Finite Group Databasebeta

← #6434296

Subgroups — #6434296

Order 512.

1807 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34639C211yesmaximal, characteristic, complemented
2C2×#2640C211yesmaximal, characteristic, complemented
3#30354C214yesmaximal, complemented
4#30352C214yesmaximal, complemented
5#30350C214yesmaximal, complemented
6#30348C214yesmaximal, complemented
7#23619C211yesmaximal, characteristic, complemented
8#23618C211yesmaximal, characteristic, complemented
9#23552C211yesmaximal, characteristic, complemented
10#23547C211yesmaximal, characteristic, complemented
11#23546C211yesmaximal, characteristic
12#23506C211yesmaximal, characteristic, complemented
13#23452C211yesmaximal, characteristic, complemented
14#23431C211yesmaximal, characteristic
15#21237C211yesmaximal, characteristic, complemented
16#21236C211yesmaximal, characteristic
17#21095C211yesmaximal, characteristic, complemented
18#21094C211yesmaximal, characteristic, complemented
19#20429C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C22×AGL(1,ℤ/8)C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C2×#418C2211yescharacteristic
35C2×#418C2211yescharacteristic
36C2×C4×M16C2211yescharacteristic
37C2×C4×M16C2211yescharacteristic
38#2645C2212yescomplemented
39#2645C2212yescomplemented
40#2644C2212yescomplemented
41#2644C2212yescomplemented
42#2643—22no—
43#2643—22no—
44#2640—22no—
45#2640—22no—
46#2640C2214yes—
47#2640C2214yes—
48#2401—22no—
49#2401—22no—
50C2×#397—22no—
51C2×#393—22no—
52#2297C2211yescharacteristic
53#2297C2211yescharacteristic
54#2297C2211yescharacteristic
55#2296C2211yescharacteristic
56#2296C2211yescharacteristic
57#2296C2211yescharacteristic
58C2×#391C2211yescharacteristic
59C2×#391C2211yescharacteristic
60C2×#390C2211yescharacteristic
61C2×#390C2211yescharacteristic
62C2×#376C2211yescharacteristic
63#2068C2214yescomplemented
64#2067C2214yes—
65#2066C2214yescomplemented
66#2065C2214yes—
67#2058C2214yescomplemented
68#2057C2214yescomplemented
69#2056C2214yescomplemented
70#2055C2214yes—
71#2055C2214yescomplemented
72#2054C2214yes—
73#2042C2214yes—
74#2042C2214yes—
75#2024C2214yescomplemented
76#2023C2214yescomplemented
77#2015C2214yes—
78#2012C2214yes—
79#1973C2212yes—
80#1973C2212yescomplemented
81#1972C2212yes—
82#1971C2212yescomplemented
83#1970C2212yes—
84#1969C2212yescomplemented
85#1968C2212yes—
86#1968C2212yescomplemented
87#1936C2214yes—
88#1935C2214yescomplemented
89#1934C2214yes—
90#1933C2214yescomplemented
91#1768C2214yescomplemented
92#1768C2214yescomplemented
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1728C2211yescharacteristic
98#1728C2211yescharacteristic
99C22×#41C2311yescharacteristic
100C23×Q8—21no—
101C23×Q8C2311yescharacteristic
102C23×D4C2311yescharacteristic
103C2×#137—24no—
104C2×#137—42no—
105C2×#137—42no—
106C2×AGL(1,ℤ/8)—24no—
107C2×AGL(1,ℤ/8)—42no—
108C2×AGL(1,ℤ/8)—42no—
109C22×Q16C2312yes—
110C22×SD16C2312yes—
111C22×SD16C2312yes—
112C22×D8C2312yes—
113C22×M16C2311yescharacteristic
114C22×M16C2311yescharacteristic
115C22×M16C2311yescharacteristic
