Finite Group Databasebeta

← #6541371

Subgroups — #6541371

Order 512.

1817 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34308C211yesmaximal, characteristic
3#29980C211yesmaximal, characteristic, complemented
4#29969C212yesmaximal, complemented
5#29276C211yesmaximal, characteristic, complemented
6#29208C211yesmaximal, characteristic, complemented
7#29162C212yesmaximal, complemented
8#26443C212yesmaximal, complemented
9#26364C214yesmaximal, complemented
10#26041C212yesmaximal, complemented
11#26012C214yesmaximal, complemented
12#24304C212yesmaximal, complemented
13#23775C211yesmaximal, characteristic, complemented
14#23554C212yesmaximal, complemented
15#22700C211yesmaximal, characteristic, complemented
16#22665C212yesmaximal, complemented
17#22500C212yesmaximal
18C2×#547C2211yescharacteristic
19C2×#533C2212yes—
20C2×#520—21no—
21C2×#478C2214yes—
22C2×#436C2212yes—
23C2×#428C2214yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2552C2212yes—
27#2542—21no—
28#2538—21no—
29#2441C2214yes—
30#2429—22no—
31#2423—22no—
32#2365—21no—
33#2365C2211yescharacteristic
34#2361C2212yes—
35#2360—21no—
36#2359—22no—
37#2354C2212yes—
38Q8×C4⋊C4—21no—
39D4×C4⋊C4C2211yescharacteristic
40D4×C4⋊C4C2211yescharacteristic
41#2344C2212yes—
42#2339—21no—
43#2323—21no—
44#2322—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#121C2212yes—
48#2293C2211yescharacteristic
49C2×#389C2212yes—
50C42×D4C2211yescharacteristic
51#2100C2211yescharacteristic
52#2100C2212yes—
53#2099C2212yes—
54#2094C2211yescharacteristic
55#2093C2212yes—
56#2088C2212yes—
57#2087C2214yes—
58#2087C2214yes—
59#2065C2212yes—
60#2059C2214yes—
61#2059C2214yes—
62#2058C2211yescharacteristic
63#2058C2212yes—
64#2057C2212yes—
65#2057C2212yes—
66#2057C2212yes—
67#2023C2212yes—
68#2023C2211yescharacteristic
69#1990C2212yes—
70#1989C2214yes—
71#1988C2212yes—
72#1948C2212yes—
73#1947C2214yes—
74#1947C2212yes—
75#1939C2212yes—
76#1920C2214yes—
77#1919C2214yes—
78#1895C2214yes—
79#1875C2214yes—
80#1862C2214yes—
81#1854C2214yes—
82#1852C2214yes—
83#1851C2214yes—
84#1845C2212yes—
85#1831C2211yescharacteristic
86#1824C2214yes—
87#1809C2214yes—
88#1802C2212yes—
89#1797C2212yes—
90C4×#106C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×#41—21no—
96C22×M16C2314yes—
97#547—24no—
98#547Q818yescomplemented
99#533—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125C2314yes—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—21no—
111C2×#121C2312yes—
112#520—48no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C2×#117—21no—
128C2×#117C2314yes—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C42C2312yes—
134#478—48no—
135#436—28no—
136#428—48no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C4×C8C2312yes—
151#399—24no—
152#398—24no—
153#396—24no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—22no—
158#390—22no—
159#390C2311yescharacteristic
160#390C2311yescharacteristic
161#389—21no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—24no—
169#389C2312yes—
170#389C2312yes—
171#389C2311yescharacteristic
172#389C2312yes—
173#389C2312yes—
174#389C2314yes—
175#389C2311yescharacteristic
176#388—24no—
177#388—21no—
178#388—22no—
179#388—21no—
180#388—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—24no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#384—22no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#381—22no—
198#380—22no—
199#380—21no—
200#380C2312yes—
201#380C2314yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C2×#95—22no—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2312yes—
235C2×#41—28no—
236C2×#41—44no—
237C2×#41—44no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×D4—22no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C2×M16—28no—
252C2×M16—28no—
253C2×M16—44no—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—82no—
262C4⋊Q8C2×Q814yes—
