Finite Group Databasebeta

← #7505413

Subgroups — #7505413

Order 512.

1918 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35679C211yesmaximal, characteristic, complemented
2#35529C211yesmaximal, characteristic, complemented
3#31656C214yesmaximal, complemented
4#31651C214yesmaximal, complemented
5#31648C214yesmaximal, complemented
6#31643C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic, complemented
9#23540C211yesmaximal, characteristic
10#23457C211yesmaximal, characteristic, complemented
11#23456C211yesmaximal, characteristic, complemented
12C2×#2012C211yesmaximal, characteristic
13#21204C211yesmaximal, characteristic, complemented
14#21095C211yesmaximal, characteristic, complemented
15#21022C211yesmaximal, characteristic, complemented
16#20936C211yesmaximal, characteristic, complemented
17#20934C211yesmaximal, characteristic
18#20933C211yesmaximal, characteristic, complemented
19#20651C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C23×D8C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#518C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36#2566C2212yescomplemented
37#2566C2212yescomplemented
38#2563C2212yescomplemented
39#2542C2212yescomplemented
40#2482—24no—
41#2454—22no—
42#2452—22no—
43#2448—22no—
44#2448—22no—
45#2448C2214yescomplemented
46#2448C2214yescomplemented
47#2437—24no—
48#2317C2211yescharacteristic
49#2310C2211yescharacteristic
50#2309C2211yescharacteristic
51#2309C2211yescharacteristic
52#2309C2211yescharacteristic
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#390C2211yescharacteristic
56#2285C2211yescharacteristic
57#2101C2212yescomplemented
58#2101C2212yescomplemented
59#2100C2212yescomplemented
60#2099C2212yescomplemented
61#2073C2214yescomplemented
62#2071C2214yescomplemented
63#2054C2214yes—
64#2054C2214yes—
65#2026C2214yescomplemented
66#2025C2214yescomplemented
67#2024C2214yescomplemented
68#2023C2214yescomplemented
69#2012C2214yes—
70#2012C2214yes—
71#1959C2212yescomplemented
72#1959C2212yescomplemented
73#1958C2212yescomplemented
74#1958C2212yescomplemented
75#1936C2214yescomplemented
76#1934C2214yescomplemented
77#1917C2214yescomplemented
78#1916C2214yescomplemented
79#1880C2214yescomplemented
80#1880C2214yescomplemented
81#1879C2214yes—
82#1879C2214yes—
83#1877C2214yescomplemented
84#1877C2214yescomplemented
85#1805C2214yescomplemented
86#1803C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—24no—
102C22×SD16—42no—
103C22×SD16C2314yes—
104C22×D8—24no—
105C22×D8—42no—
106C22×D8—42no—
107C22×D8C2314yes—
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C23×C8C2311yescharacteristic
111C23×C8C2311yescharacteristic
112#544—42no—
113#544—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8C2312yes—
116C2×#125—21no—
117C2×#125C2312yes—
118C2×#124—24no—
119C2×#124—22no—
120C2×#124—42no—
121C2×#124C2312yes—
122C2×#124C2312yes—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—22no—
125#519—24no—
126#519—42no—
127#518—42no—
128#518—42no—
129C2×C4×Q8—22no—
130C2×C4×Q8C2314yes—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×#117—22no—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C22⋊C4—22no—
144C22×C22⋊C4—22no—
145C22×C22⋊C4—22no—
146C22×C42C2311yescharacteristic
147#424—42no—
148C2×C4⋊C8C2314yes—
149#419—42no—
150#419—42no—
151#418—24no—
152#418—42no—
153#418—42no—
154#418—42no—
155C2×#103—42no—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#102—42no—
162C2×#102C2314yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2314yes—
167C2×#102C2312yes—
168C2×#102C2314yes—
169C4×M16—42no—
170C2×#97C2312yes—