116C22×M16C2311yescharacteristic
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8—42no—
119C2×C4⋊Q8—42no—
120C2×C4⋊Q8C2314yes—
121C2×#124—22no—
122C2×#124—42no—
123C2×#124—42no—
124C2×#124C2314yes—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#428—42no—
139#428—42no—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144#423—24no—
145#423—24no—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148#419—24no—
149#419—42no—
150#419—42no—
151#418—24no—
152#418—42no—
153#418—42no—
154#417—42no—
155#417—42no—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#102C2314yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C4×M16—42no—
169C4×M16—42no—
170C2×#97C2312yes—
171C2×#97C2312yes—
172C2×C4×C8C2312yes—
173C2×C4×C8C2312yes—
174#401—22no—
175#401—22no—
176#400—22no—
177#400—22no—
178#400—22no—
179#400—22no—
180#400—22no—
181#400—22no—
182#397—24no—
183#397—24no—
184#397—24no—
185#397—24no—
186#397—22no—
187#397—22no—
188#397—22no—
189#397—22no—
190#397—22no—
191#397—22no—
192#396—22no—
193#396—22no—
194#393—24no—
195#393—24no—
196#393—24no—
197#393—24no—
198#391—21no—
199#391—21no—
200#391—22no—
201#391C2312yes—
202#391C2314yes—
203#391C2312yes—
204#391C2312yes—
205#391C2311yescharacteristic
206#391C2311yescharacteristic
207#391C2312yes—
208#390—21no—
209#390—21no—
210#390—22no—
211#390C2312yes—
212#390C2314yes—
213#390C2312yes—
214#390C2312yes—
215#390C2311yescharacteristic
216#390C2311yescharacteristic
217#390C2312yes—
218#389C2312yes—
219#389C2312yes—
220#389C2312yes—
221#389C2312yes—
222#389C2312yes—
223#389C2312yes—
224#386—21no—
225#386—21no—
226#386—22no—
227#386—22no—
228#385—21no—
229#385—21no—
230#385—22no—
231#385—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—21no—
239C4×C22⋊C4—21no—
240#376D412yes—
241#376D412yes—
242#376C2312yes—
243#376C2312yes—
244C2×#95—22no—
245C2×#95—22no—
246C2×#95D412yes—
247C2×#95D412yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#336C2314yes—
253#336C2314yes—
254C25—21no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C22×Q8—21no—
262C22×Q8—24no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—22no—
267C22×Q8—22no—
268C22×Q8—22no—
269C22×Q8—44no—
270C22×Q8—84no—
271C22×Q8C2412yes—
272C22×Q8C2411yescharacteristic
273C22×D4—21no—
274C22×D4—21no—
275C22×D4—22no—
276C22×D4—22no—
277C22×D4—22no—
278C22×D4—22no—
279C22×D4—22no—
280C22×D4—22no—
281C22×D4C2412yes—
282C22×D4C2411yescharacteristic
283C23×C4—21no—
284C23×C4—21no—
285C23×C4—21no—
286C23×C4—21no—
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D412yes—
290C23×C4C2×D412yes—
291C23×C4C2411yescharacteristic
292#137—416no—
293AGL(1,ℤ/8)—416no—
294C2×Q16—24no—
295C2×Q16—42no—
296C2×Q16—42no—
297C2×SD16—24no—
298C2×SD16—24no—
299C2×SD16—42no—
300C2×SD16—42no—
301C2×SD16—42no—
302C2×SD16—42no—
303C2×D8—24no—
304C2×D8—42no—
305C2×D8—42no—
306C2×M16—22no—
307C2×M16—22no—
308C2×M16—22no—
309C2×M16—22no—
310C2×M16—22no—
311C2×M16—22no—
312C2×M16—22no—
313C2×M16—22no—
314C2×M16—22no—
315C2×M16—22no—
316C2×M16#4114yes—
317C2×M16#4114yes—
