Finite Group Databasebeta

← #6484730

Subgroups — #6484730

Order 512.

2060 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
1C2×#3239C212yesmaximal, complemented
2#36631C211yesmaximal, characteristic
3C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
4#33109C212yesmaximal, complemented
5#33107C212yesmaximal
6#29761C214yesmaximal, complemented
7#28754C212yesmaximal, complemented
8#28693C212yesmaximal, complemented
9#28658C214yesmaximal, complemented
10#28283C212yesmaximal, complemented
11#28279C212yesmaximal, complemented
12#28203C214yesmaximal, complemented
13#21867C212yesmaximal, complemented
14#20810C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C2×#529C2212yes—
17C22×C4×Q8—21no—
18#3239—216no—
19C2×#501C2214yes—
20C2×#489C2214yes—
21C2×#484C2214yes—
22C2×#481C2214yes—
23C2×#475C2214yes—
24C2×#474C2214yes—
25C2×#432C2212yes—
26#2861C2212yes—
27#2860—21no—
28#2860—21no—
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2857C2211yescharacteristic
32#2856C2211yescharacteristic
33#2854—21no—
34#2854—21no—
35#2850C2212yes—
36C4×C4⋊Q8—21no—
37C4×C4⋊Q8—21no—
38C4×C4⋊Q8—21no—
39C4×C4⋊Q8—21no—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#124C2212yes—
47C4×#121—21no—
48C4×#121—21no—
49C2×#391—21no—
50#2293—21no—
51#2293—21no—
52C2×#389—21no—
53C2×#389C2212yes—
54C2×#384C2211yescharacteristic
55C2×C4×C4⋊C4—21no—
56C2×C4×C4⋊C4C2212yes—
57C2×C43C2211yescharacteristic
58#2085C2218yes—
59#2067C2218yes—
60#2066C2218yes—
61#2042C2218yes—
62#1992C2214yes—
63#1982C2214yes—
64#1981C2214yes—
65#1959C2214yes—
66#1948C2214yes—
67#1947C2214yes—
68#1928C2218yes—
69#1926C2218yes—
70#1874C2214yes—
71#1872C2214yes—
72#1842C2212yes—
73#1831C2212yes—
74#1803C2214yes—
75#1802C2214yes—
76#1797C2212yes—
77#1797C2212yes—
78#1767C2212yes—
79#1767C2212yes—
80C23×Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8D418yescomplemented
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89#529—28no—
90#529—28no—
91C2×#126—22no—
92C2×#125C2312yes—
93C2×#124C2312yes—
94C2×#124C2312yes—
95C2×#123—22no—
96C2×C22⋊Q8C2312yes—
97C2×#121—21no—
98C2×#121—21no—
99C2×#121C2312yes—
100C2×C4×Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—48no—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4C2312yes—
110C2×#117—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C42—22no—
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#501—216no—
126#489—216no—
127#484—216no—
128#481—216no—
129#475—216no—
130#474—216no—
131#432—28no—
132C2×#107C2318yes—
133C2×#106C2318yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2318yes—
138C2×#103C2318yes—
139C2×#102C2318yes—
140C2×#102C2318yes—
141C2×C22⋊C8C2314yes—
142C2×#97C2314yes—
143#391—22no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391C2312yes—
148#391C2312yes—
149#390—21no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389C2314yes—
174#389C2314yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#386—22no—
184#386—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#384C2311yescharacteristic
191#382C2312yes—
192#380—21no—
193#380—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2314yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C2×#95—22no—
207C2×#95—22no—
208C43D411yescharacteristic
209C43D411yescharacteristic
210C43D411yescharacteristic
211C43D411yescharacteristic
212C43C2311yescharacteristic
213C43C2311yescharacteristic
214C43C2311yescharacteristic
215C43C2311yescharacteristic
216#340C2318yes—
217#340C2318yes—
218#336C2314yes—
219#336C2314yes—
220C22×Q8—28no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8C2412yes—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2412yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C22×C8C2418yes—
241C4⋊Q8—28no—
242C4⋊Q8—28no—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8C2×D418yes—
247C4⋊Q8C2×D418yes—
248#127—24no—
249#126—48no—
250#125—24no—
251#125—24no—
252#124—24no—
253#124—24no—
254#124—44no—
255#123—48no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258#121—24no—
259#121—24no—
260#121—44no—
261#121—44no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—48no—