171C2×C4×C8C2312yes—
172#399—24no—
173#399—24no—
174#399—24no—
175#397—24no—
176#397—24no—
177#396—24no—
178#396—24no—
179#393—24no—
180#391—21no—
181#391—21no—
182#391C2314yes—
183#390—21no—
184#390—21no—
185#390—22no—
186#390—22no—
187#390—22no—
188#390C2312yes—
189#390C2312yes—
190#390C2312yes—
191#390C2314yes—
192#390C2314yes—
193#390C2312yes—
194#389—22no—
195#389D412yes—
196#389D412yes—
197#389C2312yes—
198#389C2312yes—
199#388—24no—
200#388—24no—
201#388—24no—
202#388—24no—
203#385—22no—
204#385—22no—
205#385—24no—
206#385—24no—
207#385D414yes—
208#384C2311yescharacteristic
209#384C2312yes—
210#384C2311yescharacteristic
211#382C2311yescharacteristic
212#382C2312yes—
213#382C2311yescharacteristic
214#381—22no—
215#380—22no—
216#380—24no—
217#379—22no—
218#379—22no—
219#379—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236#336C2312yes—
237#336C2314yes—
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×Q8#4114yes—
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—22no—
269C23×C4—22no—
270C23×C4C2×D412yes—
271C23×C4C2411yescharacteristic
272C2×SD16—28no—
273C2×SD16—28no—
274C2×SD16—44no—
275C2×SD16—416no—
276C2×D8—28no—
277C2×D8—44no—
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—22no—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C4⋊Q8—42no—
297C4⋊Q8—42no—
298C4⋊Q8—44no—
299#125—42no—
300#125—42no—
301#125—44no—
302#124—24no—
303#124—24no—
304#124—42no—
305#124—42no—
306#124—48no—
307C22⋊Q8—48no—
308C22⋊Q8—88no—
309C4×Q8—28no—
310C4×Q8—44no—
311C4×Q8—48no—
312C4×D4—42no—
313C4×D4—42no—
314C4×D4—42no—
315C4×D4—42no—
316C4×D4—44no—
317C4×D4—44no—
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—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—48no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2×D414yes—
347C2×C4⋊C4C2×D414yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
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—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—48no—
372C2×C22⋊C4—48no—
373C2×C22⋊C4—84no—
374C2×C22⋊C4C2×D412yes—
375C2×C22⋊C4C2×D412yes—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—24no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42C2×D412yes—
384C2×C42C2412yes—
385C2×C42C2412yes—
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C4⋊C8—44no—
389#103—44no—
390#103—42no—
391#103—42no—
392#103—42no—
393#103—42no—
394#102—28no—
395#102—44no—
396#102—42no—
397#102—42no—
398#102—44no—
399#102—42no—
400#102—42no—
401#97—42no—
402C4×C8—42no—
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×D411yescharacteristic
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—28no—
428C2×Q8—28no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—44no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—84no—
439C2×Q8C2×#4114yes—
440C2×D4—28no—
441C2×D4—24no—
442C2×D4—28no—
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—22no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4C2×#4114yes—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—84no—
484C22×C4—84no—
485C22×C4—82no—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4H(5,𝔽2)12yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C42−1+412yes—
498C22×C4C2511yescharacteristic
499SD16—416no—
500D8—416no—
501M16—44no—
502M16—44no—
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—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—44no—
516C2×C8—44no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—48no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—84no—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—48no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—84no—