318C22×C8C2412yes—
319C22×C8C2412yes—
320C22×C8C2412yes—
321C22×C8C2412yes—
322C4⋊Q8—44no—
323C4⋊Q8—44no—
324C4⋊Q8—48no—
325#124—44no—
326#124—44no—
327#124—48no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—44no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4—84no—
372C2×C22⋊C4—84no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42C2×D412yes—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2412yes—
390#107—42no—
391#107—42no—
392#106—42no—
393#106—42no—
394C4⋊C8—28no—
395C4⋊C8—28no—
396#103—28no—
397#103—42no—
398#103—42no—
399#103—44no—
400#103—42no—
401#103—42no—
402#102—28no—
403#102—42no—
404#102—42no—
405#102—44no—
406#102—42no—
407#102—42no—
408#97—42no—
409#97—42no—
410C4×C8—42no—
411C4×C8—42no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95C2×D412yes—
429#95C2×D412yes—
430#95C2×D412yes—
431#95C2×D412yes—
432#95C2×D412yes—
433#95C2×D412yes—
434#95C2×D412yes—
435#95C2×D412yes—
436C24—21no—
437C24—22no—
438C24—44no—
439C24—44no—
440C24—42no—
441C24—42no—
442C24—42no—
443C24C22×D411yescharacteristic
444#41—44no—
445#41—44no—
446#41—44no—
447#41—44no—
448C2×Q8—22no—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—21no—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—21no—
458C2×Q8—22no—
459C2×Q8—21no—
460C2×Q8—48no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—44no—
465C2×Q8—44no—
466C2×Q8—44no—
467C2×Q8—44no—
468C2×Q8—84no—
469C2×Q8—84no—
470C2×Q8—816no—
471C2×D4—22no—
472C2×D4—21no—
473C2×D4—22no—
474C2×D4—21no—
475C2×D4—22no—
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—21no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—22no—
482C2×D4—21no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—44no—
490C2×D4—44no—
491C2×D4—44no—
492C2×D4—44no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—42no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—42no—
517C22×C4—42no—
518C22×C4—42no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C22×D412yes—
527C22×C4C22×D411yescharacteristic
528C22×C4C2×#4112yes—
529C22×C4C2×#4112yes—
530C22×C4H(5,𝔽2)12yes—
531C22×C4H(5,𝔽2)12yes—
532C22×C4H(5,𝔽2)12yes—
533C22×C42−1+412yes—
534C22×C4C2511yescharacteristic
535Q16—48no—
536SD16—48no—
537SD16—48no—
538D8—48no—
539M16—28no—
540M16—28no—
541M16—44no—
542M16—44no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8C2×#4114yes—
554C2×C8C2×#4114yes—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—48no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—82no—
571C4⋊C4—82no—
572C4⋊C4—82no—
573C4⋊C4—82no—
574C4⋊C4—82no—
575C4⋊C4—82no—
576C22⋊C4—48no—
577C22⋊C4—48no—
578C22⋊C4—42no—
579C22⋊C4—42no—
580C22⋊C4—42no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4—82no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C22⋊C4—82no—
587C22⋊C4—84no—
588C22⋊C4—84no—
589C22⋊C4—84no—
590C22⋊C4—84no—
591C42—22no—
592C42—22no—
593C42—24no—
594C42—24no—
595C42—22no—
596C42—22no—
597C42—21no—
598C42—21no—
599C42—22no—
600C42—22no—
601C42—21no—
602C42—21no—
603C42—42no—
604C42—42no—
605C42—42no—
606C23—22no—
607C23—22no—
