Finite Group Databasebeta

← #877564

Subgroups — #877564

Order 512.

3234 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2811C211yesmaximal, characteristic, complemented
2C22×#391C211yesmaximal, characteristic
3#17080C212yesmaximal, complemented
4#16630C212yesmaximal, complemented
5#15964C214yesmaximal, complemented
6#15510C214yesmaximal, complemented
7#14758C214yesmaximal
8#14720C212yesmaximal
9#14702C212yesmaximal
10#14467C211yesmaximal, characteristic, complemented
11#14467C212yesmaximal, complemented
12#14052C214yesmaximal
13#13922C212yesmaximal
14C24×Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2811—28no—
20C2×#395C2212yes—
21C2×#395C2214yes—
22C2×#395C2212yes—
23C2×#391—28no—
24C2×#391C2214yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2214yes—
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2211yescharacteristic
36C2×#385C2214yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C22×#95C2214yes—
41#1446C2214yes—
42#1446C2214yes—
43#1446C2214yes—
44#1446C2214yes—
45#1445C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1441C2212yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×Q8—24no—
70C23×Q8—24no—
71C23×Q8—42no—
72C23×Q8—42no—
73C23×Q8—42no—
74C23×Q8—42no—
75C23×Q8Q818yescomplemented
76C23×Q8C2312yes—
77C23×Q8C2311yescharacteristic
78C23×Q8C2312yes—
79C23×Q8C2312yes—
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C2×C22⋊Q8—44no—
86C2×C22⋊Q8—44no—
87C2×C22⋊Q8—44no—
88C2×C22⋊Q8—48no—
89C2×C22⋊Q8Q8116yescomplemented
90C22×C4⋊C4—24no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C42—24no—
109C22×C42—42no—
110C22×C42—42no—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116#395—28no—
117#395—216no—
118#395—44no—
119#391—24no—
120#391—28no—
121#391—28no—
122#391—24no—
123#391—24no—
124#391—28no—
125#391—216no—
126#391—216no—
127#391—44no—
128#391—44no—
129#391—44no—
130#391D418yes—
131#385—48no—
132#380—44no—
133#380—44no—
134C4×C22⋊C4—44no—
135C2×#95—216no—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C25C2411yescharacteristic
161C22×Q8—232no—
162C22×Q8—24no—
163C22×Q8—22no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—22no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—416no—
172C22×Q8—41no—
173C22×Q8—42no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8—41no—
177C22×Q8—41no—
178C22×Q8—41no—
179C22×Q8—42no—
180C22×Q8—41no—
181C22×Q8—41no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—44no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—44no—
189C22×Q8—42no—
190C22×Q8—416no—
191C22×Q8—416no—
192C22×Q8—416no—
193C22×Q8—416no—
194C22×Q8—416no—
195C22×Q8C2×D414yes—
196C22×Q8C2×Q818yes—
197C22×Q8C2×Q814yes—
198C22×Q8C2×Q814yes—
199C22×Q8C2×Q818yescomplemented
200C22×Q8#4114yes—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—24no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4C2×Q814yes—
225C23×C4C2×Q818yes—
226C23×C4C2×Q818yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C22⋊Q8—432no—
240C22⋊Q8—432no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—416no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×Q818yescomplemented
279C2×C4⋊C4C2×Q8116yescomplemented
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4C2×Q8116yes—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—41no—
292C2×C42—41no—
293C2×C42—42no—
294C2×C42—41no—
295C2×C42—41no—
296C2×C42—416no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42C2×D414yes—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95C2×D4116yes—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24C22×Q814yes—
333C24C22×Q812yes—
334C24H(5,𝔽2)12yes—
335C24H(5,𝔽2)14yes—
336C24H(5,𝔽2)12yes—
337C24C2511yescharacteristic
338C2×Q8—216no—
339C2×Q8—28no—
340C2×Q8—216no—
