Finite Group Databasebeta

← #383890

Subgroups — #383890

Order 512.

1880 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C212yesmaximal, complemented
2C2×#1780C212yesmaximal
3#13851C211yesmaximal, characteristic
4#11147C212yesmaximal, complemented
5#11042C212yesmaximal, complemented
6#11042C211yesmaximal, characteristic, complemented
7#11042C211yesmaximal, characteristic, complemented
8#10921C212yesmaximal, complemented
9#10921C212yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C4×C8C2212yes—
20C2×C4×C22⋊C4C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1923—28no—
25#1780—28no—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1446C411yescharacteristic
34#1446C411yescharacteristic
35#1438C411yescharacteristic
36#1438C412yes—
37#1438C412yes—
38#1438C411yescharacteristic
39C22×#41—24no—
40C22×#41—22no—
41C22×#41—22no—
42C22×#41—24no—
43C22×#41—41no—
44C22×#41—41no—
45C22×#41—42no—
46C22×#41—42no—
47C22×#41—41no—
48C22×#41—41no—
49C23×Q8C2×C411yescharacteristic
50C23×D4C2×C411yescharacteristic
51C23×D4C2×C412yes—
52C24×C4C2312yes—
53C24×C4C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2312yes—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C412yes—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—24no—
69C2×C4⋊C8—41no—
70C2×C4⋊C8—42no—
71C2×C4⋊C8—41no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—42no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8D418yes—
90C2×C4×C8—24no—
91C2×C4×C8—42no—
92C2×C4×C8—42no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—41no—
96C4×C22⋊C4—41no—
97C4×C22⋊C4—42no—
98C4×C22⋊C4—42no—
99C4×C22⋊C4—41no—
100C4×C22⋊C4—41no—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C412yes—
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C412yes—
109#336—24no—
110#336—24no—
111#336—41no—
112#336—42no—
113#336—42no—
114#336—41no—
115#336—42no—
116C25C22×C411yescharacteristic
117C25C22×C412yes—
118C2×#41—44no—
119C2×#41—48no—
120C2×#41—48no—
121C2×#41—48no—
122C2×#41—48no—
123C2×#41—48no—
124C2×#41—44no—
125C2×#41—48no—
126C2×#41—48no—
127C2×#41—44no—
128C2×#41—44no—
129C2×#41—44no—
130C2×#41—44no—
131C2×#41—48no—
132C2×#41—48no—
133C2×#41—44no—
134C2×#41—48no—
135C2×#41—44no—
136C22×Q8—22no—
137C22×Q8—24no—
138C22×Q8—41no—
139C22×Q8—42no—
140C22×Q8—41no—
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—22no—
145C22×D4—24no—
146C22×D4—42no—
147C22×D4—41no—
148C22×D4—41no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C412yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D414yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4#4114yes—
185C23×C4C2411yescharacteristic
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—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8C2×D414yes—
213C22×C8C2×D414yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—21no—
249C4⋊C8—44no—
250C4⋊C8—44no—
251C22⋊C8—48no—
252C22⋊C8—48no—
253C22⋊C8—48no—
254C22⋊C8—48no—
255C22⋊C8—48no—
256C22⋊C8—48no—
257C4×C8—48no—
258#95—24no—
259#95—24no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—42no—
264#95—42no—
265#95—41no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—22no—
290C24#11712yes—
291C24C4×D414yes—
292C24C23×C411yescharacteristic
293#41—416no—
294#41—416no—
295#41—432no—
296#41—432no—
297#41—416no—
298#41—416no—
299C2×Q8—48no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—48no—
303C2×Q8—44no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—44no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—44no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
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—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—24no—
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—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—42no—
377C22×C4—42no—
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—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4#11712yes—
403C22×C4C4×D414yes—
404C22×C4#13112yes—
405C22×C4#13114yes—
406C22×C4#13114yes—
407C22×C4#13112yes—
408C22×C4C22×D412yes—
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4112yes—
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C4H(5,𝔽2)12yes—
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
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—
444C2×C8—42no—
445C2×C8—42no—
446C22⋊C4—216no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—44no—
451C22⋊C4—48no—
452C22⋊C4—44no—
453C42—42no—
454C42—42no—
455C42—42no—
456C42—42no—
457C42—41no—
458C42—42no—
459C42—41no—
460C42—41no—
461C42—42no—
462C42—41no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
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—41no—
496C23—41no—
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—41no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C2×#11711yescharacteristic
522C23C2×C4×D412yes—
523C23#51811yescharacteristic
524C23#51811yescharacteristic
525C23#51812yes—
526C23C2×#13112yes—
527C23C2×#13112yes—
528C23C2×#13111yescharacteristic
529C23C2×#13111yescharacteristic
530C23#56811yescharacteristic
531C23#56811yescharacteristic
532Q8—416no—
533D4—416no—
534D4—432no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4#53412yes—
596C2×C4#53512yes—
597C2×C4#53811yescharacteristic
598C2×C4#53811yescharacteristic
599C2×C4#53811yescharacteristic
600C2×C4#53812yes—
601C2×C4#53811yescharacteristic
602C2×C4#53812yes—
603C2×C4D4211yescharacteristic
604C2×C4#54612yes—
605C2×C4#54612yes—
606C2×C4#54612yes—
607C2×C4#55112yes—
608C2×C4#55112yes—
609C2×C4#55111yescharacteristic
610C2×C4#56011yescharacteristic
611C2×C4#56012yes—
612C2×C4H(3,𝔽4)11yescharacteristic
613C8—44no—
614C8—44no—
615C8—82no—
616C8—82no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—42no—
628C22—42no—
629C22—82no—
630C22—82no—
631C22—81no—
632C22—82no—
633C22—82no—
634C22—81no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—81no—
640C22—81no—
641C22—82no—
642C22—82no—
643C22#231211yescharacteristic
644C22#232112yes—
645C22#234212yes—
646C22#235811yescharacteristic
647C22#237511yescharacteristic
648C22#292212yes—
649C22#292511yescharacteristic
650C22#292511yescharacteristic
651C22#292912yes—
652C22#293112yes—
653C22#293312yes—
654C22#293512yes—
655C22#297611yescharacteristic
656C22#297612yes—
657C22#297611yescharacteristic
658C22#297712yes—
659C22#297712yes—
660C22#297911yescharacteristic
661C22#297912yes—
662C22#297911yescharacteristic
663C22#297911yescharacteristic
664C22#298011yescharacteristic
665C22#298112yes—
666C4—44no—
667C4—42no—
668C4—44no—
669C4—42no—
670C4—44no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4#266811yescharacteristic
675C4#267011yescharacteristic
676C4#268512yes—
677C4#269712yes—
678C4#273511yescharacteristic
679C4#273611yescharacteristic
680C2—84no—
681C2—82no—
682C2#1436611yescharacteristic
683C2#2190311yescharacteristic
684C2#2190911yescharacteristic
685C2#2202912yes—
686C2#2203712yes—
687C2#2255612yes—
688C2#2256012yes—
689C2#2284311yescharacteristic
690C2#2285311yescharacteristic
691C2#2331011yescharacteristic
692C2#2331211yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×#41C22
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×#1923C2semidirect
4. #11147C2semidirect
5. #11042C2semidirect
6. #11042C2semidirect
7. #11042C2semidirect
8. #10921C2semidirect
9. #10921C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C2×#17802
#138512
#111472
#110422
#109212