Finite Group Databasebeta

← #364768

Subgroups — #364768

Order 512.

1528 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#446C211yesmaximal, characteristic, complemented
2C2×#2396C211yesmaximal, characteristic
3C2×#1985C211yesmaximal, characteristic, complemented
4C2×#1798C211yesmaximal, characteristic
5C2×#1750C211yesmaximal, characteristic
6#11202C212yesmaximal, complemented
7#11042C212yesmaximal
8#10922C212yesmaximal
9#10875C212yesmaximal, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C22×C4×Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C2×#446—24no—
15C2×#446—24no—
16C2×#446—24no—
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2212yes—
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2396—24no—
27#2396—24no—
28C2×#400C411yescharacteristic
29C2×#399C411yescharacteristic
30C2×#398C412yes—
31C2×#396C412yes—
32C2×#395C411yescharacteristic
33C2×#393C411yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37#1985—24no—
38#1798—24no—
39#1750—24no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44C2×#336C2211yescharacteristic
45C2×#336C2212yes—
46C2×#336C2211yescharacteristic
47C2×#336C2212yes—
48C2×#336C2212yes—
49C23×Q8C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8D414yes—
62C2×C4×Q8D414yes—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C412yes—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C412yes—
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2312yes—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#446—232no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8D414yes—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8Q814yescomplemented
103C2×#97—24no—
104C2×#97—24no—
105C2×#97—24no—
106C2×#97—22no—
107C2×#97—22no—
108C2×#97—22no—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8D414yes—
112#400—24no—
113#399—24no—
114#398—28no—
115#396—28no—
116#395—24no—
117#393—24no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C22⋊C4—24no—
123C4×C22⋊C4—24no—
124C2×#95C2×C412yes—
125C2×#95C2×C411yescharacteristic
126C2×#95C2×C412yes—
127C2×#95C2×C411yescharacteristic
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130#336—24no—
131#336—22no—
132#336—22no—
133#336—24no—
134#336—24no—
135#336—24no—
136#336—24no—
137#336—42no—
138#336—42no—
139C25C22×C411yescharacteristic
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8C22⋊C414yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×Q812yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8C2×D412yes—
205C22×C8C2×D414yes—
206C22×C8C2×D412yes—
207C22×C8C2×Q812yes—
208C22×C8C2×Q812yes—
209C22×C8#4112yes—
210C22×C8#4112yes—
211C4×Q8—28no—
212C4×Q8—28no—
213C4×Q8—28no—
214C4×Q8—28no—
215#117—44no—
216#117—44no—
217#117—44no—
218#117—44no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4C22⋊C418yes—
239C2×C4⋊C4C22⋊C414yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42C2×D414yes—
269C2×C42C2×D414yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42#4112yes—
275C2×C42#4112yes—
276C4⋊C8—216no—
277C4⋊C8—28no—
278C4⋊C8—44no—
279C22⋊C8—28no—
280C22⋊C8—28no—
281C22⋊C8—48no—
282#97—216no—
283#97—44no—
284C4×C8—28no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—22no—
289#95—22no—
290#95—24no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24C4×Q812yes—
302C24C23×C411yescharacteristic
303C2×Q8—28no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—44no—
307C2×Q8—44no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
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×C4C2×C22⋊C414yes—
382C22×C4C2×C22⋊C414yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C4×D412yes—
386C22×C4C4×D412yes—
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4C4×Q812yes—
