Finite Group Databasebeta

← #476017

Subgroups — #476017

Order 512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2024C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3#14714C211yesmaximal, characteristic, complemented
4#14708C211yesmaximal, characteristic
5#11268C212yesmaximal, complemented
6#11077C212yesmaximal, complemented
7#11076C212yesmaximal
8#11050C211yesmaximal, characteristic, complemented
9#11049C211yesmaximal, characteristic
10#11009C212yesmaximal
11#10832C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C2×#391—21no—
20C2×#391—21no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C43C2211yescharacteristic
28#2024—24no—
29#2023—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1437—21no—
38#1437—21no—
39#1437—21no—
40#1437—21no—
41C4⋊#95—21no—
42C4⋊#95—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8D414yes—
54C2×#124—22no—
55C2×#124—42no—
56C2×#124—42no—
57C2×#124Q814yescomplemented
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—24no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8—42no—
81C2×#103—24no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#102—24no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102—42no—
93#391—22no—
94#391—22no—
95#391—24no—
96#391—44no—
97#391—44no—
98#390—22no—
99#390—22no—
100#390—24no—
101#384—42no—
102#382—42no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C43—42no—
114#340—24no—
115#340—24no—
116#340—42no—
117#340—42no—
118#336—22no—
119#336—22no—
120#336—24no—
121C25—21no—
122C22×Q8—24no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8C2×Q812yes—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C2×Q812yes—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
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—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C4⋊Q8—48no—
176C4⋊Q8—48no—
177#124—48no—
178#124—48no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C2×D414yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42C2×D412yes—
246C2×C42C2×Q812yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42#4114yes—
250C4⋊C8—48no—
251#103—48no—
252#103—48no—
253#102—48no—
254#102—48no—
255#95—42no—
256#95—42no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—44no—
265C24—22no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24C22×D411yescharacteristic
281C2×Q8—24no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—22no—
285C2×Q8—48no—
286C2×Q8—48no—
287C2×Q8—42no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—41no—
291C2×Q8—48no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×D4—24no—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—22no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—41no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
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—24no—
351C22×C4—22no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4#12112yes—
386C22×C4#12112yes—
387C22×C4C22⋊Q812yes—
388C22×C4C22⋊Q812yes—
389C22×C4#12412yes—
390C22×C4#12412yes—
391C22×C4#13512yes—
392C22×C4#13512yes—
393C22×C4AGL(1,ℤ/8)12yes—
394C22×C4AGL(1,ℤ/8)12yes—
395C22×C4#13712yes—
396C22×C4#13712yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×Q811yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C4H(5,𝔽2)12yes—
406C22×C42−1+411yescharacteristic
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—48no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C42—24no—
443C42—24no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—44no—
449C42—44no—
450C42—44no—
451C42—44no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23C2×#12111yescharacteristic
488C23C2×C22⋊Q811yescharacteristic
489C23C2×#12411yescharacteristic
490C23#53411yescharacteristic
491C23#53511yescharacteristic
492C23#53511yescharacteristic
493C23#53711yescharacteristic
494C23C2×#13511yescharacteristic
495C23C2×#13511yescharacteristic
496C23C2×AGL(1,ℤ/8)11yescharacteristic
497C23C2×AGL(1,ℤ/8)11yescharacteristic
498C23C2×#13711yescharacteristic
499C23C2×#13711yescharacteristic
500C23#57511yescharacteristic
501C23#57511yescharacteristic
502Q8—44no—
503Q8—44no—
504Q8—816no—
505Q8—82no—
506Q8—82no—
507D4—44no—
508D4—44no—
509D4—82no—
510D4—82no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
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—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4#46314yes—
586C2×C4#46414yes—
587C2×C4#48914yes—
588C2×C4#53412yes—
589C2×C4#53812yes—
590C2×C4#53812yes—
591C2×C4#53812yes—
592C2×C4#53912yes—
593C2×C4#54312yes—
594C2×C4D4×Q811yescharacteristic
595C2×C4#55012yes—
596C2×C4#55011yescharacteristic
597C2×C4#55412yes—
598C2×C4#55411yescharacteristic
599C2×C4#55512yes—
600C2×C4#55611yescharacteristic
601C2×C4#55912yes—
602C2×C4H(3,𝔽4)12yes—
603C2×C4#56212yes—
604C8—82no—
605C8—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—161no—
614C22—161no—
615C22—161no—
616C22—161no—
617C22—161no—
618C22—161no—
619C22—161no—
620C22—161no—
621C22#253811yescharacteristic
622C22#254811yescharacteristic
623C22#256311yescharacteristic
624C22#261711yescharacteristic
625C22#262611yescharacteristic
626C22#263911yescharacteristic
627C22#281911yescharacteristic
628C22C2×#46311yescharacteristic
629C22C2×#46411yescharacteristic
630C22#303611yescharacteristic
631C22#303711yescharacteristic
632C22#307611yescharacteristic
633C22#307611yescharacteristic
634C22#307911yescharacteristic
635C22#307911yescharacteristic
636C22#313111yescharacteristic
637C22C2×#48911yescharacteristic
638C22#314011yescharacteristic
639C22#314111yescharacteristic
640C22#318811yescharacteristic
641C22#319011yescharacteristic
642C22#319411yescharacteristic
643C22#319611yescharacteristic
644C22#320711yescharacteristic
645C22#321011yescharacteristic
646C22#321111yescharacteristic
647C22#321111yescharacteristic
648C22#321411yescharacteristic
649C22#321611yescharacteristic
650C22#321711yescharacteristic
651C22#321811yescharacteristic
652C22#324711yescharacteristic
653C22#324811yescharacteristic
654C22#325111yescharacteristic
655C22#325211yescharacteristic
656C4—24no—
657C4—24no—
658C4—24no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—84no—
663C4—82no—
664C4—162no—
665C4—161no—
666C4—161no—
667C4#273212yes—
668C4#273312yes—
669C4#280412yes—
670C4#280812yes—
671C2—162no—
672C2#1707711yescharacteristic
673C2#2828111yescharacteristic
674C2#2828211yescharacteristic
675C2#2875011yescharacteristic
676C2#2875411yescharacteristic
677C2#2914211yescharacteristic
678C2#2914811yescharacteristic
679C2#2971811yescharacteristic
680C2#2971911yescharacteristic
681C2#2995411yescharacteristic
682C2#2995511yescharacteristic
683C2#3018811yescharacteristic
684C2#3020011yescharacteristic
685C2#3281011yescharacteristic
686C2#3281111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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×#2024C2semidirect
3. #14714C2semidirect
5. #11268C2semidirect
6. #11077C2semidirect
8. #11050C2semidirect
11. #10832C2semidirect
57. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20242
C2×#20232
#147142
#147082
#112682
#110772
#110762
#110502
#110492
#110092
#108322