Finite Group Databasebeta

← #364783

Subgroups — #364783

Order 512.

1818 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34952C211yesmaximal, characteristic
2C2×C42×Q8C211yesmaximal, characteristic, complemented
3#21303C211yesmaximal, characteristic, complemented
4C2×#1806C211yesmaximal, characteristic
5C2×C4×#104C211yesmaximal, characteristic
6#11226C212yesmaximal, complemented
7#11075C212yesmaximal
8#10923C212yesmaximal
9#10878C212yesmaximal, complemented
10#10851C211yesmaximal, characteristic, complemented
11#10839C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C22×#104C2211yescharacteristic
17C22×#104C2212yes—
18C22×#104C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2582C412yescomplemented
21#2581C412yescomplemented
22#2580C412yescomplemented
23#2579—22no—
24#2538C412yescomplemented
25#2487—21no—
26#2409—21no—
27#2396—22no—
28C4×C22⋊Q8—22no—
29#2297C2212yes—
30#2296C2212yes—
31#2285C2211yescharacteristic
32C42×Q8—24no—
33C42×Q8—24no—
34C2×C4×C4⋊C4—22no—
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C43C2212yes—
39C2×C43C2211yescharacteristic
40#1985—22no—
41#1984—22no—
42#1806—24no—
43C4×#104—24no—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2211yescharacteristic
47C2×#339C2212yes—
48C2×#339C2211yescharacteristic
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1728C2212yes—
52#1728C2212yes—
53C22×#41C2311yescharacteristic
54C23×Q8D411yescharacteristic
55C22×M16C2312yes—
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C2×C22⋊Q8—22no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41—22no—
63C2×C4×Q8—216no—
64C2×C4×Q8—28no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C412yes—
72C2×C4×Q8C2×C412yes—
73C2×C4×Q8D412yes—
74C2×C4×Q8D412yes—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2×C411yescharacteristic
79C2×#117—21no—
80C2×#117C2311yescharacteristic
81C2×#117C2312yes—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4D412yes—
85C22×C4⋊C4D411yescharacteristic
86C22×C42—22no—
87C22×C42—21no—
88C22×C42—22no—
89C22×C42—21no—
90C22×C42—21no—
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#423—22no—
97#423—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—24no—
106C2×#104—24no—
107C2×#104—42no—
108C2×#104Q814yescomplemented
109C4×M16—22no—
110C4×M16—22no—
111C2×#97—21no—
112C2×C4×C8—21no—
113#400—22no—
114#399—22no—
115#398—24no—
116#396—24no—
117#395—22no—
118#393—22no—
119#391—22no—
120#391C2×C414yes—
121#390C2×C414yes—
122#389—22no—
123#389C2×C414yes—
124#389C2×C414yes—
125#387—24no—
126#386—22no—
127#386—22no—
128#385—22no—
129#385—22no—
130#385—24no—
131#384C2×C412yes—
132#382C2×C412yes—
133#380—21no—
134#380—21no—
135#380—22no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—216no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4D414yes—
145C4×C22⋊C4—24no—
146C2×#95—22no—
147C2×#95—21no—
148C43—24no—
149C43—24no—
150C43—22no—
151C43D414yes—
152#339—24no—
153#339—24no—
154#339—22no—
155#339—22no—
156#339—24no—
157#339—42no—
158#339—42no—
159#336—24no—
160#336—24no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41C2×Q812yes—
172C2×#41#4112yes—
173C22×Q8—28no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8C22⋊C412yes—
177C22×Q8C22⋊C412yes—
178C22×Q8C22×C411yescharacteristic
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2411yescharacteristic
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16C2×Q812yes—
209C2×M16#4112yes—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—22no—
213C22⋊Q8—48no—
214C4×Q8—264no—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—24no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—42no—
232C4×D4—42no—
233C4×D4—42no—
234C4×D4—42no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—22no—
243#117—42no—
244#117—42no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4C22⋊C412yes—
260C2×C4⋊C4C22⋊C414yes—
261C2×C4⋊C4C22⋊C414yes—
262C2×C4⋊C4C22⋊C412yes—
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C414yes—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C42—28no—
273C2×C42—216no—
274C2×C42—28no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—28no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—22no—
299C2×C42—44no—
300C2×C42—48no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42C22×C414yes—
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C2×D414yes—
307C2×C42C2×D412yes—
308C2×C42C2×D414yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×Q812yes—
315C2×C42#4112yes—
316C2×C42#4112yes—
