Finite Group Databasebeta

← #364781

Subgroups — #364781

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3076C211yesmaximal, characteristic
2C2×#2962C211yesmaximal, characteristic, complemented
3C2×#1994C211yesmaximal, characteristic, complemented
4#20629C211yesmaximal, characteristic
5C2×C4×#104C211yesmaximal, characteristic
6#11221C212yesmaximal, complemented
7#11068C212yesmaximal
8#10934C212yesmaximal
9#10909C212yesmaximal, complemented
10#10853C211yesmaximal, characteristic, complemented
11#10832C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#3076—24no—
15#3076—24no—
16C2×#479C412yescomplemented
17C2×#477C411yescharacteristic, complemented
18C2×#475C411yescharacteristic, complemented
19C2×#478C412yescomplemented
20C2×#476C411yescharacteristic, complemented
21C2×#474C411yescharacteristic, complemented
22#2962—24no—
23#2962—24no—
24C2×#446—21no—
25C2×#446—22no—
26C2×C8×Q8—21no—
27C2×#427C2212yes—
28C2×#424C2212yes—
29C2×#423C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#104C2211yescharacteristic
33C22×#104C2211yescharacteristic
34C2×#416C2212yes—
35C2×#408C2212yes—
36C2×#405C2212yes—
37C2×C4×M16C2211yescharacteristic
38C2×C43C2211yescharacteristic
39#1994—24no—
40#1798—22no—
41#1797—21no—
42#1797—21no—
43C4×#104—24no—
44C2×#341C2212yes—
45C2×#341C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50#1729C2212yes—
51C2×#336C2212yes—
52C2×#336C2212yes—
53C22×#41C2311yescharacteristic
54C22×M16C2312yes—
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2312yes—
59#533—22no—
60#533—22no—
61#533D414yes—
62#533D414yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×#125C2×C412yes—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C4×#41—22no—
69C4×#41—22no—
70C4×#41—22no—
71C4×#41—22no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2×C411yescharacteristic
80C2×#117—21no—
81C2×#117C2311yescharacteristic
82C2×#117C2312yes—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#479—28no—
90#478—28no—
91#477—24no—
92#476—24no—
93#475—24no—
94#474—24no—
95#446—216no—
96#446—28no—
97C8×Q8—28no—
98#427—24no—
99#427—24no—
100C2×#107C2×C412yes—
101C2×#107C2×C412yes—
102C2×#106C2×C412yes—
103C2×#106C2×C412yes—
104#424—24no—
105#424—24no—
106#423—22no—
107#423—22no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—22no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8C2×C412yes—
117C2×C4⋊C8C2×C411yescharacteristic
118C2×C4⋊C8C2×C411yescharacteristic
119C2×C4⋊C8D411yescharacteristic
120C2×C4⋊C8D411yescharacteristic
121C2×C4⋊C8D412yes—
122C2×#104—22no—
123C2×#104—22no—
124C2×#104—22no—
125C2×#104—22no—
126C2×#104—22no—
127C2×#104Q814yescomplemented
128#416—24no—
129#416—24no—
130C2×#103C2×C412yes—
131C2×#103C2×C412yes—
132C2×#102C2×C412yes—
133C2×#102C2×C412yes—
134#408—24no—
135#408—42no—
136C2×C22⋊C8—22no—
137C2×C22⋊C8—22no—
138#405—24no—
139#405—24no—
140C4×M16—22no—
141C4×M16D414yes—
142C2×#97—22no—
143C2×#97—21no—
144C2×C4×C8—22no—
145C2×C4×C8—21no—
146C43—22no—
147C43—22no—
148#341—24no—
149#341—42no—
150#339—22no—
151#339—22no—
152#339—24no—
153#339—24no—
154#336—22no—
155#336—22no—
156#336—24no—
157#336—42no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41C2×Q812yes—
169C2×#41#4112yes—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8C22×C411yescharacteristic
173C22×D4—21no—
174C22×D4C22×C411yescharacteristic
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4#4111yescharacteristic
181C23×C4#4111yescharacteristic
182C23×C4C2411yescharacteristic
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16C2×D412yes—
198C2×M16C2×D412yes—
199C2×M16C2×Q812yes—
200C2×M16#4112yes—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8C22×C412yes—
217C22×C8C22×C412yes—
218C22×C8C2×D412yes—
219C22×C8C2×D412yes—
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8C22⋊C414yes—
223C4⋊Q8C22⋊C414yes—
224#125—24no—
225#125C22⋊C418yes—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—42no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—42no—
246#117—22no—
247#117—22no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—42no—
253#117—42no—
254#117C2×D414yes—
255#117C2×D414yes—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C22×C412yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—24no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×Q812yes—
300C2×C42#4112yes—
301C2×C42#4112yes—
302#107—24no—
303#107—24no—
304#106—24no—
305#106—24no—
306C4⋊C8—28no—
307C4⋊C8—28no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—24no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—24no—
314C4⋊C8—42no—
315C4⋊C8—42no—
316#104—28no—
