580C22semidirect
43. #2538C22semidirect
57. #2097C22semidirect
58. #2097C22semidirect
59. #2096C22semidirect
60. #2095C22semidirect
61. #2073C22semidirect
62. #2071C22semidirect
63. #2067C22semidirect
64. #2065C22semidirect
65. #2055C22semidirect
66. #2055C22semidirect
67. #2042C22semidirect
68. #2042C22semidirect
71. #1985C22semidirect
72. #1984C22semidirect
77. #1935C22semidirect
78. #1933C22semidirect
79. #1926C22semidirect
80. #1925C22semidirect
81. #1921C22semidirect
82. #1920C22semidirect
83. #1919C22semidirect
84. #1918C22semidirect
87. #1877C22semidirect
88. #1877C22semidirect

Maximal subgroups

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

subgroupindex
#355152
#353382
#326202
#326142
#326052
#325992
#237362
#236322
#236142
#235412
#235062
C2×#20122
#213042
#212332
#210882
#210602
#210322
#209342
#209332