Finite Group Databasebeta

← #949112

Subgroups — #949112

Order 512.

4450 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×#2437C214yesmaximal, complemented
2#17137C211yesmaximal, characteristic, complemented
3#17136C212yesmaximal, complemented
4#17135C211yesmaximal, characteristic, complemented
5#16323C214yesmaximal, complemented
6#16265C214yesmaximal, complemented
7#15951C212yesmaximal, complemented
8#15429C212yesmaximal, complemented
9#14914C212yesmaximal, complemented
10#14726C214yesmaximal, complemented
11#14275C212yesmaximal, complemented
12#13867C212yesmaximal
13C2×#1438C211yesmaximal, characteristic
14C24×D4C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2214yescomplemented
17C23×C22⋊C4C2212yes—
18C23×C22⋊C4C2214yes—
19#2437—232no—
20C2×#399C2214yescomplemented
21C2×#396C2214yescomplemented
22C2×#396C2214yescomplemented
23C2×#396C2214yescomplemented
24C2×#396C2214yescomplemented
25C2×#393C2214yescomplemented
26C2×#386C2214yes—
27C2×#386C2214yes—
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#385C2214yescomplemented
31C2×#385C2214yescomplemented
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2214yescomplemented
35C2×#379C2214yes—
36C2×#379C2212yes—
37C22×#95C2211yescharacteristic
38C22×#95C2211yescharacteristic
39#1446C2212yescomplemented
40#1446C2212yescomplemented
41#1444C2212yescomplemented
42#1444C2212yescomplemented
43#1444C2212yescomplemented
44#1444C2212yescomplemented
45#1442C2212yescomplemented
46#1442C2212yescomplemented
47#1441C2214yescomplemented
48#1441C2214yescomplemented
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1438C2211yescharacteristic
58#1438C2211yescharacteristic
59#1438C2214yes—
60#1438C2211yescharacteristic
61#1438C2211yescharacteristic
62#1438C2212yes—
63#1438C2212yes—
64#1438C2212yes—
65#1438C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C26C2312yes—
69C23×Q8C2312yes—
70C23×D4—42no—
71C23×D4—42no—
72C23×D4—48no—
73C23×D4Q818yescomplemented
74C23×D4C2311yescharacteristic
75C23×D4C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2314yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—44no—
84C2×C22⋊Q8—48no—
85C2×C22⋊Q8—48no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4—216no—
96C22×C22⋊C4—216no—
97C22×C22⋊C4—48no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4—48no—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115#399—216no—
116#396—216no—
117#396—216no—
118#396—48no—
119#396—48no—
120#393—216no—
121#386—48no—
122#386—48no—
123#385—216no—
124#385—216no—
125#385—216no—
126#385—216no—
127#380—216no—
128#380—44no—
129#380D4116yescomplemented
130#379—216no—
131#379—44no—
132C2×#95—42no—
133C2×#95—42no—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C25—28no—
155C25—28no—
156C25—24no—
157C25—24no—
158C25—28no—
159C25—44no—
160C25—44no—
161C25—44no—
162C25C2×Q818yes—
163C25C2412yes—
164C25C2412yes—
165C25C2411yescharacteristic
166C22×Q8—28no—
167C22×Q8—24no—
168C22×Q8—42no—
169C22×Q8—44no—
170C22×Q8—42no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—416no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—432no—
183C22×D4—432no—
184C22×D4—432no—
185C22×D4C2×Q814yes—
186C22×D4C2×Q818yes—
187C22×D4C2×Q814yes—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—24no—
194C23×C4—28no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4C2×D418yes—
208C23×C4C2×Q814yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C22⋊Q8—432no—
220C22⋊Q8—464no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C2×D4116yescomplemented
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—432no—
268C2×C22⋊C4—464no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D4116yes—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—41no—
294#95—42no—
295#95—41no—
296#95—41no—
297#95—42no—
298#95—41no—
299#95—42no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95C2×D418yescomplemented
307#95C2×D418yescomplemented
308C24—22no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—24no—
315C24—22no—
316C24—24no—
317C24—24no—
318C24—216no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—216no—
328C24—232no—
329C24—48no—
330C24—416no—