341C2×Q8—216no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—48no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—4128no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—48no—
363C2×Q8—84no—
364C2×Q8—84no—
365C2×Q8—84no—
366C2×Q8—84no—
367C2×Q8—84no—
368C2×Q8—88no—
369C2×Q8—84no—
370C2×Q8—88no—
371C2×Q8C22⋊Q8116yescomplemented
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—216no—
426C22×C4—216no—
427C22×C4—22no—
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—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
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—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—416no—
458C22×C4—48no—
459C22×C4—84no—
460C22×C4—88no—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C22×Q818yes—
466C22×C4C22×Q818yes—
467C22×C4C22×Q814yes—
468C22×C4C22×Q814yes—
469C22×C4C22×Q812yes—
470C22×C4C22×Q814yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q814yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C42−1+414yes—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—416no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C22⋊C4—232no—
492C22⋊C4—416no—
493C42—44no—
494C42—48no—
495C42—44no—
496C42—84no—
497C42—84no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23#54318yes—
517C23#54314yes—
518C23Q8214yes—
519C23Q8218yes—
520C23H(3,𝔽4)14yes—
521C23C23×D411yescharacteristic
522C23C23×Q811yescharacteristic
523C23C23×Q812yes—
524C23C22×#4111yescharacteristic
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×H(5,𝔽2)11yescharacteristic
527C23C2×H(5,𝔽2)12yes—
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×2−1+412yes—
531C23#58512yes—
532Q8—416no—
533Q8—432no—
534Q8—416no—
535Q8—816no—
536Q8—816no—
537C2×C4—232no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—432no—
577C2×C4—432no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—816no—
617C2×C4C2×C22⋊Q814yes—
618C2×C4C2×C22⋊Q818yes—
619C2×C4#54314yes—
620C2×C4D4×Q818yes—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—28no—
625C22—28no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—28no—
630C22—28no—
631C22#281118yescomplemented
632C22C22×C22⋊Q811yescharacteristic
633C22C2×#54312yes—
634C22C2×#54311yescharacteristic
635C22C2×#54311yescharacteristic
636C22C2×D4×Q812yes—
637C22C2×Q8211yescharacteristic
638C22C2×Q8212yes—
639C22Q8×#4112yes—
640C22#348411yescharacteristic
641C22#348411yescharacteristic
642C22#348812yes—
643C22#349011yescharacteristic
644C22#349011yescharacteristic
645C22#350112yes—
646C22#350212yes—
647C22#350312yes—
648C22#352212yes—
649C22C2×H(3,𝔽4)11yescharacteristic
650C22#353512yes—
651C22#353612yes—
652C22#354411yescharacteristic
653C22#354511yescharacteristic
654C22#355812yes—
655C4—28no—
656C4—24no—
657C4—24no—
658C4—28no—
659C4—48no—
660C4—44no—
661C4—48no—
662C4—44no—
663C4—44no—
664C4—84no—
665C4—82no—
666C4—84no—
667C2—28no—
668C2#3498211yescharacteristic
669C2Q8×C22⋊Q812yes—
670C2#3719411yescharacteristic
671C2#3768412yes—
672C2#3844412yes—
673C2#3962412yes—
674C2#4597711yescharacteristic
675C2#4609911yescharacteristic
676C2C2×#281111yescharacteristic
677C2#5915111yescharacteristic
678C2#5992311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2811C2semidirect
3. #17080C2semidirect
4. #16630C2semidirect
5. #15964C2semidirect
6. #15510C2semidirect
10. #14467C2semidirect
11. #14467C2semidirect
75. C23×Q8Q8semidirect
89. C2×C22⋊Q8Q8semidirect
199. C22×Q8C2×Q8semidirect
278. C2×C4⋊C4C2×Q8semidirect
279. C2×C4⋊C4C2×Q8semidirect
371. C2×Q8C22⋊Q8semidirect
631. C22#2811semidirect

Maximal subgroups

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

subgroupindex
C2×#28112
C22×#3912
#170802
#166302
#159642
#155102
#147582
#147202
#147022
#144672
#140522
#139222