Finite Group Databasebeta

← #6483194

Subgroups — #6483194

Order 512.

2164 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
1C2×#3218C212yesmaximal, complemented
2#36631C211yesmaximal, characteristic
3C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
4#32909C214yesmaximal, complemented
5#29715C212yesmaximal, complemented
6#29713C212yesmaximal
7#28723C212yesmaximal, complemented
8#28619C214yesmaximal, complemented
9#28447C212yesmaximal, complemented
10#28254C214yesmaximal, complemented
11#28161C212yesmaximal, complemented
12#28157C212yesmaximal, complemented
13#21867C212yesmaximal, complemented
14#20810C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C2×#529C2212yes—
17C22×C4×Q8C2212yes—
18#3218—216no—
19C2×#497C2214yes—
20C2×#491C2214yes—
21C2×#464C2218yes—
22C2×#462C2214yes—
23C2×#461C2214yes—
24C2×#451C2214yes—
25C2×#448C2214yes—
26C2×#432C2212yes—
27#2861C2212yes—
28#2860—21no—
29#2860—21no—
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2857C2211yescharacteristic
33#2856C2211yescharacteristic
34#2854—21no—
35#2854—21no—
36#2850C2212yes—
37C4×C4⋊Q8—21no—
38C4×C4⋊Q8—21no—
39C4×C4⋊Q8—21no—
40C4×C4⋊Q8—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×#127C2211yescharacteristic
47C4×#124C2212yes—
48C4×#121—21no—
49C4×#121—21no—
50C2×#391C2212yes—
51#2293—21no—
52#2293—21no—
53C2×#389—21no—
54C2×#389—21no—
55C2×#384C2211yescharacteristic
56C2×C4×C4⋊C4—21no—
57C2×C4×C4⋊C4—21no—
58C2×C43C2211yescharacteristic
59#2072C2214yes—
60#2072C2214yes—
61#2056C2218yes—
62#2055C2218yes—
63#2016C2214yes—
64#2014C2214yes—
65#1992C2214yes—
66#1959C2214yes—
67#1880C2218yes—
68#1878C2218yes—
69#1874C2214yes—
70#1872C2214yes—
71#1842C2212yes—
72#1831C2212yes—
73#1805C2218yes—
74#1803C2214yes—
75#1802C2214yes—
76#1797C2212yes—
77#1797C2212yes—
78#1767C2212yes—
79#1767C2212yes—
80C23×Q8C2312yes—
81C22×Q16C2318yes—
82C22×SD16C2318yes—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8D418yescomplemented
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×#127C2311yescharacteristic
91#529—28no—
92#529—28no—
93C2×#126—22no—
94C2×#125C2312yes—
95C2×#124C2312yes—
96C2×#124C2312yes—
97C2×#123—22no—
98C2×C22⋊Q8C2312yes—
99C2×#121—21no—
100C2×#121—21no—
101C2×#121C2312yes—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—48no—
105C2×C4×Q8C2314yes—
106C2×C4×Q8C2312yes—
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#497—216no—
128#491—216no—
129#464—416no—
130#462—216no—
131#461—216no—
132#451—216no—
133#448—216no—
134#432—28no—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2318yes—
139C2×#103C2318yes—
140C2×#103C2318yes—
141C2×#102C2318yes—
142C2×C22⋊C8C2314yes—
143C2×#97C2314yes—
144#391—22no—
145#391—22no—
146#391C2314yes—
147#391C2314yes—
148#391C2312yes—
149#391C2312yes—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390—21no—
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#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
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#382C2312yes—
193#380—21no—
194#380—21no—
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⋊C4—21no—
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#336C2314yes—
220#336C2314yes—
221C22×Q8—22no—
222C22×Q8—24no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8C2×D418yes—
226C22×Q8C2412yes—
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×Q16—216no—
243C2×Q16—48no—
244C2×Q16—48no—
245C2×SD16—216no—
246C2×SD16—48no—
247C2×SD16—48no—
248C22×C8C2418yes—
249C4⋊Q8—28no—
250C4⋊Q8—28no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—42no—
254C4⋊Q8C2×D418yes—
255C4⋊Q8C2×D418yes—
256#127—42no—
257#126—48no—
258#125—24no—
259#125—24no—
260#124—24no—
261#124—24no—
262#124—28no—
263#123—48no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266#121—24no—