331C24—416no—
332C24—416no—
333C24—416no—
334C24—416no—
335C24—416no—
336C24—416no—
337C24—416no—
338C24—48no—
339C24—48no—
340C24—88no—
341C24—88no—
342C24—88no—
343C24—88no—
344C24—88no—
345C24—84no—
346C24—84no—
347C24C22×D414yes—
348C24C22×Q814yes—
349C24C22×Q814yes—
350C24C22×Q812yes—
351C24H(5,𝔽2)18yes—
352C24H(5,𝔽2)12yes—
353C24H(5,𝔽2)12yes—
354C242−1+414yes—
355C24C2511yescharacteristic
356C2×Q8—48no—
357C2×Q8—416no—
358C2×Q8—88no—
359C2×Q8—84no—
360C2×Q8—84no—
361C2×D4—216no—
362C2×D4—216no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—44no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—4128no—
373C2×D4—48no—
374C2×D4—48no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—216no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—48no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—88no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—84no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4—84no—
444C22×C4—88no—
445C22×C4C22×D414yes—
446C22×C4C22×D418yes—
447C22×C4C22×D414yes—
448C22×C4C22×D418yes—
449C22×C4C22×Q812yes—
450C22×C4C22×Q812yes—
451C22×C4H(5,𝔽2)14yes—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—416no—
455C4⋊C4—416no—
456C4⋊C4—416no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C22⋊C4—232no—
462C22⋊C4—232no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—432no—
503C23—464no—
504C23—44no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—432no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—816no—
532C23—832no—
533C23—832no—
534C23—832no—
535C23—832no—
536C23—832no—
537C23#53718yes—
538C23#54318yes—
539C23#54314yes—
540C23D4214yes—
541C23D4×Q818yes—
542C23#56314yes—
543C23C23×D412yes—
544C23C23×Q811yescharacteristic
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)14yes—
548C23C2×H(5,𝔽2)12yes—
549C23C2×2−1+412yes—
550C23#58512yes—
551Q8—816no—
552D4—416no—
553D4—432no—
554D4—416no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—416no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—816no—
613C2×C4—832no—
614C2×C4—816no—
615C22—24no—
616C22—216no—
617C22—28no—
618C22—24no—
619C22—28no—
620C22—28no—
621C22—28no—
622C22—216no—
623C22—28no—
624C22—44no—
625C22—48no—
626C22—48no—
627C22—44no—
628C22—48no—
629C22—48no—
630C22—48no—
631C22—44no—
632C22—48no—
633C22—48no—
634C22—44no—
635C22—44no—
636C22—864no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22—84no—
641C22#268318yes—
642C22C2×#53712yes—
643C22C2×#54311yescharacteristic
644C22C2×#54312yes—
645C22C2×D4211yescharacteristic
646C22C2×D4×Q812yes—
647C22#348414yes—
648C22#348814yes—
649C22#350112yes—
650C22C2×#56311yescharacteristic
651C22#353212yes—
652C22#353212yes—
653C22#353412yes—
654C22#353514yes—
655C22#353614yes—
656C22#354311yescharacteristic
657C22#354511yescharacteristic
658C4—48no—
659C4—48no—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—82no—
666C4—84no—
667C4—84no—
668C2—28no—
669C2—48no—
670C2—84no—
671C2#3713411yescharacteristic
672C2#3747511yescharacteristic
673C2#3768414yes—
674C2#4058612yes—
675C2#4198312yes—
676C2#4309812yes—
677C2C2×#268311yescharacteristic
678C2#5241112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2437C2semidirect
2. #17137C2semidirect
3. #17136C2semidirect
4. #17135C2semidirect
5. #16323C2semidirect
6. #16265C2semidirect
7. #15951C2semidirect
8. #15429C2semidirect
9. #14914C2semidirect
10. #14726C2semidirect
11. #14275C2semidirect
16. C22×C22⋊Q8C22semidirect
20. C2×#399C22semidirect
21. C2×#396C22semidirect
22. C2×#396C22semidirect
23. C2×#396C22semidirect
24. C2×#396C22semidirect
25. C2×#393C22semidirect
30. C2×#385C22semidirect
31. C2×#385C22semidirect
34. C2×#380C22semidirect
39. #1446C22semidirect
40. #1446C22semidirect
41. #1444C22semidirect
42. #1444C22semidirect
43. #1444C22semidirect
44. #1444C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
73. C23×D4Q8semidirect
129. #380D4semidirect
244. C2×C4⋊C4C2×D4semidirect
306. #95C2×D4semidirect
307. #95C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24372
#171372
#171362
#171352
#163232
#162652
#159512
#154292
#149142
#147262
#142752
#138672
C2×#14382