265C4×Q8—48no—
266C4×Q8—48no—
267C4×Q8—48no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—44no—
271C4×D4—48no—
272C4×D4—44no—
273#117—48no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4C2×D418yes—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2414yes—
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×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—48no—
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×C42C2×D414yes—
348C2×C42C2×D411yescharacteristic
349C2×C42#4114yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—216no—
357#106—216no—
358C4⋊C8—28no—
359C4⋊C8—28no—
360C4⋊C8—44no—
361#103—216no—
362#103—48no—
363#102—216no—
364#102—48no—
365C22⋊C8—28no—
366#97—28no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377C24—22no—
378C24C22×D411yescharacteristic
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—432no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—28no—
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—41no—
425C22×C4—41no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4#12114yes—
435C22×C4#12114yes—
436C22×C4#13714yes—
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×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4112yes—
452C22×C4C2511yescharacteristic
453C2×C8—28no—
454C2×C8—28no—
455C2×C8—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4C22×D418yes—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
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—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—48no—
518C42—44no—
519C42—48no—
520C42#12114yes—
521C42#12114yes—
522C42AGL(1,ℤ/8)14yes—
523C42C22×D412yes—
524C42C22×D412yes—
525C42C2×#4112yes—
526C42C2×#4112yes—
527C422−1+414yes—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23C2×#12112yes—
540C23C2×#13712yes—
541C23#57512yes—
542C23C23×D411yescharacteristic
543Q8—416no—
544Q8—416no—
545Q8—44no—
546Q8—44no—
547D4—44no—
548D4—44no—
549D4—84no—
550D4—88no—
551D4—84no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—28no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
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—21no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—416no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—88no—
603C2×C4—88no—
604C2×C4C2×#12112yes—
605C2×C4C2×#12112yes—
606C2×C4C2×#12112yes—
607C2×C4C2×#12112yes—
608C2×C4C2×#12112yes—
609C2×C4#53612yes—
610C2×C4#53612yes—
611C2×C4C2×AGL(1,ℤ/8)12yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×AGL(1,ℤ/8)12yes—
614C2×C4C2×#13712yes—
615C2×C4C2×#13712yes—
616C2×C4#57512yes—
617C2×C4C23×D411yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C2×2−1+412yes—
620C2×C4#58512yes—
621C8—48no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22#303714yes—
636C22C22×#12111yescharacteristic
637C22C2×#53611yescharacteristic
638C22#344911yescharacteristic
639C22C22×AGL(1,ℤ/8)11yescharacteristic
640C22C22×#13711yescharacteristic
641C22C2×#57511yescharacteristic
642C22C2×#57511yescharacteristic
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—48no—
655C4—84no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4#303614yes—
660C4#323914yescomplemented
661C4#349712yes—
662C4#350712yes—
663C2—24no—
664C2—162no—
665C2#3517811yescharacteristic
666C2C2×#303611yescharacteristic
667C2C2×#303711yescharacteristic
668C2C2×#323911yescharacteristic
669C2#6086611yescharacteristic
670C2#6095211yescharacteristic
671C2#6096211yescharacteristic
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
1. C2×#3239C2semidirect
3. C2×C4×C4⋊Q8C2semidirect
4. #33109C2semidirect
6. #29761C2semidirect
7. #28754C2semidirect
8. #28693C2semidirect
9. #28658C2semidirect
10. #28283C2semidirect
11. #28279C2semidirect
12. #28203C2semidirect
13. #21867C2semidirect
14. #20810C2semidirect
84. C2×C4⋊Q8D4semidirect
660. C4#3239semidirect

Maximal subgroups

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

subgroupindex
C2×#32392
#366312
C2×C4×C4⋊Q82
#331092
#331072
#297612
#287542
#286932
#286582
#282832
#282792
#282032
#218672
#208102