317C2×C42#4112yes—
318C4⋊C8—42no—
319C4⋊C8—42no—
320#104—28no—
321#104—28no—
322#104—48no—
323#97—42no—
324C4×C8—42no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—24no—
333C24—21no—
334C24C22×D411yescharacteristic
335#41—22no—
336#41—22no—
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—41no—
344#41—41no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349#41—41no—
350#41—41no—
351C2×Q8—216no—
352C2×Q8—216no—
353C2×Q8—21no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—28no—
357C2×Q8—24no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8#10418yescomplemented
363C2×Q8C4×Q812yescomplemented
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4C4×Q812yescomplemented
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—82no—
413C22×C4C2×C22⋊C414yes—
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C414yes—
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4#12012yes—
420C22×C4#12012yes—
421C22×C4#12312yes—
422C22×C4#12312yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×Q811yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C42−1+411yescharacteristic
435C22×C42−1+411yescharacteristic
436C22×C42−1+411yescharacteristic
437M16—22no—
438M16—22no—
439M16—22no—
440M16—22no—
441M16—41no—
442M16—41no—
443M16—41no—
444M16—41no—
445M16—42no—
446M16—42no—
447M16—42no—
448M16—42no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C4⋊C4—232no—
459C4⋊C4—216no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—48no—
483C22⋊C4—84no—
484C42—232no—
485C42—216no—
486C42—24no—
487C42—24no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—24no—
497C42—48no—
498C42—416no—
499C42—41no—
500C42—41no—
501C42—42no—
502C42—42no—
503C42—41no—
504C42—41no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42C4×D412yes—
510C42C4×D412yes—
511C42C4×D412yes—
512C42C4×D412yes—
513C42#12112yes—
514C42#12112yes—
515C42#12112yes—
516C42#12112yes—
517C42C22⋊Q814yescomplemented
518C23—22no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23C22×C22⋊C411yescharacteristic
526C23C2×#12011yescharacteristic
527C23C2×#12311yescharacteristic
528C23#54811yescharacteristic
529C23#54811yescharacteristic
530C23D4×Q811yescharacteristic
531C23#55011yescharacteristic
532Q8—216no—
533Q8—216no—
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538D4—22no—
539D4—22no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—41no—
545D4—41no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4#38612yes—
594C2×C4#38612yes—
595C2×C4#38612yes—
596C2×C4#38612yes—
597C2×C4C2×#10418yes—
598C2×C4C2×#10414yes—
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C2×C4×D411yescharacteristic
601C2×C4C2×C4×Q811yescharacteristic
602C2×C4C4×#4111yescharacteristic
603C2×C4#51911yescharacteristic
604C2×C4#52011yescharacteristic
605C2×C4C2×#12111yescharacteristic
606C2×C4C2×#12111yescharacteristic
607C2×C4C2×C22⋊Q811yescharacteristic
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—42no—
628C8—42no—
629C8—82no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—44no—
639C22—42no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22C2×#38611yescharacteristic
644C22Q8×C22⋊C411yescharacteristic
645C22#234111yescharacteristic
646C22#240711yescharacteristic
647C22#240911yescharacteristic
648C22#242111yescharacteristic
649C22#242311yescharacteristic
650C22C22×#10412yes—
651C22#290112yes—
652C4—28no—
653C4—22no—
654C4—28no—
655C4—22no—
656C4—22no—
657C4—44no—
658C4—48no—
659C4—42no—
660C4—44no—
661C4—84no—
662C4—81no—
663C4—81no—
664C4#189614yes—
665C4C4×C22⋊Q811yescharacteristic
666C4#232311yescharacteristic
667C4#236011yescharacteristic
668C4#236211yescharacteristic
669C4#242611yescharacteristic
670C4#242911yescharacteristic
671C4#253611yescharacteristic
672C4#253811yescharacteristic
673C2—21no—
674C2—21no—
675C2—21no—
676C2—21no—
677C2—82no—
678C2#1396911yescharacteristic
679C2C2×#189611yescharacteristic
680C2#2098011yescharacteristic
681C2Q8×#10412yes—
682C2#2255012yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C42×Q8C2semidirect
3. #21303C2semidirect
6. #11226C2semidirect
9. #10878C2semidirect
10. #10851C2semidirect
11. #10839C2semidirect
20. #2582C4semidirect
21. #2581C4semidirect
22. #2580C4semidirect
24. #2538C4semidirect
108. C2×#104Q8semidirect
362. C2×Q8#104semidirect
363. C2×Q8C4×Q8semidirect
375. C2×D4C4×Q8semidirect
517. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#349522
C2×C42×Q82
#213032
C2×#18062
C2×C4×#1042
#112262
#110752
#109232
#108782
#108512
#108392