608C23—44no—
609C23—42no—
610C23—42no—
611C23—44no—
612C23—41no—
613C23—42no—
614C23—42no—
615C23—41no—
616C23—41no—
617C23—42no—
618C23—88no—
619C23—84no—
620C23—84no—
621C23—88no—
622C23—84no—
623C23#53212yes—
624C23#53512yes—
625C23#53712yes—
626C23C23×D411yescharacteristic
627Q8—41no—
628Q8—42no—
629Q8—41no—
630Q8—42no—
631Q8—42no—
632Q8—42no—
633Q8—41no—
634Q8—41no—
635Q8—816no—
636Q8—84no—
637D4—41no—
638D4—42no—
639D4—41no—
640D4—42no—
641D4—42no—
642D4—42no—
643D4—41no—
644D4—41no—
645D4—84no—
646D4—84no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—41no—
686C2×C4—42no—
687C2×C4—84no—
688C2×C4—84no—
689C2×C4—84no—
690C2×C4—84no—
691C2×C4—84no—
692C2×C4—84no—
693C2×C4#53212yes—
694C2×C4#53512yes—
695C2×C4#53712yes—
696C2×C4#53814yes—
697C2×C4#53814yes—
698C2×C4#54214yes—
699C2×C4#54214yes—
700C2×C4C22×#4111yescharacteristic
701C2×C4C22×#4111yescharacteristic
702C2×C4C2×H(5,𝔽2)11yescharacteristic
703C2×C4C2×H(5,𝔽2)11yescharacteristic
704C2×C4C2×H(5,𝔽2)11yescharacteristic
705C2×C4C2×2−1+411yescharacteristic
706C8—24no—
707C8—24no—
708C8—42no—
709C8—42no—
710C22—24no—
711C22—22no—
712C22—22no—
713C22—24no—
714C22—81no—
715C22—82no—
716C22—82no—
717C22—84no—
718C22—81no—
719C22—82no—
720C22—81no—
721C22—82no—
722C22—82no—
723C22—84no—
724C22—168no—
725C22#266214yes—
726C22C2×#53211yescharacteristic
727C22C2×#53511yescharacteristic
728C22C2×#53711yescharacteristic
729C22#358512yes—
730C22#358612yes—
731C4—42no—
732C4—42no—
733C4—42no—
734C4—42no—
735C4—42no—
736C4—42no—
737C4—81no—
738C4—82no—
739C4—82no—
740C4—81no—
741C4—82no—
742C4—82no—
743C4—81no—
744C4—82no—
745C4#266214yes—
746C4C2×#53811yescharacteristic
747C4C2×#53811yescharacteristic
748C4C2×#54211yescharacteristic
749C4C2×#54211yescharacteristic
750C2—24no—
751C2—161no—
752C2—162no—
753C2C2×#266211yescharacteristic
754C2#6070312yes—
755C2#6080611yescharacteristic
756C2#6080711yescharacteristic
757C2#6091912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34639C2semidirect
2. C2×#2640C2semidirect
3. #30354C2semidirect
4. #30352C2semidirect
5. #30350C2semidirect
6. #30348C2semidirect
7. #23619C2semidirect
8. #23618C2semidirect
9. #23552C2semidirect
10. #23547C2semidirect
12. #23506C2semidirect
13. #23452C2semidirect
15. #21237C2semidirect
17. #21095C2semidirect
18. #21094C2semidirect
19. #20429C2semidirect
38. #2645C22semidirect
39. #2645C22semidirect
40. #2644C22semidirect
41. #2644C22semidirect
63. #2068C22semidirect
65. #2066C22semidirect
67. #2058C22semidirect
68. #2057C22semidirect
69. #2056C22semidirect
71. #2055C22semidirect
75. #2024C22semidirect
76. #2023C22semidirect
80. #1973C22semidirect
82. #1971C22semidirect
84. #1969C22semidirect
86. #1968C22semidirect
88. #1935C22semidirect
90. #1933C22semidirect
91. #1768C22semidirect
92. #1768C22semidirect

Maximal subgroups

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

subgroupindex
#346392
C2×#26402
#303542
#303522
#303502
#303482
#236192
#236182
#235522
#235472
#235462
#235062
#234522
#234312
#212372
#212362
#210952
#210942
#204292