390C22×C4#13114yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×Q811yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C42−1+411yescharacteristic
403C22×C42−1+411yescharacteristic
404C22×C42−1+411yescharacteristic
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—42no—
426C2×C8—41no—
427C2×C8—42no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—42no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8C22⋊Q814yes—
447C4⋊C4—28no—
448C4⋊C4—216no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C42—24no—
459C42—24no—
460C42—24no—
461C42—24no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42#12014yes—
471C42#12114yes—
472C42#12114yes—
473C42#12314yes—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23C22×C22⋊C411yescharacteristic
498C23C2×C4×D411yescharacteristic
499C23C2×C4×D411yescharacteristic
500C23C2×C4×Q811yescharacteristic
501C23C4×#4111yescharacteristic
502C23#51911yescharacteristic
503C23#52011yescharacteristic
504C23C2×#13112yes—
505C23#56812yes—
506C23#56812yes—
507C23#56812yes—
508Q8—416no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4#38614yes—
571C2×C4#38614yes—
572C2×C4C2×#12011yescharacteristic
573C2×C4C2×#12111yescharacteristic
574C2×C4C2×#12111yescharacteristic
575C2×C4C2×C22⋊Q811yescharacteristic
576C2×C4C2×#12311yescharacteristic
577C2×C4#52511yescharacteristic
578C2×C4#52911yescharacteristic
579C2×C4#53311yescharacteristic
580C2×C4#53611yescharacteristic
581C2×C4#53711yescharacteristic
582C2×C4#54111yescharacteristic
583C2×C4#54211yescharacteristic
584C2×C4#54611yescharacteristic
585C2×C4#54711yescharacteristic
586C2×C4#54811yescharacteristic
587C2×C4#54811yescharacteristic
588C2×C4#54811yescharacteristic
589C2×C4#54811yescharacteristic
590C2×C4#54811yescharacteristic
591C2×C4D4×Q811yescharacteristic
592C2×C4D4×Q811yescharacteristic
593C2×C4D4×Q811yescharacteristic
594C2×C4#55011yescharacteristic
595C2×C4#55011yescharacteristic
596C2×C4#55211yescharacteristic
597C2×C4#55211yescharacteristic
598C2×C4#55411yescharacteristic
599C2×C4#55611yescharacteristic
600C8—24no—
601C8—42no—
602C8—44no—
603C8—42no—
604C8—42no—
605C8—82no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—44no—
611C22—82no—
612C22—82no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22C2×#38611yescharacteristic
618C22C4×C22⋊Q811yescharacteristic
619C22#232311yescharacteristic
620C22Q8×C22⋊C411yescharacteristic
621C22#234111yescharacteristic
622C22#236011yescharacteristic
623C22#236211yescharacteristic
624C22#287912yes—
625C22#288012yes—
626C22#293012yes—
627C22#293011yescharacteristic
628C22#293011yescharacteristic
629C22#293111yescharacteristic
630C22#293211yescharacteristic
631C22#293512yes—
632C22#296112yes—
633C22#296212yes—
634C22#296712yes—
635C22#296812yes—
636C22#298512yes—
637C22#298712yes—
638C22#299212yes—
639C22#299512yes—
640C4—24no—
641C4—24no—
642C4—44no—
643C4—42no—
644C4—44no—
645C4—44no—
646C4—82no—
647C4—81no—
648C4—81no—
649C4#240711yescharacteristic
650C4#240911yescharacteristic
651C4#242111yescharacteristic
652C4#242311yescharacteristic
653C4#242611yescharacteristic
654C4#242911yescharacteristic
655C4#253611yescharacteristic
656C4#253811yescharacteristic
657C2—82no—
658C2#1396911yescharacteristic
659C2#2089611yescharacteristic
660C2#2090011yescharacteristic
661C2#2160311yescharacteristic
662C2#2160411yescharacteristic
663C2#2210311yescharacteristic
664C2#2210511yescharacteristic
665C2#2251112yes—
666C2#2253312yes—
667C2#2296112yes—
668C2#2305912yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#446C2semidirect
3. C2×#1985C2semidirect
6. #11202C2semidirect
9. #10875C2semidirect
10. #10835C2semidirect
11. #10834C2semidirect
102. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4462
C2×#23962
C2×#19852
C2×#17982
C2×#17502
#112022
#110422
#109222
#108752
#108352
#108342