Finite Group Databasebeta

← #6469271

Subgroups — #6469271

Order 512.

2044 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#29500C214yesmaximal, complemented
4#29193C212yesmaximal, complemented
5#28614C212yesmaximal, complemented
6#27973C211yesmaximal, characteristic
7#27950C211yesmaximal, characteristic, complemented
8#25856C211yesmaximal, characteristic, complemented
9#25718C211yesmaximal, characteristic
10#24509C214yesmaximal, complemented
11#23683C214yesmaximal, complemented
12#21692C212yesmaximal, complemented
13#21647C212yesmaximal, complemented
14#21454C214yesmaximal, complemented
15C4×#432C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#529—21no—
18C22×C4×Q8C2212yes—
19C2×#492C2212yes—
20C2×#487C2212yes—
21C2×#448C2214yes—
22C2×#440C2218yes—
23C2×#432C2212yes—
24#2861—21no—
25#2860—21no—
26#2860C2212yes—
27#2859C2211yescharacteristic
28#2858C2211yescharacteristic
29#2857C2211yescharacteristic
30#2856C2211yescharacteristic
31#2854—21no—
32#2854C2212yes—
33#2850—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8—21no—
36C4×C4⋊Q8—21no—
37C4×C4⋊Q8—21no—
38C4×C4⋊Q8C2211yescharacteristic
39C4×C4⋊Q8C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×C4⋊Q8C2212yes—
43C4×#127C2211yescharacteristic
44C4×#124—21no—
45C4×#121—21no—
46C4×#121C2212yes—
47C2×#391C2212yes—
48#2293—21no—
49#2293C2212yes—
50C2×#389—21no—
51C2×#389—21no—
52C2×#384C2211yescharacteristic
53C2×C4×C4⋊C4—21no—
54C2×C4×C4⋊C4—21no—
55C2×C43C2211yescharacteristic
56#2083C2214yes—
57#2071C2214yes—
58#2068C2218yes—
59#2055C2218yes—
60#2043C2214yes—
61#2013C2214yes—
62#1990C2212yes—
63#1973C2218yes—
64#1957C2212yes—
65#1956C2212yes—
66#1956C2212yes—
67#1935C2218yes—
68#1880C2218yes—
69#1874C2214yes—
70#1863C2218yes—
71#1852C2214yes—
72#1845C2212yes—
73#1834C2212yes—
74#1787C2214yes—
75#1767C2212yes—
76C4×C4⋊C8C2212yes—
77#1763C2218yes—
78#1749C2212yes—
79C4×C22⋊C8C2211yescharacteristic
80C4×C22⋊C8C2211yescharacteristic
81#1745C2212yes—
82C4×#97C2212yes—
83C23×Q8C2312yes—
84C22×SD16C2318yes—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92#529—48no—
93C2×#126—22no—
94C2×#125—21no—
95C2×#124—21no—
96C2×#124—21no—
97C2×#123—22no—
98C2×C22⋊Q8—21no—
99C2×#121—21no—
100C2×#121—21no—
101C2×#121C2312yes—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—21no—
105C2×C4×Q8—48no—
106C2×C4×Q8C2314yes—
107C2×C4×Q8C2314yes—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2312yes—
119C22×C42—22no—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#492—28no—
128#487—28no—
129#448—216no—
130#440—416no—
131#432—28no—
132C2×#107C2318yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2318yes—
135C2×#103C2318yes—
136C2×#103C2318yes—
137C2×#102C2318yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×#97C2314yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#391—22no—
145#391—22no—
146#391—21no—
147#391—21no—
148#391C2314yes—
149#391C2314yes—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#386—22no—
185#386—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#382—21no—
193#380—21no—
194#380C2312yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C2×#95—22no—
208C2×#95—22no—
209C43D411yescharacteristic
210C43D411yescharacteristic
211C43D411yescharacteristic
212C43D411yescharacteristic
213C43C2311yescharacteristic
214C43C2311yescharacteristic
215C43C2311yescharacteristic
216C43C2311yescharacteristic
217#340C2318yes—
218#340C2318yes—
219#336C2312yes—
220#336C2312yes—
221#336C2314yes—
222C22×Q8—28no—
223C22×Q8—22no—
224C22×Q8—21no—
225C22×Q8—28no—
226C22×Q8—44no—
