Finite Group Databasebeta

← #6623450

Subgroups — #6623450

Order 512.

1707 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic
2C4×#557C211yesmaximal, characteristic, complemented
3#33198C211yesmaximal, characteristic, complemented
4#33073C212yesmaximal, complemented
5#29141C211yesmaximal, characteristic, complemented
6#29132C212yesmaximal, complemented
7#28252C211yesmaximal, characteristic
8#28160C212yesmaximal
9#27439C214yesmaximal, complemented
10#26919C214yesmaximal, complemented
11#24990C212yesmaximal, complemented
12#24187C212yesmaximal, complemented
13#23002C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22762C212yesmaximal, complemented
16#21505C212yesmaximal
17C4×#460C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×#554—21no—
20C2×#477C2212yes—
21C2×#460C2211yescharacteristic
22C2×#460C2212yes—
23C2×#461C2212yes—
24C2×#440C2214yes—
25C2×C4×SD16C2212yes—
26#2868C2211yescharacteristic
27#2859C2212yes—
28#2566—22no—
29#2563—21no—
30#2538—21no—
31#2538C2212yes—
32#2452—22no—
33#2452C2214yes—
34#2370—21no—
35#2370—22no—
36#2365C2212yes—
37#2365C2211yescharacteristic
38#2355C2212yes—
39#2355C2211yescharacteristic
40#2351—21no—
41Q8×C4⋊C4C2211yescharacteristic
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2212yes—
47C4×#127C2211yescharacteristic
48C4×#125—21no—
49C4×#125—22no—
50C42×Q8—21no—
51C42×Q8C2212yes—
52C42×Q8C2211yescharacteristic
53C42×Q8C2211yescharacteristic
54#2099C2214yes—
55#2071C2214yes—
56#2055C2214yes—
57#2043C2214yes—
58#2024C2214yes—
59#2013C2214yes—
60#1987C2212yes—
61#1984C2212yes—
62#1982C2211yescharacteristic
63#1956C2212yes—
64#1948C2211yescharacteristic
65#1947C2212yes—
66#1947C2212yes—
67#1935C2214yes—
68#1935C2214yes—
69#1926C2212yes—
70#1925C2214yes—
71#1920C2214yes—
72#1918C2214yes—
73#1917C2214yes—
74#1880C2214yes—
75#1880C2212yes—
76#1877C2214yes—
77#1877C2214yes—
78#1865C2214yes—
79#1863C2214yes—
80#1842C2211yescharacteristic
81#1834C2212yes—
82#1831C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1802C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1762C2214yes—
94C4×#102C2212yes—
95C22×SD16C2314yes—
96C22×SD16C2312yes—
97#557—24no—
98#557—24no—
99#554—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#125—22no—
108C2×#125—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—22no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2314yes—
128C22×C4⋊C4—22no—
129#477—44no—
130#461—28no—
131#460—24no—
132#460—28no—
133#440—48no—
134C4×SD16—28no—
135C2×#107C2314yes—
136C2×#106C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#396—24no—
154#396—24no—
155#393—24no—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#390—22no—
160#390—21no—
161#390—21no—
162#390—22no—
163#390C2312yes—
164#390C2314yes—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—22no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2314yes—
176#389C2314yes—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—24no—
184#385—24no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#382—21no—
193#382—22no—
194#382—22no—
195#380—22no—
196#380—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C43—22no—
223C43—21no—
224C43C2312yes—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2411yescharacteristic
240C22×Q8C2412yes—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4—24no—
246C2×SD16—24no—
247C2×SD16—28no—
248C2×SD16—28no—
249C2×SD16—24no—
250C2×SD16—42no—
251C2×SD16—44no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—42no—
259C4⋊Q8—42no—
260C4⋊Q8—44no—
261C4⋊Q8—82no—
262C4⋊Q8#4114yes—
263#127—22no—
264#127#4114yes—
265#125—216no—
266#125—28no—
267#125—48no—
268C22⋊Q8—416no—
