Finite Group Databasebeta

← #364812

Subgroups — #364812

Order 512.

1275 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60192C211yesmaximal, characteristic
2#35179C211yesmaximal, characteristic
3#21332C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C4×#419C211yesmaximal, characteristic
6#11225C212yesmaximal
7#11074C212yesmaximal
8#10947C212yesmaximal
9#10889C212yesmaximal
10#10837C211yesmaximal, characteristic
11#10836C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14Q8×M16—22no—
15Q8×M16—22no—
16#2962—22no—
17#2962—22no—
18#2961—22no—
19#2960—22no—
20#2905C2212yes—
21C2×#423C2211yescharacteristic
22C2×#419C2211yescharacteristic
23C2×#419C2212yes—
24C2×#419C2211yescharacteristic
25#2873C2212yes—
26C2×C4×M16C2211yescharacteristic
27#2855—22no—
28#2627—22no—
29#2582—22no—
30#2581—22no—
31#2568C412yescomplemented
32#2567C412yescomplemented
33#2565C414yescomplemented
34#2317C2212yes—
35#2310C2212yes—
36#2285C2211yescharacteristic
37#2285C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1996—22no—
40#1996—22no—
41#1805—22no—
42#1803—22no—
43#1763—22no—
44C4×#103—22no—
45#1740C2212yes—
46#1740C2212yes—
47#1736C2211yescharacteristic
48#1736C2212yes—
49#1736C2211yescharacteristic
50C2×H(3,ℤ/4)C2212yes—
51C2×H(3,ℤ/4)C2212yes—
52C2×H(3,ℤ/4)C2212yes—
53C2×H(3,ℤ/4)C2212yes—
54C23×Q8C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2312yes—
57C22×M16C2311yescharacteristic
58#533—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62#533—21no—
63#533—21no—
64#533—21no—
65#533—21no—
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×#125C2×C412yes—
69C2×C22⋊Q8D412yes—
70C2×C22⋊Q8D412yes—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8C2×C412yes—
82C2×C4×Q8C2×C412yes—
83C2×C4×Q8D412yes—
84C2×C4×Q8D412yes—
85C2×#117C2312yes—
86C2×#117C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2312yes—
89C2×#117C2312yes—
90C22×C4⋊C4—21no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94#446—24no—
95#446—24no—
96#446—28no—
97#446—28no—
98C8×Q8—24no—
99C8×Q8—24no—
100#424—24no—
101#424—24no—
102#423—21no—
103#423—21no—
104#423—21no—
105#423—21no—
106#423—24no—
107C2×C4⋊C8D412yes—
108C2×C4⋊C8D412yes—
109#419—22no—
110#419—22no—
111#419—22no—
112#419—24no—
113#419—42no—
114#419Q814yescomplemented
115C2×#103—22no—
116C2×#103—22no—
117C2×#103—24no—
118#405—24no—
119#405—24no—
120C4×M16—21no—
121C4×M16—21no—
122C4×M16—21no—
123C4×M16—21no—
124C4×M16—24no—
125C2×#97D412yes—
126C2×C4×C8D412yes—
127#400—24no—
128#398—28no—
129#395—24no—
130#391—24no—
131#391—24no—
132#391C2×C414yes—
133#391C2×C414yes—
134#389—24no—
135#389—24no—
136#387—28no—
137#385—24no—
138#384—21no—
139#384—21no—
140#384C2×C414yes—
141#382—21no—
142#382—21no—
143#382C2×C414yes—
144#381—24no—
145#380—21no—
146#380—21no—
147#380—21no—
148#380—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4C2×C412yes—
154C4×C4⋊C4C2×C412yes—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157#376—21no—
158C43—21no—
159#342—24no—
160#342—24no—
161#340—24no—
162#340—22no—
163#340—22no—
164H(3,ℤ/4)—24no—
165H(3,ℤ/4)—24no—
166H(3,ℤ/4)—42no—
167H(3,ℤ/4)—42no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8C22⋊C412yes—
180C22×Q8C22⋊C412yes—
181C22×Q8C22×C412yes—
182C22×Q8C2×Q812yes—
183C22×Q8#4112yes—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2411yescharacteristic
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—22no—
195C2×M16—21no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—22no—
201C2×M16—21no—
202C2×M16—22no—
203C2×M16—21no—
204C2×M16—22no—
205C2×M16C2×Q812yes—
206C2×M16#4112yes—
207C22×C8C2×D412yes—
208C22×C8C2×D414yes—
209C22×C8C2×D412yes—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218#125—22no—
219#125—22no—
220#125—22no—
221#125—22no—
222C22⋊Q8—22no—
223C22⋊Q8—22no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—41no—
247C4×Q8—41no—
248C4×Q8—41no—
249C4×Q8—41no—
250C4×Q8—41no—
251C4×Q8—41no—
252C4×Q8—41no—
253C4×Q8—41no—
254#117—22no—
255#117—22no—
256#117—22no—
257#117—22no—
258#117—22no—
259#117—22no—
260#117—22no—
261#117—21no—
262#117—22no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117—22no—
272#117C2×Q812yes—
273#117#4112yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—41no—
289C2×C4⋊C4—41no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4C22⋊C414yes—
295C2×C4⋊C4C22⋊C414yes—
296C2×C4⋊C4C22⋊C412yes—
297C2×C4⋊C4C22⋊C412yes—
298C2×C4⋊C4C22×C412yes—
299C2×C4⋊C4C22×C412yes—
300C2×C4⋊C4C22×C412yes—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4C2×D414yes—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—41no—
312C2×C42—41no—
313C2×C42—41no—
314C2×C42—41no—
315C2×C42—41no—
316C2×C42—41no—
317C2×C42C22×C414yes—
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C2×D414yes—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42#4111yescharacteristic
328C2×C42#4111yescharacteristic
329C2×C42#4111yescharacteristic
330C2×C42#4111yescharacteristic
331C4⋊C8—28no—
332C4⋊C8—28no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337#103—28no—
338#103—28no—
339#103—48no—
