Finite Group Databasebeta

← #1423830

Subgroups — #1423830

Order 512.

3780 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2431C211yesmaximal, characteristic
2C2×#2366C211yesmaximal, characteristic, complemented
3#15784C212yesmaximal, complemented
4#15493C214yesmaximal, complemented
5#15287C214yesmaximal, complemented
6#15260C212yesmaximal, complemented
7#15104C212yesmaximal, complemented
8#14721C214yesmaximal, complemented
9#14419C211yesmaximal, characteristic, complemented
10#14401C212yesmaximal, complemented
11#14240C212yesmaximal, complemented
12#13975C214yesmaximal, complemented
13#13956C212yesmaximal, complemented
14C23×#41C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#123C2212yes—
18C22×#121C2211yescharacteristic
19C22×C4×D4C2214yes—
20#2431—28no—
21C2×#399C2212yes—
22C2×#397C2214yes—
23C2×#394C2214yes—
24C2×#392C2214yes—
25#2366—28no—
26C2×#391C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2214yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2214yes—
32C2×#386C2212yes—
33C2×#386C2214yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#385C2214yes—
37C2×#385C2212yes—
38C2×#381C2212yes—
39C2×#380C2214yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C22×#95C2212yes—
43#1446C2214yes—
44#1445C2214yes—
45#1445C2212yes—
46#1444C2214yes—
47#1444C2214yes—
48#1442C2212yes—
49#1440C2214yes—
50#1440C2212yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2214yes—
57#1438C2212yes—
58#1438C2214yes—
59#1438C2212yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1434C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22×#41—24no—
70C22×#41—28no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41—44no—
74C23×Q8C2311yescharacteristic
75C23×D4C2312yes—
76C23×D4C2314yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C2×#127—28no—
86C2×#127D418yes—
87C2×#124—24no—
88C2×#124—44no—
89C2×#123—44no—
90C2×#123—48no—
91C2×#121—24no—
92C2×#121—42no—
93C2×#121—42no—
94C2×C4×D4—216no—
95C2×C4×D4—48no—
96C2×C4×D4—48no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121#399—28no—
122#397—48no—
123#394—48no—
124#392—216no—
125#391—24no—
126#390—216no—
127#390—24no—
128#386—28no—
129#386—28no—
130#386—216no—
131#386—28no—
132#386—216no—
133#386—44no—
134#386—44no—
135#385—44no—
136#385—48no—
137#381—44no—
138#380—48no—
139C4×C4⋊C4—28no—
140C4×C22⋊C4—24no—
141C2×#95—44no—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C25C2412yes—
161C25C2411yescharacteristic
162C25C2412yes—
163C2×#41—416no—
164C2×#41—432no—
165C2×#41—432no—
166C2×#41—416no—
167C2×#41—416no—
168C22×Q8—22no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—28no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—42no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—42no—
197C22×D4—44no—
198C22×D4C2×D4116yes—
199C22×D4C2×D414yes—
200C22×D4C2×D414yes—
201C22×D4#4118yes—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—28no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239#127—232no—
240#124—416no—
241#123—432no—
242#121—416no—
243C4×D4—464no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—22no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—42no—
309C2×C42C2×D414yes—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95#4118yescomplemented
328C24—24no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—21no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—21no—
349C24—22no—
350C24—24no—
351C24C22×D414yes—
352C24C22×D414yes—
353C24C2×#4114yes—
354C24C2×#4114yes—
355C24H(5,𝔽2)12yes—
356C24C2511yescharacteristic
357#41—4128no—
358C2×Q8—28no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×D4—216no—
364C2×D4—232no—
365C2×D4—28no—
366C2×D4—232no—
367C2×D4—416no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4—88no—
389C2×D4—84no—
390C2×D4—88no—
391C2×D4#123116yescomplemented
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—88no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C2×#4118yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4H(5,𝔽2)18yes—
478C22×C42−1+414yes—
479C4⋊C4—216no—
480C4⋊C4—44no—
481C4⋊C4—416no—
482C4⋊C4—48no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C22⋊C4—48no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—416no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—88no—
499C22⋊C4—84no—
500C42—28no—
501C42—416no—
502C42—44no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—24no—
516C23—28no—
517C23—24no—
518C23—28no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23—41no—
562C23C2×#12318yes—
563C23C23×D411yescharacteristic
564C23C22×#4112yes—
565C23C22×#4112yes—
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×H(5,𝔽2)14yes—
568C23C2×H(5,𝔽2)12yes—
569C23C2×H(5,𝔽2)11yescharacteristic
570C23C2×2−1+412yes—
571Q8—416no—
572D4—416no—
573D4—464no—
574D4—432no—
575D4—88no—
576D4—816no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—416no—
597C2×C4—432no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—816no—
638C2×C4—816no—
639C2×C4—816no—
640C2×C4C2×#12314yes—
641C2×C4#54718yes—
642C22—28no—
643C22—216no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—48no—
648C22—48no—
649C22—44no—
650C22—44no—
651C22—48no—
652C22—44no—
653C22—42no—
654C22—44no—
655C22—44no—
656C22—42no—
657C22—48no—
658C22—48no—
659C22—48no—
660C22—44no—
661C22—42no—
662C22—82no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22—84no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22C22×#12311yescharacteristic
671C22C2×#54712yes—
672C22#348114yes—
673C22#348111yescharacteristic
674C22#348411yescharacteristic
675C22#348411yescharacteristic
676C22#348412yes—
677C22#349012yes—
678C22#349012yes—
679C22#350814yes—
680C22#352014yes—
681C22#352212yes—
682C22#353212yes—
683C22#353712yes—
684C22#354814yes—
685C22#355011yescharacteristic
686C4—24no—
687C4—28no—
688C4—42no—
689C4—44no—
690C4—48no—
691C4—48no—
692C4—82no—
693C4—84no—
694C4—82no—
695C4—84no—
696C2—48no—
697C2—82no—
698C2—84no—
699C2#3535011yescharacteristic
700C2#3623712yes—
701C2#4415114yes—
702C2#4493112yes—
703C2#4970312yes—
704C2#5200411yescharacteristic
705C2#5907011yescharacteristic
706C2#6003412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2431C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2366C2semidirect
3. #15784C2semidirect
4. #15493C2semidirect
5. #15287C2semidirect
6. #15260C2semidirect
7. #15104C2semidirect
8. #14721C2semidirect
9. #14419C2semidirect
10. #14401C2semidirect
11. #14240C2semidirect
12. #13975C2semidirect
13. #13956C2semidirect
327. #95#41semidirect
391. C2×D4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24312
C2×#23662
#157842
#154932
#152872
#152602
#151042
#147212
#144192
#144012
#142402
#139752
#139562