Finite Group Databasebeta

← #387917

Subgroups — #387917

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1870C211yesmaximal, characteristic
4#11202C212yesmaximal, complemented
5#11147C212yesmaximal, complemented
6#11147C212yesmaximal, complemented
7#10875C212yesmaximal, complemented
8#10715C212yesmaximal, complemented
9#10712C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C22×#123C411yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C4×#123—24no—
21C4×#123—24no—
22C2×#388C412yes—
23C2×#387C412yes—
24C2×#386C411yescharacteristic
25C2×#385C411yescharacteristic
26C2×#381C411yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2212yes—
30#1955—24no—
31#1870—24no—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—24no—
48C22×M16—41no—
49C22×M16—41no—
50C22×M16—42no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×#123—24no—
56C2×#123—24no—
57C2×#123—24no—
58C2×C4×D4—41no—
59C2×C4×D4—41no—
60C2×C4×D4—41no—
61C2×C4×D4—41no—
62C2×C4×D4D414yes—
63C2×C4×D4D414yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—41no—
76C22×C42—41no—
77C22×C42—41no—
78C22×C42—41no—
79C22×C42C2×C412yes—
80C22×C42C2312yes—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—22no—
84C2×C4⋊C8—41no—
85C2×C4⋊C8—41no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—42no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—41no—
97C2×C22⋊C8—41no—
98C2×#97—22no—
99C2×#97—41no—
100C2×#97—41no—
101#388—28no—
102#387—28no—
103#386—24no—
104#385—24no—
105#381—24no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—24no—
108C4×C22⋊C4—24no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117#336—24no—
118#336—24no—
119#336—24no—
120#336—42no—
121#336—42no—
122#336—42no—
123#336—42no—
124#336—42no—
125#336—42no—
126C25C22×C411yescharacteristic
127C25C22×C411yescharacteristic
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—41no—
131C22×D4—41no—
132C22×D4—41no—
133C22×D4—41no—
134C22×D4C22⋊C414yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—42no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4—41no—
161C23×C4—41no—
162C23×C4M1614yescomplemented
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C412yes—
169C23×C4C22×C412yes—
170C23×C4C22×C412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2411yescharacteristic
176C2×M16—44no—
177C2×M16—48no—
178C2×M16—48no—
179C2×M16—44no—
180C2×M16—44no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8#4114yes—
208C22×C8#4114yes—
209#123—232no—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—44no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4C22⋊C414yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4C22⋊C414yes—
239C2×C22⋊C4C22⋊C414yes—
240C2×C22⋊C4M1614yescomplemented
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C4⋊C8—44no—
267C22⋊C8—48no—
268C22⋊C8—48no—
269C22⋊C8—48no—
270C22⋊C8—44no—
271#97—44no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277C24—22no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—21no—
283C24—21no—
284C24—22no—
285C24—22no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—22no—
293C24—21no—
294C24—21no—
295C24C2×C22⋊C412yes—
296C24C2×C22⋊C412yes—
297C24C2×M1612yes—
298C24C2×M1612yes—
299C24C23×C411yescharacteristic
300C2×D4—28no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×M1614yes—
403C22×C4C2×M1614yes—
404C22×C4#13114yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C4H(5,𝔽2)12yes—
414C22×C42−1+412yes—
415M16—416no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C4⋊C4—216no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C22⋊C4—216no—
448C22⋊C4—28no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—48no—
452C22⋊C4—44no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—48no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42—42no—
463C42—42no—
464C42—42no—
465C42—42no—
466C23—22no—
467C23—24no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23#40714yes—
512C23C22×C22⋊C411yescharacteristic
513C23C4×#4112yes—
514C23#51811yescharacteristic
515C23#51811yescharacteristic
516C23#52012yes—
517C23C22×M1611yescharacteristic
518C23C22×M1611yescharacteristic
519C23C2×#13112yes—
520C23#56811yescharacteristic
521C23#56811yescharacteristic
522C23#56812yes—
523D4—416no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—416no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4#53411yescharacteristic
589C2×C4#53411yescharacteristic
590C2×C4#53411yescharacteristic
591C2×C4#53411yescharacteristic
592C2×C4#53812yes—
593C2×C4#53912yes—
594C2×C4#54012yes—
595C2×C4#54212yes—
596C2×C4#54612yes—
597C2×C4#54612yes—
598C2×C4#54712yes—
599C2×C4#54712yes—
600C2×C4#55111yescharacteristic
601C2×C4#55312yes—
602C2×C4#55911yescharacteristic
603C2×C4#56011yescharacteristic
604C2×C4#56212yes—
605C2×C4#56311yescharacteristic
606C8—44no—
607C8—44no—
608C8—82no—
609C8—82no—
610C22—28no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—82no—
631C22—82no—
632C22#230811yescharacteristic
633C22#233512yes—
634C22#233912yes—
635C22#237411yescharacteristic
636C22#237611yescharacteristic
637C22C2×#40711yescharacteristic
638C22#287911yescharacteristic
639C22#288012yes—
640C22#296512yes—
641C22#296712yes—
642C22#296912yes—
643C22#297012yes—
644C22#297311yescharacteristic
645C22#297311yescharacteristic
646C22#297511yescharacteristic
647C22#297511yescharacteristic
648C22#297612yes—
649C22#297711yescharacteristic
650C22#297811yescharacteristic
651C22#298912yes—
652C22#299012yes—
653C22#299312yes—
654C22#299512yes—
655C4—42no—
656C4—42no—
657C4—44no—
658C4—44no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—82no—
663C4#269711yescharacteristic
664C4#270112yes—
665C4#270812yes—
666C4#270911yescharacteristic
667C4#271711yescharacteristic
668C4#272411yescharacteristic
669C2—44no—
670C2—82no—
671C2#1450511yescharacteristic
672C2#2174211yescharacteristic
673C2#2174411yescharacteristic
674C2#2236312yes—
675C2#2236912yes—
676C2#2247112yes—
677C2#2247212yes—
678C2#2329611yescharacteristic
679C2#2330411yescharacteristic
680C2#2331611yescharacteristic
681C2#2332411yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11202C2semidirect
5. #11147C2semidirect
6. #11147C2semidirect
7. #10875C2semidirect
8. #10715C2semidirect
9. #10712C2semidirect
162. C23×C4M16semidirect
240. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C2×#19552
C2×#18702
#112022
#111472
#108752
#107152
#107122