Finite Group Databasebeta

← #7510517

Subgroups — #7510517

Order 512.

1841 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35628C211yesmaximal, characteristic, complemented
2#35515C211yesmaximal, characteristic, complemented
3#32607C214yesmaximal, complemented
4#32604C214yesmaximal, complemented
5#32598C214yesmaximal, complemented
6#32597C214yesmaximal, complemented
7#23738C211yesmaximal, characteristic, complemented
8#23633C211yesmaximal, characteristic, complemented
9#23619C211yesmaximal, characteristic, complemented
10#23546C211yesmaximal, characteristic
11#23512C211yesmaximal, characteristic, complemented
12#23424C211yesmaximal, characteristic
13#21300C211yesmaximal, characteristic, complemented
14C2×#1968C211yesmaximal, characteristic, complemented
15#21083C211yesmaximal, characteristic, complemented
16#21063C211yesmaximal, characteristic, complemented
17#21038C211yesmaximal, characteristic, complemented
18#20933C211yesmaximal, characteristic, complemented
19#20929C211yesmaximal, characteristic
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#533C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×C4×C8C2211yescharacteristic
37#2592C2214yescomplemented
38#2580C2214yes—
39#2568C2214yescomplemented
40#2538C2214yes—
41#2492—22no—
42#2489—22no—
43#2488—24no—
44#2454—22no—
45#2452—22no—
46#2437—24no—
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2311C2211yescharacteristic
50#2310C2211yescharacteristic
51#2309C2211yescharacteristic
52#2297C2211yescharacteristic
53#2296C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#390C2211yescharacteristic
56#2274C2211yescharacteristic
57#2097C2212yescomplemented
58#2096C2212yescomplemented
59#2094C2212yescomplemented
60#2092C2212yescomplemented
61#2074C2214yescomplemented
62#2072C2214yescomplemented
63#2068C2214yescomplemented
64#2066C2214yescomplemented
65#2055C2214yes—
66#2054C2214yes—
67#2043C2214yescomplemented
68#2042C2214yescomplemented
69#2013C2212yes—
70#2013C2212yes—
71#2012C2212yes—
72#2012C2212yes—
73#1984C2214yescomplemented
74#1984C2214yescomplemented
75#1968C2214yes—
76#1968C2214yes—
77#1934C2212yescomplemented
78#1934C2212yescomplemented
79#1933C2212yescomplemented
80#1933C2212yescomplemented
81#1926C2214yes—
82#1925C2214yes—
83#1920C2214yescomplemented
84#1919C2214yescomplemented
85#1879C2214yes—
86#1877C2214yes—
87#1877C2214yes—
88#1877C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C2×AGL(1,ℤ/8)—24no—
101C2×AGL(1,ℤ/8)—42no—
102C2×AGL(1,ℤ/8)—42no—
103C22×SD16—24no—
104C22×SD16—24no—
105C22×SD16—42no—
106C22×SD16C2312yes—
107C22×SD16C2312yes—
108C22×SD16C2312yes—
109C22×D8C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#536—42no—
115#536—42no—
116#533—42no—
117#533—42no—
118C2×C4⋊Q8D412yes—
119C2×C4⋊Q8C2312yes—
120C2×#125D412yes—
121C2×#124C2312yes—
122C2×#123—22no—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8D414yes—
126#518—42no—
127#518—42no—
128C2×C4×Q8C2314yes—
129C2×C4×D4—22no—
130C2×C4×D4C2314yes—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—22no—
139C22×C22⋊C4—22no—
140C22×C42C2311yescharacteristic
141#428—42no—
142C2×#107C2312yes—
143C2×#106—24no—
144C2×#106C2312yes—
145C2×#106C2312yes—
146C2×#106C2312yes—
147#423—42no—
148C2×C4⋊C8C2314yes—
149#419—42no—
150#419—42no—
151#418—24no—
152#417—42no—
153#416—24no—
154#416—42no—
155C2×#103—42no—
156C2×#103C2311yescharacteristic
157C2×#103C2311yescharacteristic
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#102—42no—
165C2×#102C2314yes—
