Finite Group Databasebeta

← #364838

Subgroups — #364838

Order 512.

1324 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3076C211yesmaximal, characteristic
2C2×#2960C211yesmaximal, characteristic, complemented
3C2×#1986C211yesmaximal, characteristic, complemented
4C2×#1806C211yesmaximal, characteristic
5C4×#408C211yesmaximal, characteristic
6#11221C212yesmaximal, complemented
7#11068C212yesmaximal
8#10934C212yesmaximal
9#10909C212yesmaximal, complemented
10#10853C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, 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#2960—24no—
23#2960—24no—
24C2×#446—21no—
25C2×#446—21no—
26C2×C8×Q8—21no—
27C2×C8×Q8—21no—
28C2×#427C2212yes—
29C2×#424C2212yes—
30C2×#423C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#104C2212yes—
33C2×#416C2212yes—
34C2×#408C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×#405C2212yes—
37C2×C4×M16C2211yescharacteristic
38C22×C4×C8C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#1986—24no—
41#1806—24no—
42#1750—21no—
43#1750—21no—
44C4×C22⋊C8—21no—
45C4×C22⋊C8—21no—
46C2×#341C2212yes—
47C2×#341C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53C2×#336C2212yes—
54C2×#336C2212yes—
55C22×#41C2311yescharacteristic
56C22×M16C2312yes—
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C23×C8C2312yes—
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62#533—22no—
63#533—22no—
64#533D414yes—
65#533D414yes—
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×#125C2×C412yes—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C4×#41—22no—
72C4×#41—22no—
73C4×#41—22no—
74C4×#41—22no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×Q8C2×C411yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2×C411yescharacteristic
83C2×#117—21no—
84C2×#117C2311yescharacteristic
85C2×#117C2312yes—
86C22×C4⋊C4—21no—
87C22×C42—21no—
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90#479—28no—
91#478—28no—
92#477—24no—
93#476—24no—
94#475—24no—
95#474—24no—
96#446—28no—
97#446—28no—
98C8×Q8—28no—
99C8×Q8—28no—
100#427—24no—
101#427—24no—
102C2×#107C2×C412yes—
103C2×#107C2×C412yes—
104C2×#106C2×C412yes—
105C2×#106C2×C412yes—
106#424—24no—
107#424—24no—
108#423—22no—
109#423D414yes—
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×#104—24no—
120C2×#104—24no—
121C2×#104—42no—
122#416—24no—
123#416—24no—
124C2×#103C2×C412yes—
125C2×#103C2×C412yes—
126C2×#102C2×C412yes—
127C2×#102C2×C412yes—
128#408—22no—
129#408—22no—
130#408—22no—
131#408Q814yescomplemented
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136#405—24no—
137#405—24no—
138C4×M16—22no—
139C4×M16—22no—
140C2×#97—22no—
141C2×#97—21no—
142C2×C4×C8—22no—
143C2×C4×C8—22no—
144C2×C4×C8—22no—
145C2×C4×C8—21no—
146C2×C4×C8D411yescharacteristic
147C2×C4×C8D411yescharacteristic
148C2×C4×C8D411yescharacteristic
149C2×C4×C8D411yescharacteristic
150C4×C4⋊C4—22no—
151C4×C4⋊C4—22no—
152C2×#95—21no—
153#341—24no—
154#341—42no—
155#339—24no—
156#339—24no—
157#339—24no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336—24no—
163#336—42no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41C2×Q812yes—
175C2×#41#4112yes—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8C22×C411yescharacteristic
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4#4111yescharacteristic
186C23×C4#4111yescharacteristic
187C23×C4C2411yescharacteristic
188C2×M16—22no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—22no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—22no—
197C2×M16—21no—
198C2×M16—22no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—22no—
202C2×M16—21no—
203C2×M16C2×D414yes—
204C2×M16C2×Q812yes—
205C2×M16#4112yes—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—21no—
225C22×C8C22×C412yes—
226C22×C8C22×C412yes—
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C22×C8C2×D411yescharacteristic
230C22×C8C2×D411yescharacteristic
231C22×C8C2×Q812yes—
232C22×C8#4112yes—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8C22⋊C414yes—
236C4⋊Q8C22⋊C414yes—
237#125—24no—
238#125C22⋊C418yes—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242C22⋊Q8—24no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4—42no—
259#117—22no—
260#117—22no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—42no—
266#117—42no—
267#117C2×D414yes—
268#117C2×D414yes—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C412yes—
281C2×C4⋊C4C22×C412yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42#4112yes—
306#107—24no—
307#107—24no—
308#106—24no—
309#106—24no—
310C4⋊C8—28no—
311C4⋊C8—28no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8—24no—
315C4⋊C8—24no—
316C4⋊C8—24no—
317#104—48no—
318#103—24no—
319#103—24no—
320#102—24no—
321#102—24no—
322C22⋊C8—24no—
323C22⋊C8—24no—
324C22⋊C8—24no—
325C22⋊C8—24no—
326#97—28no—
