Finite Group Databasebeta

← #7505374

Subgroups — #7505374

Order 512.

1840 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35674C211yesmaximal, characteristic, complemented
2#35532C211yesmaximal, characteristic
3#31640C214yesmaximal, complemented
4#31639C214yesmaximal, complemented
5#31636C214yesmaximal, complemented
6#31635C214yesmaximal, complemented
7#23745C211yesmaximal, characteristic, complemented
8#23633C211yesmaximal, characteristic, complemented
9#23545C211yesmaximal, characteristic
10#23457C211yesmaximal, characteristic, complemented
11#23457C211yesmaximal, characteristic, complemented
12#23432C211yesmaximal, characteristic
13C2×#1956C211yesmaximal, characteristic, complemented
14#21083C211yesmaximal, characteristic
15#21020C211yesmaximal, characteristic, complemented
16#20931C211yesmaximal, characteristic
17#20931C211yesmaximal, characteristic
18#20930C211yesmaximal, characteristic
19#20651C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C22×#137C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C22×C4⋊Q8C2211yescharacteristic
24C2×#520C2211yescharacteristic
25C2×#519C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#416C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2567C2212yes—
37#2566C2212yes—
38#2565C2212yes—
39#2542C2212yes—
40#2484—24no—
41#2449—24no—
42#2446—22no—
43#2446—22no—
44#2446C2214yes—
45#2446C2214yes—
46#2441—22no—
47#2440—22no—
48#2317C2211yescharacteristic
49#2311C2211yescharacteristic
50#2310C2211yescharacteristic
51#2310C2211yescharacteristic
52#2310C2211yescharacteristic
53#2297C2211yescharacteristic
54#2296C2211yescharacteristic
55#2296C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2100C2214yes—
58#2099C2214yes—
59#2074C2214yes—
60#2072C2214yes—
61#2056C2214yes—
62#2055C2214yes—
63#2026C2214yes—
64#2026C2214yes—
65#2024C2214yes—
66#2024C2214yes—
67#2016C2212yes—
68#2016C2212yes—
69#2013C2212yes—
70#2013C2212yes—
71#1956C2214yes—
72#1956C2214yes—
73#1934C2212yes—
74#1934C2212yes—
75#1933C2212yes—
76#1933C2212yes—
77#1916C2214yes—
78#1916C2214yes—
79#1880C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1877C2214yes—
84#1877C2214yes—
85#1805C2214yes—
86#1803C2214yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C2×#137—24no—
101C2×#137—24no—
102C2×#137—24no—
103C2×#137—42no—
104C2×#137—42no—
105C2×#137—42no—
106C22×Q16C2312yes—
107C22×Q16C2312yes—
108C22×SD16C2312yes—
109C22×SD16C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#544—42no—
115#544—42no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—24no—
118C2×C4⋊Q8—22no—
119C2×C4⋊Q8—42no—
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2312yes—
123C2×#125—21no—
124C2×#125C2312yes—
125C2×C22⋊Q8—24no—
126C2×C22⋊Q8—22no—
127#520—42no—
128#520—42no—
129#519—24no—
130#519—42no—
131C2×C4×Q8—22no—
132C2×C4×Q8C2314yes—
133C2×C4×Q8C2312yes—
134C2×C4×D4—22no—
135C2×C4×D4C2312yes—
136C2×#117—21no—
137C2×#117—22no—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C22×C4⋊C4—22no—
144C22×C4⋊C4—22no—
145C22×C4⋊C4—21no—
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C42C2311yescharacteristic
149#424—42no—
150C2×C4⋊C8C2314yes—
151#419—42no—
152#419—42no—
153#416—24no—
154#416—42no—
155C2×#103—42no—
156C2×#103—42no—
157C2×#103—42no—
158C2×#103—42no—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×C4×C8—42no—
172C2×C4×C8C2312yes—
173C2×C4×C8C2312yes—
174#400—24no—
175#400—24no—
176#399—24no—
177#398—24no—
178#398—24no—
179#398—24no—
180#396—24no—
181#395—24no—
182#391—21no—
183#391—21no—
184#391—21no—
185#391—21no—
186#391—22no—
187#391C2311yescharacteristic
188#391C2312yes—
189#391C2314yes—
190#391C2314yes—
191#391C2314yes—
192#391C2311yescharacteristic
193#390C2311yescharacteristic
194#390C2312yes—
195#390C2311yescharacteristic
196#389—24no—
197#389—22no—
198#389D412yes—
199#389D412yes—
200#389C2312yes—
201#389C2312yes—
202#387—24no—
203#387—24no—
204#387—24no—
205#387—24no—
206#387—24no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—22no—
211#385—24no—
212#385D414yes—
213#384—22no—
214#384C2312yes—
215#384C2312yes—
216#384C2312yes—
217#382C2312yes—
218#381—22no—
219#381—22no—
220#381—22no—
221#380—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#336C2314yes—
235#336C2312yes—
236#336C2312yes—
237#336C2314yes—
238#336C2314yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41#4114yes—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8#4114yes—
260C22×Q8C2412yes—
261C22×Q8C2412yes—
262C22×Q8C2411yescharacteristic
263C22×D4—21no—
264C22×D4C2411yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—22no—
271C23×C4C2×D412yes—
272C23×C4C2411yescharacteristic
273#137—416no—
274#137—416no—
275C2×Q16—24no—
276C2×Q16—24no—
277C2×Q16—24no—
278C2×Q16—42no—
279C2×Q16—42no—
280C2×Q16—42no—
281C2×SD16—24no—
282C2×SD16—24no—
283C2×SD16—24no—
284C2×SD16—42no—
285C2×SD16—42no—
