Finite Group Databasebeta

← #411488

Subgroups — #411488

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1926C212yesmaximal, complemented
2#14713C211yesmaximal, characteristic
3#13755C211yesmaximal, characteristic, complemented
4#11323C211yesmaximal, characteristic, complemented
5#11267C211yesmaximal, characteristic, complemented
6#11189C212yesmaximal, complemented
7#11121C212yesmaximal, complemented
8#11008C211yesmaximal, characteristic
9#11008C211yesmaximal, characteristic
10#10931C212yesmaximal, complemented
11#10795C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#106C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C2×#390—21no—
19C2×#390—21no—
20C2×#390—21no—
21C2×#390—21no—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×#376—21no—
28C2×C43C2211yescharacteristic
29#1926—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41#1434—21no—
42#1434—21no—
43C4×#95—21no—
44C23×D4—21no—
45C23×D4—21no—
46C23×D4C2312yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#127—24no—
50C2×#127—22no—
51C2×#127—22no—
52C2×#127—42no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C42—21no—
65C22×C42—21no—
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#106—24no—
74C2×#106—42no—
75C2×#106—42no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—42no—
85C2×#102—42no—
86C2×#102—42no—
87C2×#102—42no—
88#390—24no—
89#390—24no—
90#390—42no—
91#390—42no—
92#390—44no—
93#390—44no—
94#390—44no—
95#390—44no—
96#384—42no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D414yescomplemented
101#376—44no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C43—22no—
113C43—22no—
114#340—24no—
115#340—42no—
116#340—42no—
117#336—24no—
118#336—24no—
119#336—22no—
120#336—22no—
121C25—22no—
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4#4114yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8#4114yes—
176#127—48no—
177#127—48no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4#4112yes—
205C2×C4⋊C4#4112yes—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42SD1614yes—
246C2×C42C2×D412yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250#106—48no—
251#102—216no—
252#102—48no—
253#102—48no—
254#102—48no—
255#95—24no—
256#95—24no—
257#95—42no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—42no—
265C24—24no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24C22×D411yescharacteristic
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—48no—
288C2×D4—48no—
289C2×D4—48no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—48no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—84no—
316C2×D4—84no—
317C2×D4—84no—
318C2×D4—84no—
319C2×D4—84no—
320C2×D4—84no—
321C2×D4—84no—
322C2×D4—84no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
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—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4#12112yes—
389C22×C4#12112yes—
390C22×C4#12312yes—
391C22×C4#12312yes—
392C22×C4C2×SD1612yes—
393C22×C4C2×SD1612yes—
394C22×C4C2×SD1612yes—
395C22×C4#13512yes—
396C22×C4AGL(1,ℤ/8)12yes—
397C22×C4AGL(1,ℤ/8)12yes—
398C22×C4AGL(1,ℤ/8)12yes—
399C22×C4AGL(1,ℤ/8)12yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4H(5,𝔽2)12yes—
409C22×C4H(5,𝔽2)12yes—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—24no—
447C42—24no—
448C42—28no—
449C42—44no—
450C42—44no—
451C42—48no—
452C42—48no—
453C42—42no—
454C42—42no—
455C42—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23C2×#12111yescharacteristic
492C23C2×#12311yescharacteristic
493C23#52511yescharacteristic
494C23#52511yescharacteristic
495C23#53211yescharacteristic
496C23#53511yescharacteristic
497C23#53511yescharacteristic
498C23C22×SD1611yescharacteristic
499C23C2×#13511yescharacteristic
500C23C2×AGL(1,ℤ/8)11yescharacteristic
501C23C2×AGL(1,ℤ/8)11yescharacteristic
502C23C2×AGL(1,ℤ/8)11yescharacteristic
503C23C2×AGL(1,ℤ/8)11yescharacteristic
504C23#57511yescharacteristic
505C23#57511yescharacteristic
506D4—44no—
507D4—44no—
508D4—44no—
509D4—44no—
510D4—816no—
511D4—816no—
512D4—82no—
513D4—82no—
514D4—82no—
515D4—82no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4#46014yes—
591C2×C4#48114yes—
592C2×C4#48214yes—
593C2×C4#52512yes—
594C2×C4#52911yescharacteristic
595C2×C4#52912yes—
596C2×C4#52911yescharacteristic
597C2×C4#53211yescharacteristic
598C2×C4#53211yescharacteristic
599C2×C4#53512yes—
600C2×C4#54012yes—
601C2×C4#55112yes—
602C2×C4#55112yes—
603C2×C4#55112yes—
604C2×C4#55112yes—
605C2×C4#55312yes—
606C2×C4#55312yes—
607C2×C4#55512yes—
608C2×C4#55512yes—
609C8—44no—
610C8—82no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—162no—
619C22—162no—
620C22—162no—
621C22—162no—
622C22—162no—
623C22—162no—
624C22—162no—
625C22—162no—
626C22#239211yescharacteristic
627C22#248211yescharacteristic
628C22#250011yescharacteristic
629C22#259811yescharacteristic
630C22#261011yescharacteristic
631C22#261911yescharacteristic
632C22#285411yescharacteristic
633C22C2×#46011yescharacteristic
634C22#303511yescharacteristic
635C22#303611yescharacteristic
636C22#303811yescharacteristic
637C22C2×#48211yescharacteristic
638C22C2×#48111yescharacteristic
639C22#309411yescharacteristic
640C22#309411yescharacteristic
641C22#310111yescharacteristic
642C22#310311yescharacteristic
643C22#310511yescharacteristic
644C22#310611yescharacteristic
645C22#310911yescharacteristic
646C22#310911yescharacteristic
647C22#311411yescharacteristic
648C22#311511yescharacteristic
649C22#316711yescharacteristic
650C22#317111yescharacteristic
651C22#318011yescharacteristic
652C22#318411yescharacteristic
653C22#320711yescharacteristic
654C22#320811yescharacteristic
655C22#320811yescharacteristic
656C22#321111yescharacteristic
657C22#321311yescharacteristic
658C22#321711yescharacteristic
659C22#321711yescharacteristic
660C22#321911yescharacteristic
661C4—24no—
662C4—24no—
663C4—24no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—162no—
672C4#252612yes—
673C4#253112yes—
674C4#260312yes—
675C4#261012yes—
676C2—164no—
677C2#1512011yescharacteristic
678C2#2367811yescharacteristic
679C2#2368211yescharacteristic
680C2#2758511yescharacteristic
681C2#2758811yescharacteristic
682C2#2789011yescharacteristic
683C2#2789711yescharacteristic
684C2#2936511yescharacteristic
685C2#2936611yescharacteristic
686C2#2955511yescharacteristic
687C2#2955811yescharacteristic
688C2#2979611yescharacteristic
689C2#2980011yescharacteristic
690C2#3310711yescharacteristic
691C2#3311311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1926C2semidirect
3. #13755C2semidirect
4. #11323C2semidirect
5. #11267C2semidirect
6. #11189C2semidirect
7. #11121C2semidirect
10. #10931C2semidirect
11. #10795C2semidirect
100. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#19262
#147132
#137552
#113232
#112672
#111892
#111212
#110082
#109312
#107952