166C2×#102C2311yescharacteristic
167C2×#102C2311yescharacteristic
168C2×#102C2312yes—
169C2×#102C2314yes—
170C2×#102C2312yes—
171C2×C4×C8—42no—
172C2×C4×C8C2312yes—
173C2×C4×C8C2312yes—
174#400—24no—
175#400—24no—
176#399—24no—
177#397—24no—
178#396—24no—
179#396—24no—
180#393—24no—
181#393—24no—
182#391—24no—
183#391—22no—
184#391—21no—
185#391—21no—
186#391C2314yes—
187#391C2312yes—
188#391C2312yes—
189#391C2314yes—
190#390—22no—
191#390—21no—
192#390—21no—
193#390C2312yes—
194#390C2312yes—
195#390C2312yes—
196#390C2314yes—
197#390C2312yes—
198#389—22no—
199#389—22no—
200#389C2314yes—
201#389C2312yes—
202#389C2312yes—
203#388—24no—
204#388—24no—
205#386—22no—
206#386—22no—
207#385—22no—
208#385—24no—
209#385—24no—
210#385—22no—
211#384C2312yes—
212#382—24no—
213#382C2312yes—
214#381—22no—
215#380—22no—
216#380—24no—
217#380—22no—
218#379—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4C2314yes—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—21no—
226C2×#95—22no—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
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×Q8—22no—
253C22×Q8C2×D414yes—
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#4114yes—
264C22×D4C2411yescharacteristic
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—24no—
271C23×C4C2×D412yes—
272C23×C4C2411yescharacteristic
273AGL(1,ℤ/8)—416no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—24no—
279C2×SD16—42no—
280C2×SD16—42no—
281C2×SD16—42no—
282C2×SD16—42no—
283C2×SD16—416no—
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#4114yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2411yescharacteristic
303C22×C8C2411yescharacteristic
304C4⋊Q8—44no—
305C4⋊Q8—42no—
306C4⋊Q8—42no—
307#125—44no—
308#124—42no—
309#124—42no—
310#123—48no—
311C22⋊Q8—48no—
312C22⋊Q8—48no—
313C22⋊Q8—88no—
314C4×Q8—44no—
315C4×Q8—44no—
316C4×D4—44no—
317C4×D4—44no—
318C4×D4—48no—
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—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D414yes—
343C2×C4⋊C4C2×D414yes—
344C2×C4⋊C4C2×D414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—48no—
364C2×C22⋊C4—84no—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—24no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2412yes—
376C2×C42C2412yes—
377C2×C42C2412yes—
378#107—42no—
379#106—24no—
380#106—24no—
381#106—42no—
382C4⋊C8—44no—
383#103—22no—
384#103—22no—
385#103—44no—
386#103—42no—
387#103—42no—
388#103—42no—
389#103—42no—
390#103—44no—
391#102—28no—
392#102—22no—
393#102—22no—
394#102—44no—
395#102—42no—
396#102—42no—
397C4×C8—42no—
398C4×C8—42no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—24no—
411C24—22no—
412C24—24no—
413C24—21no—
414C24—42no—
415C24—42no—
416C24—82no—
417C24—82no—
418C24C22×D411yescharacteristic
419#41—44no—
420#41—44no—
421#41—44no—
422#41—44no—
423C2×Q8—28no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×D4—28no—
446C2×D4—24no—
447C2×D4—28no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—44no—
464C2×D4—42no—
465C2×D4—44no—
466C2×D4C2×#4114yes—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—48no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4C22×D412yes—
492C22×C4C22×D414yes—
493C22×C4C22×D414yes—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C42−1+412yes—
502C22×C4C2511yescharacteristic
503SD16—48no—
504SD16—48no—
505SD16—48no—
506D8—48no—
507M16—44no—
508M16—44no—
509C2×C8—28no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
