Finite Group Databasebeta

← #6616262

Subgroups — #6616262

Order 512.

1845 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36117C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33106C211yesmaximal, characteristic
4#33095C212yesmaximal, complemented
5#27014C214yesmaximal, complemented
6#26706C214yesmaximal, complemented
7#26443C212yesmaximal, complemented
8#25903C211yesmaximal, characteristic
9#25279C212yesmaximal, complemented
10#24608C212yesmaximal, complemented
11#24301C211yesmaximal, characteristic, complemented
12#24066C212yesmaximal, complemented
13#23220C212yesmaximal, complemented
14#22970C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16C4×#483C211yesmaximal, characteristic, complemented
17#21391C212yesmaximal, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2212yes—
20C22×C4×Q8—21no—
21C2×#486C2212yes—
22C2×#483C2211yescharacteristic
23C2×#480C2212yes—
24C2×#475C2212yes—
25C2×#432C2212yes—
26#2856C2211yescharacteristic
27#2854C2212yes—
28#2552C2211yescharacteristic
29#2548C2211yescharacteristic
30#2512—21no—
31#2501—21no—
32#2479—22no—
33#2454C2212yes—
34#2448C2212yes—
35#2425—22no—
36#2366C2211yescharacteristic
37#2365—21no—
38#2365—22no—
39#2365—21no—
40#2365C2212yes—
41#2365C2214yes—
42#2365C2212yes—
43#2361—21no—
44#2354—21no—
45Q8×C4⋊C4C2211yescharacteristic
46C4×C22⋊Q8C2211yescharacteristic
47C2×#391—21no—
48C2×#389C2212yes—
49C2×#387—22no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2087C2214yes—
54#2065C2214yes—
55#2059C2214yes—
56#2058C2212yes—
57#2042C2214yes—
58#2024C2212yes—
59#2023C2214yes—
60#1990C2212yes—
61#1989C2214yes—
62#1988C2212yes—
63#1983C2214yes—
64#1982C2212yes—
65#1947C2212yes—
66#1944C2214yes—
67#1939C2212yes—
68#1936C2214yes—
69#1927C2212yes—
70#1926C2214yes—
71#1926C2214yes—
72#1925C2214yes—
73#1920C2212yes—
74#1919C2214yes—
75#1919C2214yes—
76#1917C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1864C2214yes—
83#1863C2214yes—
84#1853C2212yes—
85#1853C2211yescharacteristic
86#1823C2211yescharacteristic
87#1817C2212yes—
88#1797C2211yescharacteristic
89C4×#106C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92#1762C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102#529—28no—
103#529—44no—
104C2×#126—22no—
105C2×#125C2312yes—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×#123—22no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×#121—21no—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—24no—
116C2×C4×Q8—48no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4C2312yes—
122C2×#117—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—22no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#486—28no—
135#483—24no—
136#480—44no—
137#475—28no—
138#432—44no—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2311yescharacteristic
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#399—22no—
158#399—22no—
159#398—22no—
160#397—22no—
161#396—22no—
162#394—22no—
163#391—22no—
164#391—22no—
165#391—24no—
166#391C2311yescharacteristic
167#390—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389C2311yescharacteristic
180#389C2314yes—
181#389C2312yes—
182#389C2312yes—
183#389C2314yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—21no—
189#387—24no—
190#387—24no—
191#387—24no—
192#387—24no—
193#386—22no—
194#386—22no—
195#385—21no—
196#385—22no—
197#384—21no—
198#384—21no—
199#384C2311yescharacteristic
200#380—22no—
201#380C2311yescharacteristic
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C2×#95—21no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—28no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—44no—
242C22×Q8C2411yescharacteristic
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2411yescharacteristic
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—24no—
263C4⋊Q8#4118yes—
264#127—24no—
265#126—48no—
266#125—24no—
267#125—42no—
268#124—24no—
269#124—42no—
270#123—48no—
271#123—84no—
272C22⋊Q8—22no—
273C22⋊Q8#4114yes—
274#121—24no—
275#121—24no—
276#121—44no—
277#121—82no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—42no—
281C4×Q8—416no—
282C4×Q8—48no—
283C4×Q8—48no—
284C4×D4—24no—
285C4×D4—42no—
286C4×D4—48no—
287C4×D4—44no—
288#117—48no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—48no—
350C2×C42—44no—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—44no—
360#106—28no—
361#106—24no—
362#106—24no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365#103—24no—
366#103—28no—
367#103—24no—
368#102—28no—
369#102—28no—
370#102—28no—
371#102—44no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374C22⋊C8—24no—
375#97—44no—
376C4×C8—24no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—22no—
395C24C22×D411yescharacteristic
396C2×Q8—28no—
397C2×Q8—28no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—44no—
402C2×Q8—432no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8C2×#4112yes—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4—81no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—82no—
453C22×C4AGL(1,ℤ/8)14yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4C2511yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C4⋊C4—24no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4C2×#4118yes—
508C4⋊C4C2×#4112yes—
509C4⋊C42−1+414yes—
510C22⋊C4—24no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C42—22no—
523C42—22no—
524C42—24no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—44no—
532C42—48no—
533C42—48no—
534C42—42no—
535C42—44no—
536C42—44no—
537C42—42no—
538C42C2×#4114yes—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23#54612yes—
551C23C2×AGL(1,ℤ/8)12yes—
552C23#57712yes—
553C23C23×D411yescharacteristic
554Q8—24no—
555Q8—416no—
556Q8—416no—
557Q8—42no—
558D4—44no—
559D4—44no—
560D4—88no—
561D4—84no—
562D4—162no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—416no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—48no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4—84no—
618C2×C4#52512yes—
619C2×C4#52512yes—
620C2×C4#54612yes—
621C2×C4C2×AGL(1,ℤ/8)12yes—
622C2×C4C2×AGL(1,ℤ/8)12yes—
623C2×C4#57512yes—
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4112yes—
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×2−1+412yes—
628C8—42no—
629C8—42no—
630C8—44no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22#333214yes—
645C22C2×#52511yescharacteristic
646C22C2×#54611yescharacteristic
647C22#348611yescharacteristic
648C22C22×AGL(1,ℤ/8)11yescharacteristic
649C22C2×#57511yescharacteristic
650C22C2×#57711yescharacteristic
651C22#358611yescharacteristic
652C4—22no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—48no—
657C4—44no—
658C4—41no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—81no—
665C4—81no—
666C4—82no—
667C4—82no—
668C4#311414yes—
669C4#348212yes—
670C4#352212yes—
671C2—24no—
672C2—162no—
673C2#3642211yescharacteristic
674C2C2×#311411yescharacteristic
675C2#6058211yescharacteristic
676C2C2×#333211yescharacteristic
677C2#6116511yescharacteristic
678C2#6182711yescharacteristic
679C2#6183611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2365C2semidirect
4. #33095C2semidirect
5. #27014C2semidirect
6. #26706C2semidirect
7. #26443C2semidirect
9. #25279C2semidirect
10. #24608C2semidirect
11. #24301C2semidirect
12. #24066C2semidirect
13. #23220C2semidirect
14. #22970C2semidirect
15. #22603C2semidirect
16. C4×#483C2semidirect
17. #21391C2semidirect

Maximal subgroups

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

subgroupindex
#361172
C2×#23652
#331062
#330952
#270142
#267062
#264432
#259032
#252792
#246082
#243012
#240662
#232202
#229702
#226032
C4×#4832
#213912