Finite Group Databasebeta

← #401807

Subgroups — #401807

Order 512.

1486 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2362C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#2036C212yesmaximal
4C2×#1803C211yesmaximal, characteristic, complemented
5#15150C211yesmaximal, characteristic
6#11227C211yesmaximal, characteristic, complemented
7#11190C212yesmaximal, complemented
8#11042C211yesmaximal, characteristic
9#10907C212yesmaximal, complemented
10#10789C211yesmaximal, characteristic, complemented
11#10731C212yesmaximal
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2212yes—
20C2×#397—21no—
21C2×#397—21no—
22#2362—24no—
23#2362—24no—
24C2×#391C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388—21no—
27C2×#388—21no—
28C2×#387—21no—
29C2×#384C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#2083—24no—
35#2036—28no—
36#1803—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1445—21no—
45#1445—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8D414yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—24no—
60C2×C4×Q8—42no—
61C2×C4×Q8D414yes—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—24no—
65C2×#117—42no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C42—21no—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—24no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—42no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—42no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—42no—
100#397—44no—
101#397—44no—
102#391—22no—
103#391—22no—
104#389—22no—
105#389D414yes—
106#388—28no—
107#388—28no—
108#387—28no—
109#384—22no—
110#384—22no—
111#380—24no—
112#380—24no—
113C4×C4⋊C4—28no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95D411yescharacteristic
128C2×#95D411yescharacteristic
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—42no—
134#336—24no—
135#336—22no—
136C25C2×D411yescharacteristic
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8C2×D412yes—
145C22×Q8C2×D412yes—
146C22×Q8#4112yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8#4114yes—
183C22⋊Q8—28no—
184C22⋊Q8—28no—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C4×Q8—28no—
188C4×Q8—44no—
189C4×Q8—48no—
190#117—48no—
191#117—44no—
192#117—44no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4#4112yes—
232C2×C4⋊C4#4114yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262#107—48no—
263C4⋊C8—44no—
264#103—48no—
265#103—48no—
266C22⋊C8—48no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—22no—
272#95—22no—
273#95—42no—
274#95—44no—
275#95—44no—
276#95—42no—
277#95—44no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—22no—
287C24—22no—
288C24#12112yes—
289C24C22×D411yescharacteristic
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8#12114yes—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4#12012yes—
382C22×C4#12012yes—
383C22×C4#12112yes—
384C22×C4#13512yes—
385C22×C4AGL(1,ℤ/8)12yes—
386C22×C4AGL(1,ℤ/8)12yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4112yes—
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C42−1+412yes—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—216no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—48no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—48no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4#12114yes—
433C4⋊C4#12314yes—
434C22⋊C4—216no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C22⋊C4—88no—
439C42—22no—
440C42—22no—
441C42—24no—
442C42—48no—
443C42—48no—
444C42—48no—
445C42—41no—
446C42—41no—
447C42—42no—
448C23—22no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23C2×#12011yescharacteristic
472C23C2×#12111yescharacteristic
473C23#52511yescharacteristic
474C23#54612yes—
475C23#54712yes—
476C23C2×#13511yescharacteristic
477C23C2×AGL(1,ℤ/8)11yescharacteristic
478C23C2×AGL(1,ℤ/8)11yescharacteristic
479C23#57511yescharacteristic
480C23#57812yes—
481C23#57812yes—
482Q8—24no—
483Q8—24no—
484Q8—42no—
485Q8—42no—
486Q8—44no—
487Q8—44no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4#44914yes—
555C2×C4C2×#12111yescharacteristic
556C2×C4C2×#12111yescharacteristic
557C2×C4C2×#12311yescharacteristic
558C2×C4#52511yescharacteristic
559C2×C4#52511yescharacteristic
560C2×C4#52912yes—
561C2×C4#52912yes—
562C2×C4#53211yescharacteristic
563C2×C4#53411yescharacteristic
564C2×C4#53611yescharacteristic
565C2×C4#53912yes—
566C2×C4#54212yes—
567C2×C4#54612yes—
568C2×C4#54612yes—
569C2×C4#54712yes—
570C2×C4#54812yes—
571C2×C4#55212yes—
572C2×C4#55312yes—
573C8—44no—
574C8—82no—
575C22—44no—
576C22—44no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—81no—
581C22—81no—
582C22—82no—
583C22—81no—
584C22—82no—
585C22—81no—
586C22#240511yescharacteristic
587C22#242512yes—
588C22#243211yescharacteristic
589C22#246812yes—
590C22#284711yescharacteristic
591C22C2×#44911yescharacteristic
592C22#300611yescharacteristic
593C22#301111yescharacteristic
594C22#301111yescharacteristic
595C22#303511yescharacteristic
596C22#303811yescharacteristic
597C22#304412yes—
598C22#310611yescharacteristic
599C22#311511yescharacteristic
600C22#312712yes—
601C22#330012yes—
602C22#330812yes—
603C22#331512yes—
604C22#333012yes—
605C22#334212yes—
606C22#334312yes—
607C22#334912yes—
608C22#335312yes—
609C4—24no—
610C4—42no—
611C4—44no—
612C4—42no—
613C4—44no—
614C4—42no—
615C4—44no—
616C4—81no—
617C4—82no—
618C4—162no—
619C4—162no—
620C4#239511yescharacteristic
621C4#242712yes—
622C4#246012yes—
623C4#248611yescharacteristic
624C4#254311yescharacteristic
625C4#263811yescharacteristic
626C2—82no—
627C2#1482411yescharacteristic
628C2#2403511yescharacteristic
629C2#2405411yescharacteristic
630C2#2520112yes—
631C2#2521712yes—
632C2#2570511yescharacteristic
633C2#2571611yescharacteristic
634C2#2710312yes—
635C2#2711812yes—
636C2#3290411yescharacteristic
637C2#3291511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2083C2semidirect
4. C2×#1803C2semidirect
6. #11227C2semidirect
7. #11190C2semidirect
9. #10907C2semidirect
10. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23622
C2×#20832
C2×#20362
C2×#18032
#151502
#112272
#111902
#110422
#109072
#107892
#107312