Finite Group Databasebeta

← #6651596

Subgroups — #6651596

Order 512.

1622 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35248C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic
3#33100C212yesmaximal
4#32693C212yesmaximal, complemented
5#32682C212yesmaximal, complemented
6#32644C212yesmaximal, complemented
7#32630C212yesmaximal, complemented
8#32586C214yesmaximal, complemented
9#32579C214yesmaximal, complemented
10#29642C212yesmaximal
11#27617C212yesmaximal
12#24606C212yesmaximal
13#24602C212yesmaximal
14#24064C212yesmaximal
15#21820C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#484C2212yes—
21C2×#484C2212yes—
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24#2855C2212yes—
25#2855C2212yes—
26#2634—21no—
27#2634—21no—
28#2581C2212yes—
29#2581C2212yes—
30#2568C2211yescharacteristic
31#2568C2211yescharacteristic
32#2546—21no—
33#2546—21no—
34#2546—21no—
35#2546—21no—
36#2546C2212yes—
37#2546C2212yes—
38#2546C2212yes—
39#2546C2212yes—
40#2546C2211yescharacteristic
41#2546C2211yescharacteristic
42#2503—21no—
43#2484—21no—
44#2484—21no—
45#2434—21no—
46C2×#398—21no—
47C2×#398—21no—
48#2327—21no—
49#2327—21no—
50C2×#391—21no—
51C2×#389C2212yes—
52C2×#389C2212yes—
53#2283C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55#2101C2214yes—
56#2087C2214yes—
57#2086C2214yes—
58#2085C2214yes—
59#2068C2214yes—
60#2066C2214yes—
61#2057C2214yes—
62#2057C2214yes—
63#2057C2214yes—
64#2043C2214yes—
65#2024C2214yes—
66#2024C2214yes—
67#1985C2212yes—
68#1985C2212yes—
69#1984C2212yes—
70#1984C2212yes—
71#1944C2214yes—
72#1942C2214yes—
73#1935C2214yes—
74#1933C2214yes—
75#1928C2214yes—
76#1927C2214yes—
77#1919C2214yes—
78#1919C2214yes—
79#1918C2214yes—
80#1918C2214yes—
81#1872C2212yes—
82#1872C2212yes—
83#1863C2214yes—
84#1863C2214yes—
85#1853C2212yes—
86#1852C2212yes—
87#1824C2212yes—
88#1824C2212yes—
89#1817C2212yes—
90#1817C2212yes—
91C23×Q8—21no—
92C23×Q8D411yescharacteristic
93#533—42no—
94#533—42no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—42no—
104C2×C4⋊Q8—42no—
105C2×C4⋊Q8—48no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#125—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8—42no—
115C2×C4×Q8—42no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×#117—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#484—44no—
139#484—44no—
140#432—42no—
141#431—42no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×#97C2312yes—
155C2×C4×C8C2312yes—
156#400—21no—
157#400—21no—
158#400—21no—
159#400—21no—
160#400—22no—
161#400—22no—
162#398—21no—
163#398—21no—
164#398—21no—
165#398—21no—
166#398—22no—
167#398—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#398—22no—
172#398—22no—
173#398—22no—
174#395—22no—
175#395—22no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—22no—
181#391—22no—
182#391—22no—
183#391—22no—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389C2312yes—
193#389C2312yes—
194#389C2314yes—
195#389C2314yes—
196#389C2314yes—
197#389C2312yes—
198#389C2312yes—
199#389C2314yes—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#385—22no—
209#385—22no—
210#383—21no—
211#382—21no—
212#382—21no—
213#382C2312yes—
214#381—22no—
215#381—22no—
216#381—21no—
217#380C2312yes—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—21no—
225#376—21no—
226C2×#95—21no—
227C2×#95—22no—
228C2×#95—22no—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#336C2314yes—
238#336C2314yes—
239#336C2314yes—
240#336C2314yes—
241C22×Q8—24no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8—44no—
248C22×Q8—44no—
249C22×Q8—44no—
250C22×Q8C2×D411yescharacteristic
251C22×Q8C2×D411yescharacteristic
252C22×Q8C2×D411yescharacteristic
253C22×Q8C2×D411yescharacteristic
254C22×Q8C2412yes—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2411yescharacteristic
264C22×C8C2414yes—
265C22×C8C2414yes—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8—42no—
270C4⋊Q8—41no—
271C4⋊Q8—41no—
272C4⋊Q8—41no—
273C4⋊Q8—41no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—48no—
278C4⋊Q8—48no—
279C4⋊Q8—48no—
280C4⋊Q8—44no—
281C4⋊Q8—48no—
282#125—44no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—44no—
286C4×Q8—42no—
287C4×Q8—42no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×Q8—41no—
291C4×Q8—41no—
292C4×Q8—41no—
293C4×Q8—41no—
294C4×Q8—41no—
295C4×Q8—41no—
296C4×Q8—41no—
297C4×Q8—41no—
298C4×Q8—42no—
299C4×Q8—42no—
300C4×Q8—44no—
301#117—44no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—48no—
327C2×C4⋊C4—48no—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2414yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—44no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—28no—
372#107—44no—
373#107—44no—
374#106—28no—
375C4⋊C8—42no—
376C4⋊C8—42no—
377#103—28no—
378#103—28no—
379#103—44no—
380#103—44no—
381C22⋊C8—24no—
382C22⋊C8—24no—
383#97—42no—
384C4×C8—42no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408C24C22×D411yescharacteristic
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—41no—
419C2×Q8—41no—
420C2×Q8—41no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8—41no—
424C2×Q8—416no—
425C2×Q8—42no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—42no—
433C2×Q8—44no—
434C2×Q8—816no—
435C2×Q8AGL(1,ℤ/8)14yes—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4C2511yescharacteristic
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C4⋊C4—28no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C42−1+418yes—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—44no—
544C42—44no—
545C42—42no—
546C42—44no—
547C42—42no—
548C23—22no—
549C23—21no—
550C23—21no—
551C23#53412yes—
552C23#57814yes—
553C23C23×D411yescharacteristic
554Q8—44no—
555Q8—44no—
556Q8—44no—
557Q8—44no—
558Q8—44no—
559Q8—44no—
560Q8—84no—
561Q8—88no—
562Q8—88no—
563C2×C4—28no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—28no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—416no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—88no—
618C2×C4#53412yes—
619C2×C4C2×AGL(1,ℤ/8)12yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4C2×AGL(1,ℤ/8)12yes—
622C2×C4#57512yes—
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×2−1+414yes—
626C8—44no—
627C8—44no—
628C22—22no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22#320214yes—
633C22C2×#53411yescharacteristic
634C22#348612yes—
635C22C22×AGL(1,ℤ/8)11yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57812yes—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—48no—
652C4—84no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4#354412yes—
658C4#354412yes—
659C2—24no—
660C2#3725811yescharacteristic
661C2C2×#320211yescharacteristic
662C2#6075011yescharacteristic
663C2#6183712yes—
664C2#6184212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32693C2semidirect
5. #32682C2semidirect
6. #32644C2semidirect
7. #32630C2semidirect
8. #32586C2semidirect
9. #32579C2semidirect

Maximal subgroups

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

subgroupindex
#352482
C2×#25462
#331002
#326932
#326822
#326442
#326302
#325862
#325792
#296422
#276172
#246062
#246022
#240642
#218202