267#121—24no—
268#121—44no—
269#121—44no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—216no—
273C4×Q8—48no—
274C4×Q8—48no—
275C4×Q8—48no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—44no—
279C4×D4—48no—
280C4×D4—44no—
281#117—48no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—48no—
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×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D414yes—
355C2×C42C2×D411yescharacteristic
356C2×C42#4114yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C4⋊C8—28no—
364C4⋊C8—28no—
365C4⋊C8—44no—
366#103—216no—
367#103—216no—
368#103—48no—
369#102—48no—
370C22⋊C8—28no—
371#97—28no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382C24—22no—
383C24C22×D411yescharacteristic
384C2×Q8—28no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—28no—
389C2×Q8—24no—
390C2×Q8—28no—
391C2×Q8—24no—
392C2×Q8—416no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—416no—
396C2×Q8C22×D418yes—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4#12114yes—
442C22×C4#12114yes—
443C22×C4#13714yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4112yes—
459C22×C4C2511yescharacteristic
460Q16—816no—
461SD16—816no—
462C2×C8—28no—
463C2×C8—28no—
464C2×C8—28no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—28no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
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—22no—
515C42—22no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—48no—
525C42—44no—
526C42—48no—
527C42#12114yes—
528C42#12114yes—
529C42#13714yes—
530C42C22×D412yes—
531C42C22×D412yes—
532C42C2×#4112yes—
533C42C2×#4112yes—
534C42H(5,𝔽2)14yes—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23C2×#12112yes—
547C23C2×#13712yes—
548C23#57512yes—
549C23C23×D411yescharacteristic
550Q8—216no—
551Q8—416no—
552Q8—48no—
553Q8—44no—
554Q8—44no—
555D4—44no—
556D4—44no—
557D4—84no—
558D4—84no—
559D4—88no—
560C2×C4—28no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—28no—
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—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—416no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—88no—
611C2×C4—88no—
612C2×C4C2×#12112yes—
613C2×C4C2×#12112yes—
614C2×C4C2×#12112yes—
615C2×C4C2×#12112yes—
616C2×C4C2×#12112yes—
617C2×C4#53512yes—
618C2×C4#53512yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4#57512yes—
625C2×C4C23×D411yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)12yes—
628C2×C4#58512yes—
629C8—48no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22#303714yes—
644C22C22×#12111yescharacteristic
645C22C2×#53511yescharacteristic
646C22#344911yescharacteristic
647C22C22×#13711yescharacteristic
648C22C22×#13711yescharacteristic
649C22C2×#57511yescharacteristic
650C22C2×#57511yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—41no—
662C4—48no—
663C4—41no—
664C4—82no—
665C4—84no—
666C4—82no—
667C4—82no—
668C4#303714yes—
669C4#349612yes—
670C4#350612yes—
671C2—24no—
672C2—162no—
673C2#3516511yescharacteristic
674C2C2×#303711yescharacteristic
675C2C2×#303711yescharacteristic
676C2#6079011yescharacteristic
677C2#6080011yescharacteristic
678C2#6095011yescharacteristic
679C2#6096011yescharacteristic
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×#3218C2semidirect
3. C2×C4×C4⋊Q8C2semidirect
4. #32909C2semidirect
5. #29715C2semidirect
7. #28723C2semidirect
8. #28619C2semidirect
9. #28447C2semidirect
10. #28254C2semidirect
11. #28161C2semidirect
12. #28157C2semidirect
13. #21867C2semidirect
14. #20810C2semidirect
86. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32182
#366312
C2×C4×C4⋊Q82
#329092
#297152
#297132
#287232
#286192
#284472
#282542
#281612
#281572
#218672
#208102