Finite Group Databasebeta

← #390407

Subgroups — #390407

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1870C212yesmaximal, complemented
2C2×#1870C212yesmaximal, complemented
3C2×#1869C212yesmaximal, complemented
4C4×#392C211yesmaximal, characteristic
5#11298C212yesmaximal, complemented
6#11042C211yesmaximal, characteristic, complemented
7#11042C212yesmaximal, complemented
8#10715C212yesmaximal, complemented
9#10714C211yesmaximal, characteristic, complemented
10C23×M16C2212yes—
11C24×C8C2211yescharacteristic
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C2×#392C411yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23#1870—28no—
24#1870—28no—
25#1869—28no—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34#1444C411yescharacteristic
35#1439C412yes—
36#1439C411yescharacteristic
37#1438C412yes—
38#1438C411yescharacteristic
39C4×#95C2211yescharacteristic
40C23×D4C2×C412yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2312yes—
44C22×M16—24no—
45C22×M16—24no—
46C22×M16—24no—
47C22×M16—42no—
48C22×M16—42no—
49C22×M16—42no—
50C22×M16—42no—
51C23×C8—22no—
52C23×C8—24no—
53C23×C8—42no—
54C23×C8—41no—
55C23×C8—41no—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—42no—
67C2×C4×D4—41no—
68C2×C4×D4—42no—
69C2×C4×D4—41no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2312yes—
82C2×C4⋊C8—41no—
83C2×C4⋊C8—42no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8—41no—
93C2×C22⋊C8—41no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8D418yescomplemented
98C2×C22⋊C8D414yescomplemented
99#392—24no—
100C4×C22⋊C4—41no—
101C4×C22⋊C4—42no—
102C4×C22⋊C4—41no—
103C4×C22⋊C4—42no—
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C412yes—
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C412yes—
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C412yes—
111#336—24no—
112#336—24no—
113#336—24no—
114#336—24no—
115#336—24no—
116#336—24no—
117#336—42no—
118#336—41no—
119C25C22×C412yes—
120C25C22×C411yescharacteristic
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—22no—
127C22×D4—24no—
128C22×D4—42no—
129C22×D4—41no—
130C22×D4—41no—
131C22×D4—42no—
132C22×D4—42no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C2×D414yes—
162C23×C4C2×D412yes—
163C23×C4#4112yes—
164C23×C4#4114yes—
165C23×C4C2411yescharacteristic
166C2×M16—48no—
167C2×M16—48no—
168C2×M16—48no—
169C2×M16—48no—
170C2×M16—48no—
171C2×M16—48no—
172C2×M16—48no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
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—44no—
199C22×C8—48no—
200C22×C8—48no—
201C22×C8—44no—
202C22×C8—44no—
203C22×C8C2×D414yes—
204C22×C8C2×D414yes—
205C22×C8C2×D414yes—
206C4×D4—48no—
207C4×D4—44no—
208C4×D4—48no—
209C4×D4—48no—
210C4×D4—48no—
211C4×D4—48no—
212C4×D4—44no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4M1614yescomplemented
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42—22no—
253C4⋊C8—44no—
254C22⋊C8—48no—
255C22⋊C8—44no—
256C22⋊C8—48no—
257C22⋊C8—48no—
258C22⋊C8—48no—
259#95—22no—
260#95—24no—
261#95—41no—
262#95—41no—
263#95—42no—
264#95—41no—
265#95—42no—
266C24—22no—
267C24—21no—
268C24—22no—
269C24—21no—
270C24—22no—
271C24—21no—
272C24—21no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24C4×D414yes—
289C24C4×D412yes—
290C24C2×M1612yes—
291C24#13112yes—
292C24C23×C411yescharacteristic
293C2×D4—216no—
294C2×D4—48no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—44no—
298C2×D4—48no—
299C2×D4—44no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—44no—
304C2×D4—48no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4C4×D412yes—
393C22×C4C4×D414yes—
394C22×C4C2×M1612yes—
395C22×C4C2×M1612yes—
396C22×C4#13114yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4112yes—
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C4H(5,𝔽2)12yes—
406C22×C42−1+411yescharacteristic
407M16—432no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—416no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
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—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—44no—
445C22⋊C4—48no—
446C42—42no—
447C42—41no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—42no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—24no—
464C23—22no—
465C23—24no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23#43614yes—
514C23C2×C4×D411yescharacteristic
515C23C2×C4×D412yes—
516C23#51811yescharacteristic
517C23#51812yes—
518C23#52011yescharacteristic
519C23C22×M1611yescharacteristic
520C23C2×#13111yescharacteristic
521C23C2×#13112yes—
522C23#56811yescharacteristic
523C23#56812yes—
524C23#56811yescharacteristic
525D4—432no—
526D4—416no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4#53412yes—
588C2×C4#53411yescharacteristic
589C2×C4#53712yes—
590C2×C4#53711yescharacteristic
591C2×C4#53812yes—
592C2×C4#53811yescharacteristic
593C2×C4#53912yes—
594C2×C4#53911yescharacteristic
595C2×C4D4211yescharacteristic
596C2×C4D4212yes—
597C2×C4#54612yes—
598C2×C4#54612yes—
599C2×C4#54612yes—
600C2×C4#55111yescharacteristic
601C2×C4#55112yes—
602C2×C4#56011yescharacteristic
603C2×C4#56012yes—
604C2×C4#56011yescharacteristic
605C8—44no—
606C8—44no—
607C8—44no—
608C8—82no—
609C22—28no—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—42no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22#232512yes—
636C22#232511yescharacteristic
637C22#235811yescharacteristic
638C22#235812yes—
639C22#238011yescharacteristic
640C22C2×#43611yescharacteristic
641C22#292912yes—
642C22#293012yes—
643C22#293211yescharacteristic
644C22#293312yes—
645C22D4×M1612yes—
646C22#293512yes—
647C22#297311yescharacteristic
648C22#297412yes—
649C22#297611yescharacteristic
650C22#297612yes—
651C22#297812yes—
652C22#297811yescharacteristic
653C22#297911yescharacteristic
654C22#298112yes—
655C22#298911yescharacteristic
656C22#299211yescharacteristic
657C22#299312yes—
658C4—42no—
659C4—44no—
660C4—42no—
661C4—42no—
662C4—44no—
663C4—81no—
664C4—81no—
665C4—82no—
666C4#266811yescharacteristic
667C4#266812yes—
668C4#267511yescharacteristic
669C4#269311yescharacteristic
670C4#269312yes—
671C4#276011yescharacteristic
672C2—44no—
673C2—84no—
674C2#1457211yescharacteristic
675C2#2217312yes—
676C2#2217412yes—
677C2#2217611yescharacteristic
678C2#2217911yescharacteristic
679C2#2284311yescharacteristic
680C2#2284812yes—
681C2#2284912yes—
682C2#2285011yescharacteristic
683C2#2335611yescharacteristic
684C2#2335911yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC2×#392C4
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
1. C2×#1870C2semidirect
2. C2×#1870C2semidirect
3. C2×#1869C2semidirect
5. #11298C2semidirect
6. #11042C2semidirect
7. #11042C2semidirect
8. #10715C2semidirect
9. #10714C2semidirect
97. C2×C22⋊C8D4semidirect
98. C2×C22⋊C8D4semidirect
238. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#18702
C2×#18692
C4×#3922
#112982
#110422
#107152
#107142