Finite Group Databasebeta

← #399972

Subgroups — #399972

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic, complemented
3C2×#1928C211yesmaximal, characteristic, complemented
4C2×#1872C212yesmaximal, complemented
5#15177C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic, complemented
7#11096C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10964C211yesmaximal, characteristic
10#10807C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal, complemented
12C22×C22⋊Q8C2212yes—
13C22×C4×D4—21no—
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2212yes—
16C22×#107C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22D4×C4⋊C4—24no—
23D4×C4⋊C4—24no—
24C2×#391C2211yescharacteristic
25C2×#389—21no—
26C2×#385—22no—
27C2×#384C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2072—24no—
32#1928—24no—
33#1872—28no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1444—21no—
43#1439—22no—
44#1438—21no—
45C4⋊#95C2211yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×D4D411yescharacteristic
48C24×C4—22no—
49C24×C4C2312yes—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8—42no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—28no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—48no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D412yes—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C2×#107—22no—
88C2×#107—42no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—22no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103D414yescomplemented
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—42no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—42no—
103#391—22no—
104#391—22no—
105#389—28no—
106#385—216no—
107#384—22no—
108#384D414yes—
109#380—24no—
110#380—24no—
111#380—24no—
112#380—42no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4D414yes—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—21no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—22no—
126#340—24no—
127#340—42no—
128#336—22no—
129#336—22no—
130C25C2×D412yes—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—21no—
140C22×Q8C2×D412yes—
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4Q1614yes—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—28no—
164C23×C4—21no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—82no—
170C23×C4—82no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4#4114yes—
175C23×C4#4114yes—
176C23×C4C2411yescharacteristic
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8C2×D412yes—
194C22⋊Q8—216no—
195C22⋊Q8—48no—
196C22⋊Q8—48no—
197C22⋊Q8—48no—
198C4×D4—28no—
199C4×D4—28no—
200C4×D4—216no—
201C4×D4—432no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4#4114yes—
243C2×C4⋊C4#4112yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—82no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—44no—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262#107—44no—
263#103—28no—
264#103—48no—
265#103—44no—
266C22⋊C8—48no—
267C22⋊C8—48no—
268#95—28no—
269#95—24no—
270#95—44no—
271#95—42no—
272#95—42no—
273#95—42no—
274#95—81no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24#12314yes—
288C24C2×Q1614yes—
289C24AGL(1,ℤ/8)14yes—
290C24C22×D411yescharacteristic
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×D4—28no—
310C2×D4—28no—
311C2×D4—216no—
312C2×D4—216no—
313C2×D4—28no—
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—22no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—416no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—416no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
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—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—88no—
391C22×C4—88no—
392C22×C4—88no—
393C22×C4—88no—
394C22×C4—88no—
395C22×C4#12012yes—
396C22×C4#12012yes—
397C22×C4#12314yes—
398C22×C4C2×Q1612yes—
399C22×C4#13712yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4H(5,𝔽2)11yescharacteristic
408C22×C4H(5,𝔽2)12yes—
409C22×C4H(5,𝔽2)11yescharacteristic
410C2×C8—22no—
411C2×C8—24no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—41no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C4⋊C4—28no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—416no—
438C4⋊C4—48no—
439C4⋊C4—416no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—42no—
448C4⋊C4—84no—
449C4⋊C4#12114yes—
450C22⋊C4—216no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—88no—
454C42—22no—
455C42—22no—
456C42—24no—
457C42—48no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23#48418yes—
493C23C2×#12011yescharacteristic
494C23C2×#12312yes—
495C23D4211yescharacteristic
496C23#54611yescharacteristic
497C23#54612yes—
498C23C22×Q1611yescharacteristic
499C23C2×AGL(1,ℤ/8)12yes—
500C23C2×#13711yescharacteristic
501C23#57712yes—
502C23#57711yescharacteristic
503C23#57811yescharacteristic
504Q8—42no—
505Q8—44no—
506Q8—42no—
507Q8—81no—
508Q8—81no—
509Q8—82no—
510D4—232no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—832no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4#45114yes—
571C2×C4C2×#12111yescharacteristic
572C2×C4#52511yescharacteristic
573C2×C4#52511yescharacteristic
574C2×C4#52512yes—
575C2×C4#52512yes—
576C2×C4#52912yes—
577C2×C4#53211yescharacteristic
578C2×C4#53511yescharacteristic
579C2×C4#53511yescharacteristic
580C2×C4#53812yes—
581C2×C4#53812yes—
582C2×C4D4211yescharacteristic
583C2×C4#54612yes—
584C2×C4#54611yescharacteristic
585C2×C4#55012yes—
586C2×C4#55112yes—
587C2×C4#55312yes—
588C2×C4#55312yes—
589C8—42no—
590C8—81no—
591C8—82no—
592C22—28no—
593C22—28no—
594C22—28no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22#240811yescharacteristic
608C22#241812yes—
609C22#243111yescharacteristic
610C22#244412yes—
611C22#284911yescharacteristic
612C22C2×#45111yescharacteristic
613C22#300311yescharacteristic
614C22#300911yescharacteristic
615C22#301311yescharacteristic
616C22C2×#48412yes—
617C22#309212yes—
618C22#309712yes—
619C22#309912yes—
620C22#326512yes—
621C22#327811yescharacteristic
622C22D4×Q1611yescharacteristic
623C22#329912yes—
624C22#329911yescharacteristic
625C22#330612yes—
626C22#330812yes—
627C22#331012yes—
628C22#332811yescharacteristic
629C22#333112yes—
630C4—24no—
631C4—44no—
632C4—44no—
633C4—42no—
634C4—84no—
635C4—84no—
636C4—82no—
637C4—82no—
638C4—161no—
639C4#239211yescharacteristic
640C4#245612yes—
641C4#246712yes—
642C4#253511yescharacteristic
643C4#260311yescharacteristic
644C4#263111yescharacteristic
645C2—48no—
646C2#1479511yescharacteristic
647C2#2424311yescharacteristic
648C2#2425211yescharacteristic
649C2#2486712yes—
650C2#2487912yes—
651C2#2557311yescharacteristic
652C2#2557711yescharacteristic
653C2#2620412yes—
654C2#2621612yes—
655C2#3298311yescharacteristic
656C2#3298811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2072C2semidirect
3. C2×#1928C2semidirect
4. C2×#1872C2semidirect
6. #11371C2semidirect
7. #11096C2semidirect
8. #10970C2semidirect
10. #10807C2semidirect
11. #10730C2semidirect
96. C2×#103D4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#20722
C2×#19282
C2×#18722
#151772
#113712
#110962
#109702
#109642
#108072
#107302