Finite Group Databasebeta

← #402452

Subgroups — #402452

Order 512.

1528 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2C2×#2101C212yesmaximal, complemented
3#14697C211yesmaximal, characteristic
4#11372C212yesmaximal, complemented
5#11267C211yesmaximal, characteristic, complemented
6#11267C211yesmaximal, characteristic, complemented
7#11207C211yesmaximal, characteristic, complemented
8#11207C211yesmaximal, characteristic, complemented
9#10731C212yesmaximal
10#10731C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×#117C2212yes—
13C23×C42C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C2×#389C2211yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#385—21no—
18#2285—28no—
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#382C2212yes—
22C2×#380—21no—
23C2×#380—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4—21no—
28#2101—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1436—21no—
41#1436—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C22⋊#95—21no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#117—24no—
51C2×#117—24no—
52C2×#117—24no—
53C2×#117—24no—
54C2×#117—42no—
55C2×#117—42no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—24no—
72C22×C42—24no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83#389—24no—
84#389—22no—
85#389—22no—
86#385—44no—
87#384—24no—
88#384—24no—
89#382—24no—
90#382D418yes—
91#380—28no—
92#380—28no—
93#380—22no—
94#380—24no—
95#380Q814yes—
96C4×C4⋊C4—28no—
97C4×C4⋊C4—28no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—42no—
113#340—42no—
114#340—42no—
115#340—42no—
116#336—24no—
117#336—22no—
118#336—22no—
119C25C2×D411yescharacteristic
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×Q812yes—
152C23×C4#4112yes—
153C23×C4#4114yes—
154C23×C4#4114yes—
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170#117—48no—
171#117—48no—
172#117—48no—
173#117—48no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D414yes—
220C2×C4⋊C4C2×Q812yes—
221C2×C4⋊C4C2×Q812yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C42—216no—
232C2×C42—216no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—28no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42C2×D414yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250C22⋊C8—48no—
251#95—28no—
252#95—28no—
253#95—24no—
254#95—42no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—42no—
260#95—42no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24C22⋊Q812yes—
274C24#12312yes—
275C24#12412yes—
276C24C22×D411yescharacteristic
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4#12012yes—
372C22×C4#12012yes—
373C22×C4C22⋊Q812yes—
374C22×C4#12312yes—
375C22×C4#12312yes—
376C22×C4#12312yes—
377C22×C4#12412yes—
378C22×C4#12518yes—
379C22×C4#12618yes—
380C22×C4#13512yes—
381C22×C4#13512yes—
382C22×C4AGL(1,ℤ/8)12yes—
383C22×C4#13712yes—
384C22×C4C22×D412yes—
385C22×C4C22×Q811yescharacteristic
386C22×C4C2×#4112yes—
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4C2×#4111yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C42−1+412yes—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C22⋊C4—48no—
433C22⋊C4—416no—
434C22⋊C4—84no—
435C42—28no—
436C42—28no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—416no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23#40014yes—
477C23C2×#12011yescharacteristic
478C23C2×C22⋊Q811yescharacteristic
479C23C2×#12311yescharacteristic
480C23C2×#12311yescharacteristic
481C23C2×#12411yescharacteristic
482C23#53511yescharacteristic
483C23#53611yescharacteristic
484C23C2×#13511yescharacteristic
485C23C2×#13511yescharacteristic
486C23C2×AGL(1,ℤ/8)11yescharacteristic
487C23C2×#13711yescharacteristic
488C23#57511yescharacteristic
489C23#57511yescharacteristic
490C23#57511yescharacteristic
491C23#57511yescharacteristic
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—416no—
521C2×C4—416no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4#40014yes—
573C2×C4#48214yes—
574C2×C4#48514yes—
575C2×C4#52512yes—
576C2×C4#52512yes—
577C2×C4C2×#12512yes—
578C2×C4C2×#12612yes—
579C2×C4#53912yes—
580C2×C4#53912yes—
581C2×C4#54612yes—
582C2×C4#54612yes—
583C2×C4#54812yes—
584C2×C4D4×Q812yes—
585C2×C4#55212yes—
586C2×C4#55212yes—
587C2×C4#55311yescharacteristic
588C2×C4#55311yescharacteristic
589C2×C4#55411yescharacteristic
590C2×C4#55411yescharacteristic
591C8—82no—
592C8—82no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22C2×#40011yescharacteristic
609C22#240911yescharacteristic
610C22#243511yescharacteristic
611C22#248711yescharacteristic
612C22#250311yescharacteristic
613C22#262311yescharacteristic
614C22#284911yescharacteristic
615C22#300211yescharacteristic
616C22#300211yescharacteristic
617C22#300611yescharacteristic
618C22#300711yescharacteristic
619C22#307611yescharacteristic
620C22#307611yescharacteristic
621C22#307711yescharacteristic
622C22#307811yescharacteristic
623C22C2×#48211yescharacteristic
624C22C2×#48511yescharacteristic
625C22#309111yescharacteristic
626C22#309111yescharacteristic
627C22#309311yescharacteristic
628C22#309311yescharacteristic
629C22#309411yescharacteristic
630C22#309511yescharacteristic
631C22#313111yescharacteristic
632C22#313511yescharacteristic
633C22#314111yescharacteristic
634C22#314211yescharacteristic
635C22#321211yescharacteristic
636C22#321211yescharacteristic
637C22#321411yescharacteristic
638C22#321611yescharacteristic
639C22#322811yescharacteristic
640C22#323111yescharacteristic
641C22#324011yescharacteristic
642C22#324011yescharacteristic
643C4—24no—
644C4—24no—
645C4—24no—
646C4—42no—
647C4—42no—
648C4—84no—
649C4—84no—
650C4—82no—
651C4—82no—
652C4—161no—
653C4—161no—
654C4#247312yes—
655C4#247312yes—
656C4#248112yes—
657C4#248112yes—
658C2—44no—
659C2#1484311yescharacteristic
660C2#2374411yescharacteristic
661C2#2374711yescharacteristic
662C2#2430411yescharacteristic
663C2#2430611yescharacteristic
664C2#2585811yescharacteristic
665C2#2586311yescharacteristic
666C2#2767911yescharacteristic
667C2#2768211yescharacteristic
668C2#2796211yescharacteristic
669C2#2796711yescharacteristic
670C2#2991011yescharacteristic
671C2#2991111yescharacteristic
672C2#3298111yescharacteristic
673C2#3298511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2101C2semidirect
4. #11372C2semidirect
5. #11267C2semidirect
6. #11267C2semidirect
7. #11207C2semidirect
8. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#21012
#146972
#113722
#112672
#112072
#107312
#107122