Finite Group Databasebeta

← #7505380

Subgroups — #7505380

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35679C211yesmaximal, characteristic, complemented
2#35532C211yesmaximal, characteristic, complemented
3#31656C214yesmaximal, complemented
4#31655C214yesmaximal, complemented
5#31652C214yesmaximal, complemented
6#31651C214yesmaximal, complemented
7#23745C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic, complemented
9#23544C211yesmaximal, characteristic
10#23457C211yesmaximal, characteristic, complemented
11#23456C211yesmaximal, characteristic, complemented
12#23431C211yesmaximal, characteristic
13C2×#1959C211yesmaximal, characteristic, complemented
14#21086C211yesmaximal, characteristic, complemented
15#21021C211yesmaximal, characteristic, complemented
16#20932C211yesmaximal, characteristic
17#20931C211yesmaximal, characteristic, complemented
18#20930C211yesmaximal, characteristic, complemented
19#20651C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C22×AGL(1,ℤ/8)C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#520C2211yescharacteristic
25C2×#519C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#418C2211yescharacteristic
29C2×#416C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#97C2211yescharacteristic
36#2567C2212yescomplemented
37#2566C2212yescomplemented
38#2565C2212yescomplemented
39#2542C2212yescomplemented
40#2482—24no—
41#2449—24no—
42#2448—22no—
43#2448—22no—
44#2448C2214yes—
45#2448C2214yes—
46#2441—22no—
47#2440—22no—
48#2317C2211yescharacteristic
49#2311C2211yescharacteristic
50#2310C2211yescharacteristic
51#2309C2211yescharacteristic
52#2309C2211yescharacteristic
53#2297C2211yescharacteristic
54#2296C2211yescharacteristic
55#2296C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2100C2214yescomplemented
58#2099C2214yescomplemented
59#2073C2214yes—
60#2071C2214yes—
61#2055C2214yes—
62#2054C2214yes—
63#2026C2214yescomplemented
64#2025C2214yes—
65#2024C2214yescomplemented
66#2023C2214yes—
67#2015C2212yes—
68#2015C2212yes—
69#2012C2212yes—
70#2012C2212yes—
71#1959C2214yes—
72#1959C2214yes—
73#1936C2212yescomplemented
74#1936C2212yescomplemented
75#1935C2212yescomplemented
76#1935C2212yescomplemented
77#1917C2214yescomplemented
78#1917C2214yescomplemented
79#1880C2214yes—
80#1880C2214yescomplemented
81#1879C2214yes—
82#1878C2214yescomplemented
83#1878C2214yescomplemented
84#1877C2214yescomplemented
85#1805C2214yescomplemented
86#1803C2214yescomplemented
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×D4C2311yescharacteristic
100C2×#137—24no—
101C2×#137—24no—
102C2×#137—42no—
103C2×AGL(1,ℤ/8)—24no—
104C2×AGL(1,ℤ/8)—42no—
105C2×AGL(1,ℤ/8)—42no—
106C22×Q16C2312yes—
107C22×SD16C2312yes—
108C22×SD16C2312yes—
109C22×D8C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#544—42no—
115#544—42no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8C2312yes—
118C2×#125—21no—
119C2×#125C2312yes—
120C2×#124—24no—
121C2×#124—22no—
122C2×#124—42no—
123C2×#124C2312yes—
124C2×#124C2312yes—
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—21no—
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C22⋊C4—22no—
148C22×C42C2311yescharacteristic
149#424—42no—
150C2×C4⋊C8C2314yes—
151#419—42no—
152#418—42no—
153#416—24no—
154#416—42no—
155C2×#103—42no—
156C2×#103—42no—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#102—42no—
164C2×#102—42no—
165C2×#102C2312yes—
166C2×#102C2314yes—
167C2×#102C2312yes—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#97—42no—
172C2×#97C2312yes—
173C2×#97C2312yes—
174#399—24no—
175#399—24no—
176#399—24no—
177#398—24no—
178#397—24no—
179#397—24no—
180#396—24no—
181#395—24no—
182#391—21no—
183#391—21no—
184#391C2311yescharacteristic
185#391C2312yes—
186#391C2314yes—
187#391C2311yescharacteristic
188#390—21no—
189#390—21no—
190#390—22no—
191#390C2314yes—
192#390C2314yes—
193#390C2312yes—
194#390C2311yescharacteristic
195#390C2311yescharacteristic
196#389—24no—
197#389—22no—
198#389D412yes—
199#389D412yes—
200#389C2312yes—
201#389C2312yes—
202#388—24no—
203#388—24no—
204#388—24no—
205#388—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#380—22no—
220#379—22no—
221#379—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—
239C25—21no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41#4114yes—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8#4114yes—
254C22×Q8C2412yes—
255C22×Q8C2411yescharacteristic
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4C2412yes—
265C22×D4C2411yescharacteristic
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—22no—
271C23×C4C2×D412yes—
272C23×C4C2411yescharacteristic
273#137—416no—
274AGL(1,ℤ/8)—416no—
275C2×Q16—24no—
276C2×Q16—24no—
277C2×Q16—42no—
278C2×SD16—24no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—42no—
282C2×SD16—42no—
283C2×SD16—42no—
284C2×D8—24no—
285C2×D8—42no—
286C2×D8—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—42no—
304C4⋊Q8—42no—
305C4⋊Q8—44no—
306#125—42no—
307#125—42no—
308#125—44no—
309#124—24no—
310#124—24no—
311#124—42no—
312#124—42no—
313#124—48no—
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—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4—84no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D414yes—