286C2×SD16—42no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C4⋊Q8—24no—
304C4⋊Q8—24no—
305C4⋊Q8—42no—
306C4⋊Q8—42no—
307C4⋊Q8—42no—
308C4⋊Q8—42no—
309C4⋊Q8—48no—
310C4⋊Q8—44no—
311#125—42no—
312#125—42no—
313#125—44no—
314C22⋊Q8—48no—
315C22⋊Q8—88no—
316C4×Q8—28no—
317C4×Q8—44no—
318C4×Q8—42no—
319C4×Q8—42no—
320C4×Q8—48no—
321C4×D4—42no—
322C4×D4—42no—
323C4×D4—48no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—48no—
335#117—44no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—44no—
355C2×C4⋊C4—48no—
356C2×C4⋊C4—84no—
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2×D414yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2411yescharacteristic
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4C2×D412yes—
376C2×C22⋊C4C2×D412yes—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—24no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42C2×D412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C4⋊C8—44no—
390#103—24no—
391#103—44no—
392#103—44no—
393#103—44no—
394#103—44no—
395#103—42no—
396#103—42no—
397#103—42no—
398#103—42no—
399#103—42no—
400#102—24no—
401#102—42no—
402C4×C8—42no—
403C4×C8—42no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95C2×D414yes—
417#95C2×D414yes—
418C24—21no—
419C24C22×D411yescharacteristic
420#41—28no—
421#41—28no—
422#41—44no—
423#41—44no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—28no—
428C2×Q8—28no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—44no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8—84no—
456C2×Q8C2×#4112yes—
457C2×Q8C2×#4114yes—
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4C2×#4112yes—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—84no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4H(5,𝔽2)12yes—
504C22×C4H(5,𝔽2)12yes—
505C22×C42−1+412yes—
506C22×C4C2511yescharacteristic
507Q16—48no—
508Q16—48no—
509SD16—48no—
510SD16—48no—
511M16—44no—
512M16—44no—
513C2×C8—22no—
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×C8—44no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—28no—
531C4⋊C4—28no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—48no—
545C4⋊C4—48no—
546C4⋊C4—48no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—44no—
554C4⋊C4—48no—
555C4⋊C4—44no—
556C4⋊C4—84no—
557C4⋊C4—84no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—44no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—84no—
567C42—24no—
568C42—21no—
569C42—21no—
570C42—22no—
571C42—21no—
572C42—21no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—48no—
577C42—44no—
578C42—42no—
579C42—42no—
580C42—42no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—41no—
585C23—41no—
586C23—42no—
587C23—42no—
588C23#53512yes—
589C23#54612yes—
590C23#54812yes—
591C23C23×D411yescharacteristic
592Q8—22no—
593Q8—24no—
594Q8—24no—
595Q8—22no—
596Q8—42no—
597Q8—41no—
598Q8—48no—
599Q8—42no—
600Q8—41no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—42no—
605Q8—84no—
606Q8—164no—
607D4—22no—
608D4—22no—
609D4—48no—
610D4—41no—
611D4—41no—
612C2×C4—216no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—48no—
630C2×C4—42no—
631C2×C4—48no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—84no—
650C2×C4—164no—
651C2×C4—162no—
652C2×C4—162no—
653C2×C4#53512yes—
654C2×C4#53814yes—
655C2×C4#53914yes—
656C2×C4#54612yes—
657C2×C4#54812yes—
658C2×C4#55114yes—
659C2×C4#56214yes—
660C2×C4#57714yes—
661C2×C4#57814yes—
662C2×C4C22×#4111yescharacteristic
663C2×C4C22×#4111yescharacteristic
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C2×C4C2×2−1+411yescharacteristic
668C8—42no—
669C8—42no—
670C8—42no—
671C8—42no—
672C22—22no—
673C22—24no—
674C22—22no—
675C22—24no—
676C22—44no—
677C22—42no—
678C22—81no—
679C22—81no—
680C22—82no—
681C22#269614yes—
682C22C2×#53511yescharacteristic
683C22C2×#54611yescharacteristic
684C22C2×#54811yescharacteristic
685C22C2×#57712yes—
686C22C2×#57812yes—
687C4—24no—
688C4—24no—
689C4—42no—
690C4—42no—
691C4—44no—
692C4—42no—
693C4—44no—
694C4—82no—
695C4—82no—
696C4—81no—
697C4—82no—
698C4—82no—
699C4—81no—
700C4—82no—
701C4#269614yes—
702C4C2×#53811yescharacteristic
703C4C2×#53911yescharacteristic
704C4C2×#55111yescharacteristic
705C4C2×#56211yescharacteristic
706C2—24no—
707C2—82no—
708C2C2×#269611yescharacteristic
709C2#6081811yescharacteristic
710C2#6082111yescharacteristic
711C2#6116812yes—
712C2#6133012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. #35674C2semidirect
3. #31640C2semidirect
4. #31639C2semidirect
5. #31636C2semidirect
6. #31635C2semidirect
7. #23745C2semidirect
8. #23633C2semidirect
10. #23457C2semidirect
11. #23457C2semidirect
13. C2×#1956C2semidirect
15. #21020C2semidirect
19. #20651C2semidirect

Maximal subgroups

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

subgroupindex
#356742
#355322
#316402
#316392
#316362
#316352
#237452
#236332
#235452
#234572
#234322
C2×#19562
#210832
#210202
#209312
#209302
#206512