Finite Group Databasebeta

← #375754

Subgroups — #375754

Order 512.

2124 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#435C212yesmaximal, complemented
2C2×#1870C212yesmaximal, complemented
3C2×#1851C212yesmaximal, complemented
4C4×#392C211yesmaximal, characteristic
5#11042C211yesmaximal, characteristic, complemented
6#10964C212yesmaximal, complemented
7#10922C212yesmaximal, complemented
8#10877C212yesmaximal, complemented
9#10714C211yesmaximal, characteristic, complemented
10C23×M16C2212yes—
11C24×C8C2211yescharacteristic
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C2×#435—28no—
15C2×#435—28no—
16C2×#435—28no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×#97C2212yes—
25C2×#392C411yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1870—28no—
29#1851—28no—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36#1444C411yescharacteristic
37#1439C411yescharacteristic
38#1439C412yes—
39#1438C411yescharacteristic
40#1438C412yes—
41C4×#95C2211yescharacteristic
42C23×D4C2×C412yes—
43C23×D4C2×C411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2312yes—
46C22×M16—24no—
47C22×M16—24no—
48C22×M16—24no—
49C22×M16—24no—
50C22×M16—24no—
51C22×M16—42no—
52C22×M16—42no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8—41no—
56C23×C8—41no—
57C23×C8D414yescomplemented
58C23×C8C2312yes—
59C23×C8C2312yes—
60C23×C8C2312yes—
61C23×C8C2312yes—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—24no—
69C2×C4×D4—24no—
70C2×C4×D4—41no—
71C2×C4×D4—41no—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2312yes—
84#435—264no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—41no—
89C2×C4⋊C8D418yescomplemented
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—42no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—41no—
103C2×C22⋊C8—41no—
104C2×C22⋊C8D418yescomplemented
105C2×#97—24no—
106C2×#97—24no—
107C2×#97—24no—
108#392—24no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—24no—
111C4×C22⋊C4—41no—
112C4×C22⋊C4—41no—
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117C2×#95C2×C411yescharacteristic
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C411yescharacteristic
120#336—24no—
121#336—24no—
122#336—24no—
123#336—24no—
124#336—24no—
125#336—41no—
126C25C22×C412yes—
127C25C22×C411yescharacteristic
128C22×D4—24no—
129C22×D4—24no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4—22no—
135C22×D4—41no—
136C22×D4—42no—
137C22×D4—41no—
138C22×D4—42no—
139C22×D4—42no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C412yes—
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D414yes—
170C23×C4#4112yes—
171C23×C4#4114yes—
172C23×C4C2411yescharacteristic
173C2×M16—216no—
174C2×M16—48no—
175C2×M16—48no—
176C2×M16—48no—
177C2×M16—48no—
178C2×M16—48no—
179C2×M16—48no—
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—216no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—44no—
205C22×C8—44no—
206C22×C8—48no—
207C22×C8—44no—
208C22×C8C2×D414yes—
209C22×C8C2×D414yes—
210C22×C8C2×D414yes—
211C22×C8C2×D414yes—
212C22×C8C2×D414yes—
213C4×D4—216no—
214C4×D4—216no—
215C4×D4—216no—
216C4×D4—216no—
217C4×D4—44no—
218C4×D4—48no—
219C4×D4—44no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—41no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4M1614yescomplemented
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—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42C2×D414yes—
259C2×C42#4114yes—
260C4⋊C8—216no—
261C4⋊C8—44no—
262C22⋊C8—216no—
263C22⋊C8—216no—
264C22⋊C8—48no—
265C22⋊C8—48no—
266C22⋊C8—44no—
267#97—216no—
268#95—24no—
269#95—24no—
270#95—22no—
271#95—41no—
272#95—42no—
273#95—41no—
274#95—41no—
275C24—22no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24C4×D414yes—
298C24C4×D412yes—
299C24C2×M1612yes—
300C24#13112yes—