358C2×C4⋊C4C2×D414yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—48no—
379C2×C22⋊C4C2×D412yes—
380C2×C22⋊C4C2×D412yes—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—24no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42C2×D412yes—
389C2×C42C2412yes—
390C2×C42C2412yes—
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C4⋊C8—44no—
394#103—24no—
395#103—44no—
396#103—44no—
397#103—42no—
398#103—42no—
399#103—42no—
400#102—24no—
401#102—44no—
402#102—44no—
403#102—42no—
404#102—42no—
405#102—42no—
406#97—42no—
407#97—42no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95C2×D414yes—
417#95C2×D414yes—
418C24—21no—
419C24—22no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—82no—
424C24—82no—
425C24C22×D411yescharacteristic
426#41—28no—
427#41—28no—
428#41—44no—
429#41—44no—
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—28no—
434C2×Q8—28no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—44no—
446C2×Q8—42no—
447C2×Q8—44no—
448C2×Q8—84no—
449C2×Q8C2×#4112yes—
450C2×Q8C2×#4114yes—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4C2×#4112yes—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—84no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4C22×D414yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4H(5,𝔽2)12yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C4H(5,𝔽2)12yes—
507C22×C42−1+412yes—
508C22×C4C2511yescharacteristic
509Q16—48no—
510SD16—48no—
511SD16—48no—
512D8—48no—
513M16—44no—
514M16—44no—
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—22no—
526C2×C8—22no—
527C2×C8—44no—
528C2×C8—44no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—48no—
547C4⋊C4—44no—
548C4⋊C4—48no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—84no—
557C4⋊C4—84no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—48no—
561C22⋊C4—48no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C22⋊C4—42no—
568C22⋊C4—84no—
569C22⋊C4—84no—
570C22⋊C4—84no—
571C22⋊C4—84no—
572C22⋊C4—84no—
573C42—24no—
574C42—21no—
575C42—21no—
576C42—22no—
577C42—21no—
578C42—21no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—48no—
583C42—44no—
584C42—42no—
585C42—42no—
586C42—42no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—42no—
591C23—42no—
592C23—41no—
593C23—41no—
594C23—44no—
595C23—44no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—88no—
600C23—84no—
601C23—164no—
602C23—162no—
603C23—162no—
604C23#53512yes—
605C23#54612yes—
606C23#54812yes—
607C23C23×D411yescharacteristic
608Q8—22no—
609Q8—24no—
610Q8—24no—
611Q8—22no—
612Q8—42no—
613Q8—41no—
614Q8—48no—
615Q8—42no—
616Q8—41no—
617Q8—164no—
618D4—22no—
619D4—22no—
620D4—48no—
621D4—42no—
622D4—41no—
623D4—42no—
624D4—42no—
625D4—41no—
626D4—42no—
627D4—84no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—48no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—48no—
648C2×C4—42no—
649C2×C4—48no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—164no—
665C2×C4—162no—
666C2×C4—162no—
667C2×C4#53512yes—
668C2×C4#53814yes—
669C2×C4#53914yes—
670C2×C4#54612yes—
671C2×C4#54812yes—
672C2×C4#55114yes—
673C2×C4#56214yes—
674C2×C4C22×#4111yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C2×C4C2×H(5,𝔽2)11yescharacteristic
678C2×C4C2×H(5,𝔽2)11yescharacteristic
679C2×C4C2×2−1+411yescharacteristic
680C8—42no—
681C8—42no—
682C8—42no—
683C8—42no—
684C22—24no—
685C22—22no—
686C22—22no—
687C22—24no—
688C22—42no—
689C22—44no—
690C22—81no—
691C22—81no—
692C22—82no—
693C22—82no—
694C22—84no—
695C22—82no—
696C22—82no—
697C22—84no—
698C22—168no—
699C22#269614yes—
700C22C2×#53511yescharacteristic
701C22C2×#54611yescharacteristic
702C22C2×#54811yescharacteristic
703C22#358512yes—
704C22#358612yes—
705C4—42no—
706C4—44no—
707C4—42no—
708C4—42no—
709C4—44no—
710C4—42no—
711C4—42no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—81no—
716C4—81no—
717C4—82no—
718C4#269614yes—
719C4C2×#53811yescharacteristic
720C4C2×#53911yescharacteristic
721C4C2×#55111yescharacteristic
722C4C2×#56211yescharacteristic
723C2—24no—
724C2—82no—
725C2—162no—
726C2C2×#269611yescharacteristic
727C2#6082711yescharacteristic
728C2#6082811yescharacteristic
729C2#6117812yes—
730C2#6133612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
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. #35532C2semidirect
3. #31656C2semidirect
4. #31655C2semidirect
5. #31652C2semidirect
6. #31651C2semidirect
7. #23745C2semidirect
8. #23632C2semidirect
10. #23457C2semidirect
11. #23456C2semidirect
13. C2×#1959C2semidirect
14. #21086C2semidirect
15. #21021C2semidirect
17. #20931C2semidirect
18. #20930C2semidirect
19. #20651C2semidirect
36. #2567C22semidirect
37. #2566C22semidirect
38. #2565C22semidirect
39. #2542C22semidirect
57. #2100C22semidirect
58. #2099C22semidirect
63. #2026C22semidirect
65. #2024C22semidirect
73. #1936C22semidirect
74. #1936C22semidirect
75. #1935C22semidirect
76. #1935C22semidirect
77. #1917C22semidirect
78. #1917C22semidirect
80. #1880C22semidirect
82. #1878C22semidirect
83. #1878C22semidirect
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
#355322
#316562
#316552
#316522
#316512
#237452
#236322
#235442
#234572
#234562
#234312
C2×#19592
#210862
#210212
#209322
#209312
#209302
#206512