558C22⋊C4—84no—
559C22⋊C4—84no—
560C22⋊C4—84no—
561C22⋊C4—84no—
562C22⋊C4—84no—
563C22⋊C4—84no—
564C42—24no—
565C42—21no—
566C42—21no—
567C42—22no—
568C42—21no—
569C42—21no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—48no—
574C42—42no—
575C42—42no—
576C42—42no—
577C42—42no—
578C23—22no—
579C23—22no—
580C23—24no—
581C23—44no—
582C23—44no—
583C23—42no—
584C23—42no—
585C23—48no—
586C23—44no—
587C23—44no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—88no—
592C23—88no—
593C23—84no—
594C23—164no—
595C23—162no—
596C23—162no—
597C23—162no—
598C23—162no—
599C23—164no—
600C23#53512yes—
601C23#54612yes—
602C23#54812yes—
603C23C23×D411yescharacteristic
604Q8—24no—
605Q8—24no—
606Q8—42no—
607Q8—48no—
608Q8—42no—
609Q8—164no—
610D4—24no—
611D4—24no—
612D4—42no—
613D4—42no—
614D4—42no—
615D4—48no—
616D4—42no—
617D4—42no—
618D4—42no—
619D4—84no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—42no—
635C2×C4—48no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—84no—
654C2×C4—164no—
655C2×C4—164no—
656C2×C4—162no—
657C2×C4#53512yes—
658C2×C4#53814yes—
659C2×C4#53914yes—
660C2×C4#54612yes—
661C2×C4#54812yes—
662C2×C4#55114yes—
663C2×C4#56214yes—
664C2×C4C22×#4111yescharacteristic
665C2×C4C22×#4111yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C2×C4C2×H(5,𝔽2)11yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×2−1+411yescharacteristic
670C8—42no—
671C8—42no—
672C8—42no—
673C8—42no—
674C22—24no—
675C22—22no—
676C22—22no—
677C22—24no—
678C22—44no—
679C22—48no—
680C22—84no—
681C22—82no—
682C22—82no—
683C22—84no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—84no—
688C22—168no—
689C22—168no—
690C22#269614yes—
691C22C2×#53511yescharacteristic
692C22C2×#54611yescharacteristic
693C22C2×#54811yescharacteristic
694C22#358412yes—
695C22#358512yes—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—42no—
702C4—82no—
703C4—81no—
704C4—81no—
705C4—81no—
706C4—82no—
707C4—81no—
708C4—82no—
709C4#269614yes—
710C4C2×#53811yescharacteristic
711C4C2×#53911yescharacteristic
712C4C2×#55111yescharacteristic
713C4C2×#56211yescharacteristic
714C2—24no—
715C2—84no—
716C2—162no—
717C2C2×#269611yescharacteristic
718C2#6081911yescharacteristic
719C2#6082711yescharacteristic
720C2#6118912yes—
721C2#6134312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35679C2semidirect
2. #35529C2semidirect
3. #31656C2semidirect
4. #31651C2semidirect
5. #31648C2semidirect
6. #31643C2semidirect
7. #23744C2semidirect
8. #23632C2semidirect
10. #23457C2semidirect
11. #23456C2semidirect
13. #21204C2semidirect
14. #21095C2semidirect
15. #21022C2semidirect
16. #20936C2semidirect
18. #20933C2semidirect
19. #20651C2semidirect
36. #2566C22semidirect
37. #2566C22semidirect
38. #2563C22semidirect
39. #2542C22semidirect
45. #2448C22semidirect
46. #2448C22semidirect
57. #2101C22semidirect
58. #2101C22semidirect
59. #2100C22semidirect
60. #2099C22semidirect
61. #2073C22semidirect
62. #2071C22semidirect
65. #2026C22semidirect
66. #2025C22semidirect
67. #2024C22semidirect
68. #2023C22semidirect
71. #1959C22semidirect
72. #1959C22semidirect
73. #1958C22semidirect
74. #1958C22semidirect
75. #1936C22semidirect
76. #1934C22semidirect
77. #1917C22semidirect
78. #1916C22semidirect
79. #1880C22semidirect
80. #1880C22semidirect
83. #1877C22semidirect
84. #1877C22semidirect
85. #1805C22semidirect
86. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#356792
#355292
#316562
#316512
#316482
#316432
#237442
#236322
#235402
#234572
#234562
C2×#20122
#212042
#210952
#210222
#209362
#209342
#209332
#206512