Finite Group Databasebeta

← #6624347

Subgroups — #6624347

Order 512.

1695 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#36561C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#33148C212yesmaximal, complemented
4#32954C211yesmaximal, characteristic, complemented
5#29111C212yesmaximal, complemented
6#28118C212yesmaximal
7#27960C211yesmaximal, characteristic, complemented
8#27644C211yesmaximal, characteristic
9#27439C214yesmaximal, complemented
10#27375C214yesmaximal, complemented
11#26026C212yesmaximal, complemented
12#24678C212yesmaximal, complemented
13#23068C212yesmaximal, complemented
14#22762C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21578C212yesmaximal
17C4×#448C211yesmaximal, characteristic
18C2×#554—21no—
19C22×C4×Q8C2211yescharacteristic
20C2×#480C2212yes—
21C2×#475C2212yes—
22C2×#448C2211yescharacteristic
23C2×C4×SD16C2212yes—
24#2857C2211yescharacteristic
25#2857C2212yes—
26#2564—22no—
27#2556—22no—
28#2551C2211yescharacteristic
29#2542C2211yescharacteristic
30#2506C2212yes—
31#2492—21no—
32#2452C2212yes—
33#2417—21no—
34#2365C2211yescharacteristic
35#2360—21no—
36#2355—21no—
37#2355—21no—
38#2355—22no—
39#2355C2212yes—
40#2355C2214yes—
41#2355C2212yes—
42D4×C4⋊C4C2211yescharacteristic
43C4×#121C2212yes—
44C4×#121C2211yescharacteristic
45C2×#391C2212yes—
46C2×#387—22no—
47C2×#384C2211yescharacteristic
48C2×#382—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2214yes—
53#2083C2212yes—
54#2072C2214yes—
55#2071C2212yes—
56#2068C2214yes—
57#2055C2214yes—
58#2024C2214yes—
59#1990C2212yes—
60#1987C2214yes—
61#1948C2214yes—
62#1947C2212yes—
63#1939C2212yes—
64#1935C2214yes—
65#1935C2214yes—
66#1928C2214yes—
67#1926C2214yes—
68#1921C2214yes—
69#1920C2214yes—
70#1920C2214yes—
71#1917C2214yes—
72#1880C2212yes—
73#1877C2214yes—
74#1874C2211yescharacteristic
75#1865C2214yes—
76#1865C2212yes—
77#1853C2212yes—
78#1853C2211yescharacteristic
79#1834C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1823C2212yes—
83#1803C2212yes—
84#1802C2212yes—
85#1797C2212yes—
86#1797C2211yescharacteristic
87#1787C2211yescharacteristic
88#1787C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1762C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×SD16C2312yes—
97#554—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#125—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—21no—
107C2×#121C2311yescharacteristic
108C2×C4×Q8—28no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C42—21no—
127C22×C42—22no—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#480—28no—
131#475—28no—
132#448—24no—
133C4×SD16—28no—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×#106C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2311yescharacteristic
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#400—22no—
152#399—22no—
153#398—22no—
154#396—22no—
155#395—22no—
156#393—22no—
157#391—21no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2314yes—
163#391C2314yes—
164#390C2312yes—
165#390C2311yescharacteristic
166#389—21no—
167#389—22no—
168#389—21no—
169#389C2312yes—
170#389C2311yescharacteristic
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#388—22no—
175#388—21no—
176#387—22no—
177#387—22no—
178#387—24no—
179#387—24no—
180#387—24no—
181#387—24no—
182#386—21no—
183#386—22no—
184#385—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#382—24no—
194#382—22no—
195#380—21no—
196#380—22no—
197#380—21no—
198#380C2312yes—
199#380C2311yescharacteristic
200C4×C4⋊C4—22no—
201C4×C4⋊C4—24no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C2×#95—22no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—24no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8C2412yes—
242C22×D4—21no—
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2411yescharacteristic
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—24no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263#125—48no—
264C22⋊Q8—44no—
265C22⋊Q8—48no—
266C22⋊Q8—48no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269#121—22no—
270#121#4114yes—
271C4×Q8—28no—
272C4×Q8—28no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×Q8#4118yes—
278C4×D4—24no—
279C4×D4—44no—
280C4×D4—48no—
281C4×D4—44no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C42—28no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—44no—
341C2×C42—84no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350#107—28no—
351#106—24no—
352#106—28no—
353C4⋊C8—24no—
354C4⋊C8—44no—
355#103—24no—
356#103—28no—
357#103—24no—
358#103—44no—
359#103—42no—
360#102—28no—
361#102—24no—
362C22⋊C8—22no—
363C22⋊C8—24no—
364C22⋊C8—22no—
365#97—44no—
366C4×C8—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—21no—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×Q8C2×#4114yes—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4C2×#4112yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4AGL(1,ℤ/8)14yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4112yes—
459C22×C42−1+412yes—
460C22×C4C2511yescharacteristic
461SD16—48no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C4⋊C4—24no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4#13714yes—
504C4⋊C4C2×#4114yes—
505C4⋊C4C2×#4112yes—
506C22⋊C4—24no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—81no—
514C22⋊C4—82no—
515C42—28no—
516C42—28no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—24no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—42no—
526C42—84no—
527C42—84no—
528C42C2×#4114yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23#54812yes—
541C23C2×AGL(1,ℤ/8)12yes—
542C23#57712yes—
543C23C23×D411yescharacteristic
544Q8—24no—
545Q8—24no—
546Q8—48no—
547Q8—42no—
548Q8—42no—
549Q8—84no—
550Q8—164no—
551D4—24no—
552D4—48no—
553D4—42no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—84no—
606C2×C4—88no—
607C2×C4—84no—
608C2×C4—164no—
609C2×C4#52512yes—
610C2×C4#52512yes—
611C2×C4#54812yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×AGL(1,ℤ/8)12yes—
614C2×C4C2×#13714yes—
615C2×C4C2×#13712yes—
616C2×C4C22×#4111yescharacteristic
617C2×C4C22×#4112yes—
618C2×C4C2×2−1+411yescharacteristic
619C2×C4#58512yes—
620C8—44no—
621C8—42no—
622C8—42no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22#337314yes—
637C22C2×#52511yescharacteristic
638C22C2×#54811yescharacteristic
639C22#348511yescharacteristic
640C22C22×AGL(1,ℤ/8)11yescharacteristic
641C22C22×#13711yescharacteristic
642C22C2×#57711yescharacteristic
643C22#358611yescharacteristic
644C4—24no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—41no—
654C4—44no—
655C4—82no—
656C4—81no—
657C4—84no—
658C4—81no—
659C4—82no—
660C4#311214yes—
661C4#347512yes—
662C4#349512yes—
663C2—24no—
664C2—82no—
665C2#3651611yescharacteristic
666C2C2×#311211yescharacteristic
667C2#6058511yescharacteristic
668C2C2×#337311yescharacteristic
669C2#6132211yescharacteristic
670C2#6172211yescharacteristic
671C2#6173511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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×#2355C2semidirect
3. #33148C2semidirect
4. #32954C2semidirect
5. #29111C2semidirect
7. #27960C2semidirect
9. #27439C2semidirect
10. #27375C2semidirect
11. #26026C2semidirect
12. #24678C2semidirect
13. #23068C2semidirect
14. #22762C2semidirect
15. #22629C2semidirect

Maximal subgroups

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

subgroupindex
#365612
C2×#23552
#331482
#329542
#291112
#281182
#279602
#276442
#274392
#273752
#260262
#246782
#230682
#227622
#226292
#215782
C4×#4482