263#125—28no—
264#125—44no—
265#123—216no—
266C22⋊Q8—28no—
267C22⋊Q8—48no—
268C22⋊Q8—48no—
269C22⋊Q8—84no—
270#121—24no—
271#121C2×Q818yes—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—48no—
275C4×Q8—48no—
276C4×Q8—44no—
277C4×D4—24no—
278C4×D4—22no—
279C4×D4—42no—
280C4×D4—44no—
281C4×D4—48no—
282C4×D4—48no—
283C4×D4—44no—
284C4×D4C2×Q814yes—
285C4×D4C2×Q814yes—
286C4×D4#4114yes—
287#117—28no—
288#117—48no—
289#117—44no—
290#117—44no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×Q818yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—44no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—44no—
359#107—44no—
360#107—44no—
361#106—24no—
362#106—24no—
363#106—28no—
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367#102—24no—
368#102—24no—
369#102—44no—
370C22⋊C8—28no—
371C4×C8—24no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—24no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—21no—
389C24C22×D412yes—
390#41—432no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×D4—28no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4C22⋊Q814yes—
409C2×D4C22×Q812yes—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×Q814yes—
448C22×C4C2×#4114yes—
449C22×C4C2511yescharacteristic
450M16—416no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C82−1+414yes—
460C4⋊C4—24no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4C22⋊Q814yes—
495C4⋊C4C22×Q812yes—
496C4⋊C4C22×Q814yes—
497C4⋊C4H(5,𝔽2)14yes—
498C22⋊C4—24no—
499C22⋊C4—28no—
500C22⋊C4—24no—
501C22⋊C4—42no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—81no—
508C22⋊C4—82no—
509C22⋊C4—84no—
510C22⋊C4C22⋊Q818yes—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—42no—
517C42—42no—
518C42—41no—
519C42—48no—
520C42—44no—
521C42—44no—
522C42—42no—
523C42—44no—
524C42AGL(1,ℤ/8)14yes—
525C42C22×Q812yes—
526C42C2×#4112yes—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C2×C22⋊Q814yes—
541C23C23×D411yescharacteristic
542Q8—48no—
543Q8—48no—
544Q8—88no—
545D4—28no—
546D4—24no—
547D4—416no—
548D4—42no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—416no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—164no—
599C2×C4—164no—
600C2×C4C2×C22⋊Q812yes—
601C2×C4C2×C22⋊Q814yes—
602C2×C4C2×C22⋊Q812yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4#57512yes—
607C2×C4#57812yes—
608C2×C4C23×Q811yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)12yes—
611C2×C4C2×2−1+412yes—
612C8—24no—
613C8—42no—
614C8—44no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C22×C22⋊Q811yescharacteristic
630C22#348411yescharacteristic
631C22#349911yescharacteristic
632C22C22×AGL(1,ℤ/8)11yescharacteristic
633C22C2×#57511yescharacteristic
634C22C2×#57811yescharacteristic
635C22#358511yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—24no—
640C4—22no—
641C4—41no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—81no—
648C4—82no—
649C4—81no—
650C4—82no—
651C4—84no—
652C4#307714yes—
653C4#350312yes—
654C4#352012yes—
655C2—44no—
656C2—82no—
657C2#3573911yescharacteristic
658C2C2×#307711yescharacteristic
659C2#6049811yescharacteristic
660C2#6158711yescharacteristic
661C2#6159511yescharacteristic
662C2#6247911yescharacteristic
663C2#6251211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36134C2semidirect
3. #29980C2semidirect
4. #29969C2semidirect
5. #29276C2semidirect
6. #29208C2semidirect
7. #29162C2semidirect
8. #26443C2semidirect
9. #26364C2semidirect
10. #26041C2semidirect
11. #26012C2semidirect
12. #24304C2semidirect
13. #23775C2semidirect
14. #23554C2semidirect
15. #22700C2semidirect
16. #22665C2semidirect
98. #547Q8semidirect

Maximal subgroups

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

subgroupindex
#361342
#343082
#299802
#299692
#292762
#292082
#291622
#264432
#263642
#260412
#260122
#243042
#237752
#235542
#227002
#226652
#225002