Finite Group Databasebeta

← #360992

Subgroups — #360992

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#135C211yesmaximal, characteristic
2C2×#1929C212yesmaximal, complemented
3#20684C211yesmaximal, characteristic, complemented
4#20378C211yesmaximal, characteristic, complemented
5#11125C212yesmaximal, complemented
6#11025C212yesmaximal
7#10909C212yesmaximal, complemented
8#10864C211yesmaximal, characteristic, complemented
9#10797C211yesmaximal, characteristic, complemented
10#10790C211yesmaximal, characteristic, complemented
11#10790C211yesmaximal, characteristic, complemented
12C22×#135C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C4×#135—28no—
16C2×C4×Q16C411yescharacteristic, complemented
17C2×C4×Q16C411yescharacteristic, complemented
18C2×C4×SD16C411yescharacteristic, complemented
19C2×C4×SD16C411yescharacteristic, complemented
20C2×C4×SD16C411yescharacteristic, complemented
21C2×C4×SD16C411yescharacteristic, complemented
22C2×C4×D8C411yescharacteristic, complemented
23C2×C4×D8C411yescharacteristic, complemented
24C22×#108C2212yes—
25C2×#427C2211yescharacteristic
26C22×#104C2212yes—
27C22×#104C2212yes—
28C2×#416C2211yescharacteristic
29C2×#416C2211yescharacteristic
30C2×#408C2212yes—
31C2×#408C2212yes—
32C2×#405C2212yes—
33C22×C4×C8C2211yescharacteristic
34#1929—28no—
35#1818—22no—
36#1817—21no—
37#1817—21no—
38#1750—22no—
39#1749—21no—
40C4×C22⋊C8—21no—
41C2×#341C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44#1729C2212yes—
45#1728C2211yescharacteristic
46#1728C2211yescharacteristic
47C2×#336C2212yes—
48C2×#336C2212yes—
49C2×#336C2211yescharacteristic
50C4×#95C2211yescharacteristic
51C22×#41C2311yescharacteristic
52C22×#41C2311yescharacteristic
53C2×#135—24no—
54C2×#135—24no—
55C2×#135—22no—
56C2×#135—22no—
57C22×Q16C2×C411yescharacteristic
58C22×SD16C2×C411yescharacteristic
59C22×SD16C2×C411yescharacteristic
60C22×D8C2×C411yescharacteristic
61C22×M16C2312yes—
62C22×M16C2312yes—
63C23×C8C2312yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C4×#41—24no—
67C4×#41—24no—
68C4×#41—42no—
69C4×#41—42no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×#117—21no—
83C2×#117—21no—
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90C4×Q16—24no—
91C4×Q16—24no—
92C4×SD16—24no—
93C4×SD16—24no—
94C4×SD16—24no—
95C4×SD16—24no—
96C4×D8—24no—
97C4×D8—24no—
98C2×#108—24no—
99C2×#108—24no—
100C2×#108—42no—
101#427—24no—
102C2×#107C2×C411yescharacteristic
103C2×#107C2×C411yescharacteristic
104C2×#106C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
106C2×#104—24no—
107C2×#104—24no—
108C2×#104—24no—
109C2×#104—42no—
110C2×#104—42no—
111C2×#104—42no—
112#416—24no—
113#416D418yes—
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122#408—24no—
123#408—24no—
124#408—24no—
125#408—42no—
126C2×C22⋊C8—22no—
127C2×C22⋊C8—22no—
128C2×C22⋊C8—22no—
129C2×C22⋊C8—22no—
130#405—24no—
131#405—42no—
132C2×#97—22no—
133C2×C4×C8—24no—
134C2×C4×C8—22no—
135C2×C4×C8C2×C411yescharacteristic
136C2×C4×C8C2×C411yescharacteristic
137C2×C4×C8C2×C411yescharacteristic
138C2×C4×C8C2×C411yescharacteristic
139C2×#95—21no—
140C2×#95—21no—
141#341—24no—
142#339—24no—
143#339—42no—
144#336—22no—
145#336—22no—
146#336—24no—
147#336—24no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336D411yescharacteristic
153#336D411yescharacteristic
154#336D412yes—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—22no—
164C2×#41—22no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41C2×D414yes—
170C2×#41#4114yes—
171C22×Q8C22×C411yescharacteristic
172C22×Q8C22×C411yescharacteristic
173C22×D4—21no—
174C22×D4—21no—
175C22×D4C22×C411yescharacteristic
176C22×D4C22×C411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4#4111yescharacteristic
185C23×C4#4111yescharacteristic
186C23×C4C2411yescharacteristic
187#135—216no—
188#135—216no—
189C2×Q16—22no—
190C2×Q16—22no—
191C2×Q16—22no—
192C2×SD16—22no—
193C2×SD16—22no—
194C2×SD16—22no—
195C2×SD16—22no—
196C2×SD16—22no—
197C2×SD16—22no—
198C2×D8—22no—
199C2×D8—22no—
200C2×D8—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16#4114yes—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8C22×C411yescharacteristic
227C22×C8C22×C411yescharacteristic
228C22×C8C22×C411yescharacteristic
229C22×C8C22×C411yescharacteristic
230C22×C8C2×D412yes—
231C22×C8C2×D412yes—
232C22×C8C2×D414yes—
233C22×C8#4112yes—
234C22×C8#4112yes—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—41no—
240C4×Q8—41no—
241C4×Q8—41no—
242C4×Q8—41no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—41no—
248C4×D4—41no—
249C4×D4—41no—
250C4×D4—41no—
251C4×D4—44no—