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—44no—
524C2×C8C2×#4112yes—
525C2×C8C2×#4112yes—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
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—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—48no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—82no—
548C4⋊C4—84no—
549C4⋊C4—84no—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—44no—
558C22⋊C4—48no—
559C22⋊C4—84no—
560C22⋊C4—82no—
561C22⋊C4—84no—
562C22⋊C4—84no—
563C22⋊C4—84no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—42no—
573C42—42no—
574C42—42no—
575C42—84no—
576C23—22no—
577C23—22no—
578C23—24no—
579C23—44no—
580C23—42no—
581C23—42no—
582C23—44no—
583C23—48no—
584C23—42no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—42no—
589C23—88no—
590C23—164no—
591C23—162no—
592C23—162no—
593C23#53412yes—
594C23#54812yes—
595C23#55012yes—
596C23C23×D411yescharacteristic
597Q8—41no—
598Q8—42no—
599Q8—42no—
600Q8—41no—
601Q8—42no—
602Q8—41no—
603Q8—42no—
604Q8—41no—
605Q8—88no—
606Q8—84no—
607D4—24no—
608D4—24no—
609D4—41no—
610D4—42no—
611D4—41no—
612D4—41no—
613D4—48no—
614D4—42no—
615D4—41no—
616D4—84no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—48no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—168no—
652C2×C4—162no—
653C2×C4—162no—
654C2×C4#53412yes—
655C2×C4#53814yes—
656C2×C4#54214yes—
657C2×C4#54812yes—
658C2×C4#55012yes—
659C2×C4#55114yes—
660C2×C4#56214yes—
661C2×C4#57714yes—
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—22no—
669C8—22no—
670C8—42no—
671C8—42no—
672C8—42no—
673C22—24no—
674C22—24no—
675C22—22no—
676C22—22no—
677C22—48no—
678C22—44no—
679C22—84no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—168no—
688C22#273014yes—
689C22C2×#53411yescharacteristic
690C22C2×#54811yescharacteristic
691C22C2×#55011yescharacteristic
692C22C2×#57712yes—
693C22#358412yes—
694C4—24no—
695C4—42no—
696C4—44no—
697C4—44no—
698C4—82no—
699C4—82no—
700C4—81no—
701C4—82no—
702C4—81no—
703C4—82no—
704C4—81no—
705C4—82no—
706C4#273014yes—
707C4C2×#53811yescharacteristic
708C4C2×#54211yescharacteristic
709C4C2×#55111yescharacteristic
710C4C2×#56211yescharacteristic
711C2—24no—
712C2—84no—
713C2—161no—
714C2C2×#273011yescharacteristic
715C2#6076912yes—
716C2#6132612yes—
717C2#6141011yescharacteristic
718C2#6141211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
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. #35628C2semidirect
2. #35515C2semidirect
3. #32607C2semidirect
4. #32604C2semidirect
5. #32598C2semidirect
6. #32597C2semidirect
7. #23738C2semidirect
8. #23633C2semidirect
9. #23619C2semidirect
11. #23512C2semidirect
13. #21300C2semidirect
14. C2×#1968C2semidirect
15. #21083C2semidirect
16. #21063C2semidirect
17. #21038C2semidirect
18. #20933C2semidirect
37. #2592C22semidirect
39. #2568C22semidirect
57. #2097C22semidirect
58. #2096C22semidirect
59. #2094C22semidirect
60. #2092C22semidirect
61. #2074C22semidirect
62. #2072C22semidirect
63. #2068C22semidirect
64. #2066C22semidirect
67. #2043C22semidirect
68. #2042C22semidirect
73. #1984C22semidirect
74. #1984C22semidirect
77. #1934C22semidirect
78. #1934C22semidirect
79. #1933C22semidirect
80. #1933C22semidirect
83. #1920C22semidirect
84. #1919C22semidirect

Maximal subgroups

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

subgroupindex
#356282
#355152
#326072
#326042
#325982
#325972
#237382
#236332
#236192
#235462
#235122
#234242
#213002
C2×#19682
#210832
#210632
#210382
#209332
#209292