Finite Group Databasebeta

← #474765

Subgroups — #474765

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14712C211yesmaximal, characteristic, complemented
2#14708C211yesmaximal, characteristic
3#11268C212yesmaximal, complemented
4#11067C212yesmaximal, complemented
5#11066C212yesmaximal
6#11050C211yesmaximal, characteristic, complemented
7#11050C211yesmaximal, characteristic, complemented
8#11049C211yesmaximal, characteristic
9#11049C211yesmaximal, characteristic
10#11009C212yesmaximal
11#10795C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389—21no—
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C43C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1437—21no—
37#1437—21no—
38#1437—21no—
39#1437—21no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95C2212yes—
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#124—22no—
48C2×#124—42no—
49C2×#124—42no—
50C2×#124Q814yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#103—24no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#102—24no—
80C2×#102—42no—
81C2×#102—42no—
82C2×#102—42no—
83C2×#102—42no—
84C2×#102—42no—
85#391—22no—
86#391—22no—
87#391—42no—
88#390—22no—
89#390—22no—
90#390—42no—
91#389—42no—
92#389—42no—
93#389—44no—
94#389—44no—
95#384—24no—
96#384—42no—
97#382—42no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C43—42no—
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336—22no—
114#336—22no—
115#336—42no—
116C25—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8C2×Q812yes—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C2×Q812yes—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163#124—48no—
164#124—48no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
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—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4#4114yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
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×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42C2×Q812yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242#103—48no—
243#103—48no—
244#102—48no—
245#102—48no—
246#95—42no—
247#95—42no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—42no—
255C24—22no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24C22×D411yescharacteristic
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—24no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C22×C4—21no—
304C22×C4—21no—
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—21no—
318C22×C4—21no—
319C22×C4—21no—
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—22no—
336C22×C4—24no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
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—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4C22⋊Q812yes—
371C22×C4C22⋊Q812yes—
372C22×C4#12412yes—
373C22×C4#12412yes—
374C22×C4#13512yes—
375C22×C4#13512yes—
376C22×C4AGL(1,ℤ/8)12yes—
377C22×C4AGL(1,ℤ/8)12yes—
378C22×C4#13712yes—
379C22×C4#13712yes—
380C22×C4C22×Q811yescharacteristic
381C22×C4C2×#4112yes—
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4H(5,𝔽2)12yes—
386C22×C4H(5,𝔽2)11yescharacteristic
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)12yes—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
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—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—88no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—44no—
432C42—44no—
433C42—44no—
434C42—44no—
435C42—42no—
436C42—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
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—
472C23C2×C22⋊Q811yescharacteristic
473C23#52511yescharacteristic
474C23C2×#12411yescharacteristic
475C23#53411yescharacteristic
476C23#53511yescharacteristic
477C23#53511yescharacteristic
478C23#53511yescharacteristic
479C23C2×#13511yescharacteristic
480C23C2×#13511yescharacteristic
481C23C2×AGL(1,ℤ/8)11yescharacteristic
482C23C2×AGL(1,ℤ/8)11yescharacteristic
483C23C2×#13711yescharacteristic
484C23C2×#13711yescharacteristic
485C23#57511yescharacteristic
486C23#57511yescharacteristic
487Q8—44no—
488Q8—44no—
489Q8—82no—
490Q8—82no—
491D4—44no—
492D4—44no—
493D4—82no—
494D4—82no—
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—22no—
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—41no—
520C2×C4—41no—
521C2×C4—41no—
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—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4#48814yes—
571C2×C4#48914yes—
572C2×C4#53812yes—
573C2×C4#53812yes—
574C2×C4#53812yes—
575C2×C4#54012yes—
576C2×C4#54012yes—
577C2×C4#54312yes—
578C2×C4#55312yes—
579C2×C4#55312yes—
580C2×C4#55411yescharacteristic
581C2×C4#55411yescharacteristic
582C2×C4#55412yes—
583C2×C4#55511yescharacteristic
584C2×C4#55511yescharacteristic
585C2×C4#55912yes—
586C2×C4#56012yes—
587C2×C4#56012yes—
588C8—82no—
589C8—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—161no—
598C22—161no—
599C22—161no—
600C22—161no—
601C22—161no—
602C22—161no—
603C22—161no—
604C22—161no—
605C22#256311yescharacteristic
606C22#258011yescharacteristic
607C22#259111yescharacteristic
608C22#262611yescharacteristic
609C22#263111yescharacteristic
610C22#264111yescharacteristic
611C22#281511yescharacteristic
612C22#307611yescharacteristic
613C22#307611yescharacteristic
614C22#307911yescharacteristic
615C22#307911yescharacteristic
616C22#310511yescharacteristic
617C22#310811yescharacteristic
618C22#311411yescharacteristic
619C22#311711yescharacteristic
620C22C2×#48811yescharacteristic
621C22C2×#48911yescharacteristic
622C22#314111yescharacteristic
623C22#314211yescharacteristic
624C22#318811yescharacteristic
625C22#319011yescharacteristic
626C22#319411yescharacteristic
627C22#319611yescharacteristic
628C22#320711yescharacteristic
629C22#320711yescharacteristic
630C22#321011yescharacteristic
631C22#321011yescharacteristic
632C22#321211yescharacteristic
633C22#321211yescharacteristic
634C22#321311yescharacteristic
635C22#321411yescharacteristic
636C22#321511yescharacteristic
637C22#321611yescharacteristic
638C22#321711yescharacteristic
639C22#321811yescharacteristic
640C4—24no—
641C4—24no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—84no—
647C4—82no—
648C4—161no—
649C4—161no—
650C4—162no—
651C4#278012yes—
652C4#278012yes—
653C4#280312yes—
654C4#280312yes—
655C2—162no—
656C2#1700711yescharacteristic
657C2#2914011yescharacteristic
658C2#2914811yescharacteristic
659C2#2926011yescharacteristic
660C2#2926211yescharacteristic
661C2#2933011yescharacteristic
662C2#2933611yescharacteristic
663C2#2995411yescharacteristic
664C2#2995511yescharacteristic
665C2#3004011yescharacteristic
666C2#3004311yescharacteristic
667C2#3021911yescharacteristic
668C2#3022711yescharacteristic
669C2#3275911yescharacteristic
670C2#3276111yescharacteristic
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. #14712C2semidirect
3. #11268C2semidirect
4. #11067C2semidirect
6. #11050C2semidirect
7. #11050C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
#147122
#147082
#112682
#110672
#110662
#110502
#110492
#110092
#107952