301C24C23×C411yescharacteristic
302C2×D4—216no—
303C2×D4—48no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—44no—
313C2×D4—48no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
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—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4C4×D414yes—
401C22×C4C4×D414yes—
402C22×C4C4×D412yes—
403C22×C4C4×D414yes—
404C22×C4C2×M1612yes—
405C22×C4C2×M1612yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C42−1+411yescharacteristic
416M16—432no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—416no—
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#12018yes—
446C4⋊C4—216no—
447C4⋊C4—44no—
448C22⋊C4—216no—
449C22⋊C4—216no—
450C22⋊C4—44no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—44no—
455C42—24no—
456C42—24no—
457C42—24no—
458C42—24no—
459C42—41no—
460C42—41no—
461C42—42no—
462C42—42no—
463C42—41no—
464C42—41no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—41no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—41no—
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—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23#43614yes—
523C23C2×C4×D411yescharacteristic
524C23C2×C4×D412yes—
525C23C2×C4×D412yes—
526C23#51811yescharacteristic
527C23#52011yescharacteristic
528C23C22×M1611yescharacteristic
529C23C2×#13111yescharacteristic
530C23#56812yes—
531C23#56811yescharacteristic
532C23#56811yescharacteristic
533C23#56812yes—
534D4—432no—
535D4—416no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
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—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4C2×#12012yes—
597C2×C4#52512yes—
598C2×C4#52512yes—
599C2×C4#52512yes—
600C2×C4#53411yescharacteristic
601C2×C4#53711yescharacteristic
602C2×C4#53811yescharacteristic
603C2×C4#53911yescharacteristic
604C2×C4D4211yescharacteristic
605C2×C4D4212yes—
606C2×C4D4211yescharacteristic
607C2×C4#54612yes—
608C2×C4#54712yes—
609C2×C4#54812yes—
610C2×C4#55012yes—
611C2×C4#55111yescharacteristic
612C2×C4#55312yes—
613C2×C4#55911yescharacteristic
614C8—28no—
615C8—44no—
616C8—44no—
617C8—82no—
618C22—24no—
619C22—28no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22C4×#12012yes—
645C22#232511yescharacteristic
646C22#235811yescharacteristic
647C22#236112yes—
648C22#236611yescharacteristic
649C22C2×#43611yescharacteristic
650C22#293012yes—
651C22#293012yes—
652C22#293012yes—
653C22#293212yes—
654C22#293212yes—
655C22#293211yescharacteristic
656C22#293312yes—
657C22#293312yes—
658C22D4×M1612yes—
659C22D4×M1612yes—
660C22#297311yescharacteristic
661C22#297611yescharacteristic
662C22#297811yescharacteristic
663C22#298011yescharacteristic
664C22#298911yescharacteristic
665C22#299211yescharacteristic
666C22#299512yes—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—42no—
671C4—44no—
672C4—44no—
673C4—81no—
674C4—81no—
675C4#240311yescharacteristic
676C4#240712yes—
677C4#245511yescharacteristic
678C4#245612yes—
679C4#246311yescharacteristic
680C4#260011yescharacteristic
681C2—44no—
682C2—84no—
683C2#1417111yescharacteristic
684C2#2144112yes—
685C2#2144412yes—
686C2#2217611yescharacteristic
687C2#2217711yescharacteristic
688C2#2285011yescharacteristic
689C2#2285111yescharacteristic
690C2#2298812yes—
691C2#2299312yes—
692C2#2317511yescharacteristic
693C2#2317811yescharacteristic
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. C22×#435C2semidirect
2. C2×#1870C2semidirect
3. C2×#1851C2semidirect
5. #11042C2semidirect
6. #10964C2semidirect
7. #10922C2semidirect
8. #10877C2semidirect
9. #10714C2semidirect
57. C23×C8D4semidirect
89. C2×C4⋊C8D4semidirect
104. C2×C22⋊C8D4semidirect
245. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C22×#4352
C2×#18702
C2×#18512
C4×#3922
#110422
#109642
#109222
#108772
#107142