269C22⋊Q8—48no—
270C4×Q8—28no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—24no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—24no—
282C4×Q8—22no—
283C4×Q8—24no—
284C4×Q8—42no—
285C4×Q8—41no—
286C4×Q8—44no—
287C4×Q8#4114yes—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—48no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—84no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—28no—
363#106—24no—
364C4⋊C8—22no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367C4⋊C8—22no—
368C4⋊C8—42no—
369#103—24no—
370#103—28no—
371#103—44no—
372#103—44no—
373#103—44no—
374#102—28no—
375#102—24no—
376#102—24no—
377#102—24no—
378#97—44no—
379C4×C8—24no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—44no—
410C2×Q8—82no—
411C2×Q8—82no—
412C2×Q8AGL(1,ℤ/8)14yes—
413C2×Q8C2×#4114yes—
414C2×Q8C2×#4112yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4C2×#4114yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2511yescharacteristic
452SD16—48no—
453SD16—416no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—48no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4C2×#4112yes—
503C4⋊C4H(5,𝔽2)12yes—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C42—28no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—21no—
521C42—22no—
522C42—21no—
523C42—41no—
524C42—42no—
525C42—42no—
526C42—44no—
527C42—42no—
528C42—44no—
529C42—44no—
530C42—42no—
531C42—42no—
532C42—81no—
533C42—82no—
534C42#13714yes—
535C42C2×#4112yes—
536C42C2×#4112yes—
537C42C2×#4112yes—
538C422−1+412yes—
539C23—24no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23C23×D411yescharacteristic
547Q8—28no—
548Q8—22no—
549Q8—24no—
550Q8—22no—
551Q8—24no—
552Q8—42no—
553Q8—41no—
554Q8—42no—
555Q8—41no—
556Q8—164no—
557D4—24no—
558D4—24no—
559D4—48no—
560D4—48no—
561D4—42no—
562D4—42no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—88no—
608C2×C4—168no—
609C2×C4#52512yes—
610C2×C4#52512yes—
611C2×C4#54812yes—
612C2×C4#54812yes—
613C2×C4C2×AGL(1,ℤ/8)14yes—
614C2×C4C2×AGL(1,ℤ/8)12yes—
615C2×C4C2×#13712yes—
616C2×C4C2×#13712yes—
617C2×C4C2×#13712yes—
618C2×C4#57712yes—
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C2×C4#58511yescharacteristic
625C8—42no—
626C8—44no—
627C8—42no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22C2×#52511yescharacteristic
636C22C2×#54811yescharacteristic
637C22#348811yescharacteristic
638C22C22×AGL(1,ℤ/8)11yescharacteristic
639C22C22×#13711yescharacteristic
640C22C2×#57711yescharacteristic
641C22#358611yescharacteristic
642C4—22no—
643C4—24no—
644C4—22no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—41no—
649C4—42no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—84no—
655C4—82no—
656C4—81no—
657C4—82no—
658C4#311214yes—
659C4#337214yes—
660C4#347511yescharacteristic
661C4#348111yescharacteristic
662C4#351111yescharacteristic
663C4#351511yescharacteristic
664C2—84no—
665C2#3649211yescharacteristic
666C2C2×#311211yescharacteristic
667C2#6059911yescharacteristic
668C2C2×#337211yescharacteristic
669C2#6131711yescharacteristic
670C2#6196611yescharacteristic
671C2#6199911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#557C2semidirect
3. #33198C2semidirect
4. #33073C2semidirect
5. #29141C2semidirect
6. #29132C2semidirect
9. #27439C2semidirect
10. #26919C2semidirect
11. #24990C2semidirect
12. #24187C2semidirect
13. #23002C2semidirect
14. #22825C2semidirect
15. #22762C2semidirect

Maximal subgroups

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

subgroupindex
#365592
C4×#5572
#331982
#330732
#291412
#291322
#282522
#281602
#274392
#269192
#249902
#241872
#230022
#228252
#227622
#215052
C4×#4602