317#104—28no—
318#103—24no—
319#103—24no—
320#102—24no—
321#102—24no—
322C22⋊C8—44no—
323C22⋊C8—44no—
324#97—28no—
325#97—24no—
326C4×C8—28no—
327C4×C8—24no—
328C24—21no—
329C24C2×#4111yescharacteristic
330#41—22no—
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—22no—
336#41—22no—
337#41—22no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—41no—
345#41—41no—
346C2×Q8—21no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—81no—
356C2×Q8—81no—
357C2×Q8C4×Q812yes—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4C4×Q812yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4#12112yes—
410C22×C4#12112yes—
411C22×C4#12312yes—
412C22×C4#12312yes—
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×Q811yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4H(5,𝔽2)11yescharacteristic
424C22×C42−1+411yescharacteristic
425C22×C42−1+411yescharacteristic
426C22×C42−1+411yescharacteristic
427M16—22no—
428M16—22no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16—41no—
433M16—41no—
434M16—42no—
435M16—42no—
436M16—42no—
437M16—42no—
438M16C22⋊Q814yes—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4C2×C22⋊C414yes—
481C4⋊C4C2×C22⋊C414yes—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—48no—
495C42—48no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—42no—
503C42—42no—
504C42C2×C22⋊C412yes—
505C42C2×C22⋊C412yes—
506C42C4×D412yes—
507C42C4×D412yes—
508C42#12012yes—
509C42#12012yes—
510C42#12112yes—
511C42#12112yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C2×C4×D411yescharacteristic
524C23C2×#12111yescharacteristic
525C23C2×#12311yescharacteristic
526C23#55211yescharacteristic
527C23#55211yescharacteristic
528C23#55411yescharacteristic
529C23#55611yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—48no—
535Q8—84no—
536D4—22no—
537D4—22no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—41no—
543D4—41no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4#38612yes—
590C2×C4#38612yes—
591C2×C4#38612yes—
592C2×C4#38612yes—
593C2×C4#44312yes—
594C2×C4#44412yes—
595C2×C4C22×C22⋊C411yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×Q811yescharacteristic
598C2×C4C4×#4111yescharacteristic
599C2×C4#51911yescharacteristic
600C2×C4#52011yescharacteristic
601C2×C4C2×#12011yescharacteristic
602C2×C4C2×#12111yescharacteristic
603C2×C4C2×C22⋊Q811yescharacteristic
604C2×C4#52511yescharacteristic
605C2×C4#52911yescharacteristic
606C2×C4#53311yescharacteristic
607C2×C4#53611yescharacteristic
608C2×C4#53711yescharacteristic
609C2×C4#54111yescharacteristic
610C2×C4#54211yescharacteristic
611C2×C4#54611yescharacteristic
612C2×C4#54711yescharacteristic
613C2×C4#54811yescharacteristic
614C2×C4#54811yescharacteristic
615C2×C4#54811yescharacteristic
616C2×C4#54811yescharacteristic
617C2×C4#54811yescharacteristic
618C2×C4D4×Q811yescharacteristic
619C2×C4D4×Q811yescharacteristic
620C2×C4D4×Q811yescharacteristic
621C2×C4#55011yescharacteristic
622C2×C4#55011yescharacteristic
623C8—24no—
624C8—42no—
625C8—42no—
626C8—42no—
627C8—42no—
628C8—42no—
629C8—82no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22C2×#38611yescharacteristic
650C22#236011yescharacteristic
651C22#236211yescharacteristic
652C22#242111yescharacteristic
653C22#242311yescharacteristic
654C22#253611yescharacteristic
655C22#253811yescharacteristic
656C22C2×#44311yescharacteristic
657C22C2×#44411yescharacteristic
658C22#295611yescharacteristic
659C22#295611yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—44no—
664C4—44no—
665C4—42no—
666C4—44no—
667C4—44no—
668C4—81no—
669C4—81no—
670C4#187614yes—
671C4C4×C22⋊Q811yescharacteristic
672C4#232311yescharacteristic
673C4Q8×C22⋊C411yescharacteristic
674C4#234111yescharacteristic
675C4#240711yescharacteristic
676C4#240911yescharacteristic
677C4#242611yescharacteristic
678C4#242911yescharacteristic
679C2—21no—
680C2—21no—
681C2—21no—
682C2—21no—
683C2—82no—
684C2#1396911yescharacteristic
685C2C2×#187611yescharacteristic
686C2#2091811yescharacteristic
687C2#2298211yescharacteristic
688C2#2298311yescharacteristic
689C2#2308011yescharacteristic
690C2#2308111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2962C2semidirect
3. C2×#1994C2semidirect
6. #11221C2semidirect
9. #10909C2semidirect
10. #10853C2semidirect
11. #10832C2semidirect
16. C2×#479C4semidirect
17. C2×#477C4semidirect
18. C2×#475C4semidirect
19. C2×#478C4semidirect
20. C2×#476C4semidirect
21. C2×#474C4semidirect
127. C2×#104Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30762
C2×#29622
C2×#19942
#206292
C2×C4×#1042
#112212
#110682
#109342
#109092
#108532
#108322