340#97—28no—
341#97—22no—
342#97—22no—
343C4×C8—28no—
344C4×C8—22no—
345C4×C8—22no—
346#95—22no—
347#95—22no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—41no—
352#95—41no—
353#95—41no—
354#95—41no—
355C24C22×D411yescharacteristic
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—44no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×Q8—82no—
389C2×Q8C4×Q814yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—81no—
406C22×C4—82no—
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C414yes—
409C22×C4C2×C22⋊C414yes—
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12111yescharacteristic
415C22×C4#12111yescharacteristic
416C22×C4#12111yescharacteristic
417C22×C4#12111yescharacteristic
418C22×C4#12111yescharacteristic
419C22×C4#12111yescharacteristic
420C22×C4#12111yescharacteristic
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×Q811yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4H(5,𝔽2)11yescharacteristic
432C22×C42−1+411yescharacteristic
433C22×C42−1+411yescharacteristic
434C22×C42−1+411yescharacteristic
435M16—22no—
436M16—22no—
437M16—22no—
438M16—22no—
439M16—42no—
440M16—42no—
441M16—41no—
442M16—41no—
443M16—41no—
444M16—42no—
445M16—42no—
446M16—41no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8C22⋊Q814yes—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—44no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—88no—
488C4⋊C4—82no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—42no—
494C22⋊C4—41no—
495C22⋊C4—42no—
496C22⋊C4—41no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C42—24no—
500C42—24no—
501C42—24no—
502C42—24no—
503C42—22no—
504C42—22no—
505C42—41no—
506C42—41no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—82no—
514C42—82no—
515C42C4×D411yescharacteristic
516C42C4×D411yescharacteristic
517C42C4×D411yescharacteristic
518C42C4×D411yescharacteristic
519C42C4×D411yescharacteristic
520C42C4×D411yescharacteristic
521C42C4×D411yescharacteristic
522C42C4×D411yescharacteristic
523C42#12011yescharacteristic
524C42#12011yescharacteristic
525C42#12011yescharacteristic
526C42#12011yescharacteristic
527C42#12311yescharacteristic
528C42#12311yescharacteristic
529C42#12311yescharacteristic
530C42#12311yescharacteristic
531C23—41no—
532C23—41no—
533C23C22×C22⋊C411yescharacteristic
534C23C2×#12111yescharacteristic
535C23C2×#12111yescharacteristic
536C23#54811yescharacteristic
537C23#54811yescharacteristic
538C23D4×Q811yescharacteristic
539C23#55011yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—44no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—44no—
548Q8—42no—
549Q8—42no—
550Q8—82no—
551Q8—82no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4#38611yescharacteristic
592C2×C4#38611yescharacteristic
593C2×C4#38611yescharacteristic
594C2×C4#38611yescharacteristic
595C2×C4#38611yescharacteristic
596C2×C4#38611yescharacteristic
597C2×C4#38611yescharacteristic
598C2×C4#38611yescharacteristic
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C2×C4×D411yescharacteristic
601C2×C4C2×C4×Q811yescharacteristic
602C2×C4C4×#4111yescharacteristic
603C2×C4#51911yescharacteristic
604C2×C4#52011yescharacteristic
605C2×C4C2×#12011yescharacteristic
606C2×C4C2×C22⋊Q811yescharacteristic
607C2×C4C2×#12311yescharacteristic
608C2×C4#52511yescharacteristic
609C2×C4#52911yescharacteristic
610C2×C4#53311yescharacteristic
611C2×C4#53611yescharacteristic
612C2×C4#53711yescharacteristic
613C2×C4#54111yescharacteristic
614C2×C4#54211yescharacteristic
615C2×C4#54611yescharacteristic
616C2×C4#54711yescharacteristic
617C2×C4#54811yescharacteristic
618C2×C4#54811yescharacteristic
619C2×C4#54811yescharacteristic
620C2×C4D4×Q811yescharacteristic
621C2×C4D4×Q811yescharacteristic
622C2×C4#55011yescharacteristic
623C2×C4#55211yescharacteristic
624C2×C4#55211yescharacteristic
625C2×C4#55411yescharacteristic
626C2×C4#55611yescharacteristic
627C8—24no—
628C8—42no—
629C8—44no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22C2×#38611yescharacteristic
641C22Q8×C22⋊C411yescharacteristic
642C22#234111yescharacteristic
643C22#242611yescharacteristic
644C22#242911yescharacteristic
645C22#253611yescharacteristic
646C22#253811yescharacteristic
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—84no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—162no—
662C4#190912yes—
663C4C4×C22⋊Q811yescharacteristic
664C4#232311yescharacteristic
665C4#236011yescharacteristic
666C4#236211yescharacteristic
667C4#240711yescharacteristic
668C4#240911yescharacteristic
669C4#242111yescharacteristic
670C4#242311yescharacteristic
671C2—21no—
672C2—21no—
673C2—21no—
674C2—21no—
675C2—21no—
676C2—21no—
677C2#1396911yescharacteristic
678C2C2×#190911yescharacteristic
679C2#2100911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #2568C4semidirect
32. #2567C4semidirect
33. #2565C4semidirect
114. #419Q8semidirect

Maximal subgroups

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

subgroupindex
#601922
#351792
#213322
#206472
C4×#4192
#112252
#110742
#109472
#108892
#108372
#108362