252C4×D4—44no—
253C4×D4—44no—
254C4×D4—44no—
255#117—22no—
256#117—22no—
257#117—22no—
258#117—44no—
259#117—44no—
260#117C2×D414yes—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—22no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295#108—48no—
296#107—22no—
297#107—22no—
298#106—22no—
299#106—22no—
300#104—48no—
301#104—48no—
302#103—22no—
303#103—22no—
304#103C22⋊C414yes—
305#103C22⋊C414yes—
306#102—22no—
307#102—22no—
308#102C22⋊C414yes—
309#102C22⋊C414yes—
310C22⋊C8—28no—
311C22⋊C8—28no—
312C22⋊C8—44no—
313C22⋊C8—44no—
314#97—44no—
315C4×C8—22no—
316C4×C8—22no—
317C4×C8—22no—
318C4×C8—22no—
319C4×C8—44no—
320#95—24no—
321#95—42no—
322C24—21no—
323C24—21no—
324C24C2×#4111yescharacteristic
325#41—24no—
326#41—24no—
327#41—24no—
328#41—24no—
329#41—24no—
330#41—24no—
331#41—24no—
332#41—24no—
333#41—42no—
334#41—42no—
335#41—42no—
336#41—42no—
337#41—42no—
338#41—42no—
339#41—42no—
340#41—42no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8C2×C22⋊C412yes—
352C2×Q8#11712yes—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4C2×C22⋊C412yes—
372C2×D4#11712yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4C4×D411yescharacteristic
413C22×C4C4×D411yescharacteristic
414C22×C4C4×D411yescharacteristic
415C22×C4C4×D411yescharacteristic
416C22×C4C23×C411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C2×#4112yes—
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4112yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C4H(5,𝔽2)11yescharacteristic
425C22×C42−1+411yescharacteristic
426Q16—28no—
427SD16—28no—
428SD16—28no—
429D8—28no—
430M16—24no—
431M16—24no—
432M16—42no—
433M16—42no—
434M16—42no—
435M16—42no—
436M16—42no—
437M16—42no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8C2×C22⋊C412yes—
467C2×C8C2×C22⋊C412yes—
468C2×C8#11712yes—
469C2×C8#11712yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4C2×C22⋊C412yes—
481C4⋊C4C2×C22⋊C412yes—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—24no—
493C42—22no—
494C42—22no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—42no—
503C42—42no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C2×C4×D411yescharacteristic
523C23#52912yes—
524C23#54612yes—
525C23#55511yescharacteristic
526C23#55611yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—22no—
530Q8—22no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535D4—22no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4—44no—
540D4—44no—
541D4—44no—
542D4—44no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4#44312yes—
593C2×C4#44412yes—
594C2×C4C22×C22⋊C411yescharacteristic
595C2×C4C2×#11711yescharacteristic
596C2×C4C2×#11711yescharacteristic
597C2×C4C4×#4111yescharacteristic
598C2×C4#51811yescharacteristic
599C2×C4#52011yescharacteristic
600C2×C4#52512yes—
601C2×C4#52512yes—
602C2×C4#52512yes—
603C2×C4#53812yes—
604C2×C4#53912yes—
605C2×C4D4212yes—
606C2×C4#54612yes—
607C2×C4#54712yes—
608C2×C4#54711yescharacteristic
609C2×C4#55011yescharacteristic
610C2×C4#55312yes—
611C2×C4#55912yes—
612C8—22no—
613C8—22no—
614C8—22no—
615C8—22no—
616C8—42no—
617C8—42no—
618C8—44no—
619C8—82no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22#229011yescharacteristic
643C22D4×C22⋊C411yescharacteristic
644C22#235411yescharacteristic
645C22#245912yes—
646C22#246612yes—
647C22C2×#44311yescharacteristic
648C22C2×#44411yescharacteristic
649C22#295711yescharacteristic
650C22#295711yescharacteristic
651C4—22no—
652C4—22no—
653C4—42no—
654C4—42no—
655C4—41no—
656C4—42no—
657C4—41no—
658C4—42no—
659C4—84no—
660C4—81no—
661C4—81no—
662C4#228011yescharacteristic
663C4#231811yescharacteristic
664C4#233911yescharacteristic
665C4#235711yescharacteristic
666C4#245512yes—
667C4#245612yes—
668C2—22no—
669C2—22no—
670C2—82no—
671C2—82no—
672C2#1388611yescharacteristic
673C2#2113711yescharacteristic
674C2#2114011yescharacteristic
675C2#2245211yescharacteristic
676C2#2245411yescharacteristic
677C2#2280211yescharacteristic
678C2#2280311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#135C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1929C2semidirect
3. #20684C2semidirect
4. #20378C2semidirect
5. #11125C2semidirect
7. #10909C2semidirect
8. #10864C2semidirect
9. #10797C2semidirect
10. #10790C2semidirect
11. #10790C2semidirect
16. C2×C4×Q16C4semidirect
17. C2×C4×Q16C4semidirect
18. C2×C4×SD16C4semidirect
19. C2×C4×SD16C4semidirect
20. C2×C4×SD16C4semidirect
21. C2×C4×SD16C4semidirect
22. C2×C4×D8C4semidirect
23. C2×C4×D8C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1352
C2×#19292
#206842
#203782
#111252
#110252
#109092
#108642
#107972
#107902