Finite Group Databasebeta

← #6626451

Subgroups — #6626451

Order 512.

1799 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33149C212yesmaximal, complemented
4#33147C211yesmaximal, characteristic, complemented
5#29301C212yesmaximal, complemented
6#29238C212yesmaximal
7#27015C214yesmaximal, complemented
8#26873C214yesmaximal, complemented
9#26444C212yesmaximal, complemented
10#25813C211yesmaximal, characteristic, complemented
11#25449C212yesmaximal, complemented
12#24298C211yesmaximal, characteristic
13#23161C212yesmaximal, complemented
14#22968C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21580C212yesmaximal
17C4×#481C211yesmaximal, characteristic
18C2×#554—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8—21no—
21C2×#500C2212yes—
22C2×#481C2211yescharacteristic
23C2×#474C2212yes—
24#2857C2212yes—
25#2857C2211yescharacteristic
26#2564—22no—
27#2556—22no—
28#2551C2211yescharacteristic
29#2542C2211yescharacteristic
30#2506C2212yes—
31#2492—21no—
32#2452C2212yes—
33#2417—21no—
34#2365—21no—
35#2365—21no—
36#2365—22no—
37#2365C2212yes—
38#2365C2212yes—
39#2365C2214yes—
40#2365C2211yescharacteristic
41#2360—21no—
42D4×C4⋊C4C2211yescharacteristic
43C4×#121C2212yes—
44C4×#121C2211yescharacteristic
45C2×#391—21no—
46C2×#389C2212yes—
47C2×#387—22no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2088C2214yes—
52#2083C2214yes—
53#2067C2214yes—
54#2058C2214yes—
55#2057C2214yes—
56#2057C2212yes—
57#2023C2212yes—
58#1990C2212yes—
59#1988C2212yes—
60#1988C2212yes—
61#1987C2214yes—
62#1948C2212yes—
63#1947C2214yes—
64#1943C2214yes—
65#1943C2214yes—
66#1939C2212yes—
67#1926C2212yes—
68#1925C2214yes—
69#1921C2214yes—
70#1921C2214yes—
71#1921C2214yes—
72#1920C2214yes—
73#1919C2212yes—
74#1919C2214yes—
75#1917C2214yes—
76#1874C2211yescharacteristic
77#1874C2212yes—
78#1864C2214yes—
79#1853C2212yes—
80#1853C2211yescharacteristic
81#1852C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1824C2211yescharacteristic
85#1824C2212yes—
86#1797C2212yes—
87#1797C2211yescharacteristic
88#1776C2214yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1762C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96#554—48no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#125—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—21no—
111C2×#121C2311yescharacteristic
112C2×C4×Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—24no—
115C2×C4×Q8—22no—
116C2×C4×Q8—48no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C42—22no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#500—28no—
134#481Q818yescomplemented
135#474—28no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#400—22no—
153#399—22no—
154#398—22no—
155#396—22no—
156#395—22no—
157#393—22no—
158#391—22no—
159#391—24no—
160#391—22no—
161#391—21no—
162#391—21no—
163#391—22no—
164#390C2312yes—
165#390C2311yescharacteristic
166#389—21no—
167#389—22no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389C2312yes—
172#389C2311yescharacteristic
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#388—22no—
179#388—21no—
180#387—22no—
181#387—22no—
182#387—24no—
183#387—24no—
184#387—24no—
185#387—24no—
186#386—21no—
187#386—22no—
188#385—22no—
189#385—22no—
190#384C2312yes—
191#384C2311yescharacteristic
192#380—21no—
193#380—22no—
194#380—21no—
195#380C2312yes—
196#380C2311yescharacteristic
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C2×#95—21no—
216C2×#95—22no—
217C2×#95—22no—
218C2×#95D412yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8—28no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×D4—21no—
242C22×D4C2411yescharacteristic
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2411yescharacteristic