227C22×Q8C2414yes—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2412yes—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×SD16—216no—
243C2×SD16—216no—
244C2×SD16—48no—
245C22×C8C2414yes—
246C22×C8C2414yes—
247C4⋊Q8—28no—
248C4⋊Q8—28no—
249C4⋊Q8—28no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8#4118yes—
253#127—24no—
254#126—48no—
255#125—44no—
256#124—44no—
257#124—44no—
258#123—48no—
259C22⋊Q8—44no—
260#121—28no—
261#121—44no—
262#121—44no—
263C4×Q8—216no—
264C4×Q8—48no—
265C4×Q8—48no—
266C4×Q8—48no—
267C4×Q8—44no—
268C4×D4—28no—
269C4×D4—44no—
270C4×D4—44no—
271C4×D4—48no—
272#117—48no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4C2414yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—42no—
324C2×C42—48no—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—48no—
354C4⋊C8—28no—
355#103—216no—
356#103—216no—
357#103—48no—
358#102—48no—
359C22⋊C8—24no—
360C22⋊C8#4118yes—
361#97—28no—
362#97—28no—
363C4×C8—24no—
364C4×C8—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375C24—22no—
376C24C22×D411yescharacteristic
377C2×Q8—28no—
378C2×Q8—24no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—432no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8C2×#4118yes—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4AGL(1,ℤ/8)14yes—
433C22×C4#13714yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4114yes—
448C22×C4C2511yescharacteristic
449SD16—432no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8C2×#4118yes—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—41no—
515C42—41no—
516C42—48no—
517C42—44no—
518C42—48no—
519C42#13714yes—
520C42C2×#4114yes—
521C42H(5,𝔽2)14yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23#53212yes—
534C23C2×AGL(1,ℤ/8)12yes—
535C23C2×#13712yes—
536C23C23×D411yescharacteristic
537Q8—216no—
538Q8—416no—
539Q8—48no—
540Q8—84no—
541D4—48no—
542D4—88no—
543D4—84no—
544D4—84no—
545C2×C4—28no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—416no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—88no—
595C2×C4—88no—
596C2×C4#52512yes—
597C2×C4#52512yes—
598C2×C4#53212yes—
599C2×C4#53512yes—
600C2×C4#53512yes—
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4C2×#13712yes—
606C2×C4C2×#13712yes—
607C2×C4C2×#13712yes—
608C2×C4#57512yes—
609C2×C4C22×#4112yes—
610C2×C4C22×#4112yes—
611C2×C4C2×H(5,𝔽2)12yes—
612C8—28no—
613C8—44no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22#318214yes—
628C22C2×#52511yescharacteristic
629C22C2×#53211yescharacteristic
630C22C2×#53511yescharacteristic
631C22C22×AGL(1,ℤ/8)11yescharacteristic
632C22C22×#13711yescharacteristic
633C22C22×#13711yescharacteristic
634C22C2×#57511yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—48no—
647C4—82no—
648C4—82no—
649C4—84no—
650C4—82no—
651C4#311214yes—
652C4#311714yes—
653C4#348114yes—
654C2—24no—
655C2—162no—
656C2#3496511yescharacteristic
657C2C2×#311711yescharacteristic
658C2C2×#311211yescharacteristic
659C2C2×#318211yescharacteristic
660C2#6070111yescharacteristic
661C2#6080011yescharacteristic
662C2#6080311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Q8C2semidirect
3. #29500C2semidirect
4. #29193C2semidirect
5. #28614C2semidirect
7. #27950C2semidirect
8. #25856C2semidirect
10. #24509C2semidirect
11. #23683C2semidirect
12. #21692C2semidirect
13. #21647C2semidirect
14. #21454C2semidirect
15. C4×#432C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C2×C4×C4⋊Q82
#295002
#291932
#286142
#279732
#279502
#258562
#257182
#245092
#236832
#216922
#216472
#214542
C4×#4322