327#97—42no—
328C4×C8—28no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8—22no—
332C4×C8—22no—
333C4×C8—42no—
334#95—42no—
335C24—21no—
336C24C2×#4111yescharacteristic
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—22no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—41no—
353C2×Q8—21no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8C4×Q812yes—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4C4×Q812yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
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—24no—
428M16—24no—
429M16—22no—
430M16—22no—
431M16—22no—
432M16—22no—
433M16—42no—
434M16—41no—
435M16—42no—
436M16—41no—
437M16—41no—
438M16—41no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
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—21no—
454C2×C8—21no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—22no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—42no—
465C2×C8—41no—
466C2×C8—42no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—41no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—41no—
476C2×C8C22⋊Q812yes—
477C2×C8C22⋊Q812yes—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—48no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4C2×C22⋊C414yes—
492C4⋊C4C2×C22⋊C414yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C42—24no—
500C42—24no—
501C42—41no—
502C42—41no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42C2×C22⋊C412yes—
510C42C2×C22⋊C412yes—
511C42C4×D412yes—
512C42C4×D412yes—
513C42#12012yes—
514C42#12012yes—
515C42#12112yes—
516C42#12112yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C2×C4×D411yescharacteristic
529C23C2×#12111yescharacteristic
530C23C2×#12311yescharacteristic
531C23#52911yescharacteristic
532C23#53311yescharacteristic
533C23#54111yescharacteristic
534C23#54211yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—48no—
538Q8—41no—
539Q8—41no—
540Q8—48no—
541D4—22no—
542D4—22no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—41no—
548D4—41no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—84no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—84no—
587C2×C4#38612yes—
588C2×C4#38612yes—
589C2×C4#38612yes—
590C2×C4#38612yes—
591C2×C4#44312yes—
592C2×C4#44412yes—
593C2×C4C22×C22⋊C411yescharacteristic
594C2×C4C2×C4×D411yescharacteristic
595C2×C4C2×C4×Q811yescharacteristic
596C2×C4C4×#4111yescharacteristic
597C2×C4#51911yescharacteristic
598C2×C4#52011yescharacteristic
599C2×C4C2×#12011yescharacteristic
600C2×C4C2×#12111yescharacteristic
601C2×C4C2×C22⋊Q811yescharacteristic
602C2×C4#52511yescharacteristic
603C2×C4#53611yescharacteristic
604C2×C4#53711yescharacteristic
605C2×C4#54611yescharacteristic
606C2×C4#54711yescharacteristic
607C2×C4#54811yescharacteristic
608C2×C4#54811yescharacteristic
609C2×C4#54811yescharacteristic
610C2×C4#54811yescharacteristic
611C2×C4#54811yescharacteristic
612C2×C4D4×Q811yescharacteristic
613C2×C4D4×Q811yescharacteristic
614C2×C4D4×Q811yescharacteristic
615C2×C4#55011yescharacteristic
616C2×C4#55011yescharacteristic
617C2×C4#55211yescharacteristic
618C2×C4#55211yescharacteristic
619C2×C4#55411yescharacteristic
620C2×C4#55611yescharacteristic
621C8—22no—
622C8—22no—
623C8—42no—
624C8—42no—
625C8—41no—
626C8—41no—
627C8—44no—
628C8—42no—
629C8—42no—
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
650C22C4×C22⋊Q811yescharacteristic
651C22#232311yescharacteristic
652C22#242111yescharacteristic
653C22#242311yescharacteristic
654C22#242611yescharacteristic
655C22#242911yescharacteristic
656C22C2×#44311yescharacteristic
657C22C2×#44411yescharacteristic
658C22#295511yescharacteristic
659C22#295511yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—42no—
664C4—44no—
665C4—44no—
666C4—84no—
667C4—81no—
668C4—81no—
669C4#187614yes—
670C4Q8×C22⋊C411yescharacteristic
671C4#234111yescharacteristic
672C4#236011yescharacteristic
673C4#236211yescharacteristic
674C4#240711yescharacteristic
675C4#240911yescharacteristic
676C4#253611yescharacteristic
677C4#253811yescharacteristic
678C2—21no—
679C2—21no—
680C2—21no—
681C2—21no—
682C2—82no—
683C2#1396911yescharacteristic
684C2C2×#187611yescharacteristic
685C2#2091711yescharacteristic
686C2#2162711yescharacteristic
687C2#2162811yescharacteristic
688C2#2213011yescharacteristic
689C2#2213111yescharacteristic
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×#2960C2semidirect
3. C2×#1986C2semidirect
6. #11221C2semidirect
9. #10909C2semidirect
10. #10853C2semidirect
11. #10834C2semidirect
16. C2×#479C4semidirect
17. C2×#477C4semidirect
18. C2×#475C4semidirect
19. C2×#478C4semidirect
20. C2×#476C4semidirect
21. C2×#474C4semidirect
131. #408Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30762
C2×#29602
C2×#19862
C2×#18062
C4×#4082
#112212
#110682
#109342
#109092
#108532
#108342