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—28no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263C4⋊Q8#4118yes—
264#125—48no—
265C22⋊Q8—44no—
266C22⋊Q8—48no—
267C22⋊Q8—48no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270#121—22no—
271#121C2×Q814yes—
272C4×Q8—48no—
273C4×Q8—416no—
274C4×Q8—44no—
275C4×Q8—48no—
276C4×D4—24no—
277C4×D4—44no—
278C4×D4—48no—
279C4×D4—44no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×Q814yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2414yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42—48no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348#107—28no—
349#107—28no—
350#107—28no—
351#106—42no—
352#106—44no—
353#106C2×Q818yescomplemented
354C4⋊C8—24no—
355C4⋊C8—44no—
356#102—28no—
357#102—42no—
358#102C2×Q818yescomplemented
359C22⋊C8—22no—
360C22⋊C8—24no—
361C22⋊C8C2×Q814yes—
362#97—44no—
363C4×C8—24no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—21no—
384C24C22×D411yescharacteristic
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—44no—
388C2×Q8—432no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4C22×Q812yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4AGL(1,ℤ/8)14yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×Q812yes—
452C22×C42−1+412yes—
453C22×C4C2511yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8C22×Q814yes—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—81no—
494C4⋊C4—82no—
495C4⋊C4AGL(1,ℤ/8)14yes—
496C4⋊C4C22×Q814yes—
497C4⋊C4C22×Q812yes—
498C4⋊C4C2×#4118yes—
499C22⋊C4—24no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—81no—
507C22⋊C4—82no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—24no—
512C42—24no—
513C42—42no—
514C42—42no—
515C42—44no—
516C42—48no—
517C42—44no—
518C42—48no—
519C42—44no—
520C42—42no—
521C42C2×#4114yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23D4×Q812yes—
534C23C2×AGL(1,ℤ/8)12yes—
535C23#57812yes—
536C23C23×D411yescharacteristic
537Q8—416no—
538Q8—416no—
539Q8—84no—
540Q8—164no—
541D4—24no—
542D4—48no—
543D4—42no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—416no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—164no—
599C2×C4#52512yes—
600C2×C4#52512yes—
601C2×C4D4×Q812yes—
602C2×C4C2×AGL(1,ℤ/8)14yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C23×Q811yescharacteristic
607C2×C4C22×#4112yes—
608C2×C4C2×2−1+411yescharacteristic
609C2×C4#58512yes—
610C8—24no—
611C8—42no—
612C8—44no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22#338614yes—
627C22C2×#52511yescharacteristic
628C22C2×D4×Q811yescharacteristic
629C22#348511yescharacteristic
630C22C22×AGL(1,ℤ/8)11yescharacteristic
631C22C22×AGL(1,ℤ/8)11yescharacteristic
632C22C2×#57811yescharacteristic
633C22#358611yescharacteristic
634C4—24no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—48no—
642C4—41no—
643C4—42no—
644C4—44no—
645C4—82no—
646C4—81no—
647C4—82no—
648C4—81no—
649C4—82no—
650C4#310914yes—
651C4Q8×#4112yes—
652C4#351612yes—
653C2—24no—
654C2—82no—
655C2#3654711yescharacteristic
656C2C2×#310911yescharacteristic
657C2#6058311yescharacteristic
658C2C2×#338611yescharacteristic
659C2Q8×AGL(1,ℤ/8)11yescharacteristic
660C2#6172211yescharacteristic
661C2#6173711yescharacteristic
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×#2365C2semidirect
3. #33149C2semidirect
4. #33147C2semidirect
5. #29301C2semidirect
7. #27015C2semidirect
8. #26873C2semidirect
9. #26444C2semidirect
10. #25813C2semidirect
11. #25449C2semidirect
13. #23161C2semidirect
14. #22968C2semidirect
15. #22629C2semidirect
134. #481Q8semidirect
353. #106C2×Q8semidirect
358. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365612
C2×#23652
#331492
#331472
#293012
#292382
#270152
#268732
#264442
#258132
#254492
#242982
#231612
#229682
#226292
#215802
C4×#4812