Finite Group Databasebeta

← #6621065

Subgroups — #6621065

Order 512.

1903 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3375C212yesmaximal
2Q8×C4⋊Q8C211yesmaximal, characteristic, complemented
3C4×#559C211yesmaximal, characteristic
4#33198C211yesmaximal, characteristic, complemented
5#33195C212yesmaximal, complemented
6#28726C212yesmaximal, complemented
7#28716C211yesmaximal, characteristic
8#28254C212yesmaximal, complemented
9#28252C211yesmaximal, characteristic
10#25336C214yesmaximal, complemented
11#25322C212yesmaximal
12#24696C214yesmaximal, complemented
13#24682C212yesmaximal, complemented
14#22722C211yesmaximal, characteristic, complemented
15#22525C212yesmaximal, complemented
16#21504C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×Q82C2212yes—
20#3375—216no—
21C2×#489C2214yes—
22C2×#486C2212yes—
23C2×#477C2212yes—
24C2×#474C2214yes—
25C2×#464C2214yes—
26C2×#462C2211yescharacteristic
27C2×#460C2212yes—
28C2×#461C2212yes—
29C2×#446C2212yes—
30C2×#440C2214yes—
31C2×C4×Q16C2212yes—
32#2868C2211yescharacteristic
33#2868C2212yes—
34#2567C2212yes—
35#2567C2214yes—
36#2551—21no—
37#2551—22no—
38#2551—21no—
39#2551—22no—
40#2374—22no—
41#2367—21no—
42#2367C2212yes—
43#2366C2211yescharacteristic
44#2357—21no—
45#2357C2212yes—
46#2355C2211yescharacteristic
47#2355C2211yescharacteristic
48Q8×C4⋊C4—21no—
49Q8×C4⋊C4—21no—
50C4×C4⋊Q8C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52C4×#127C2211yescharacteristic
53C4×#124—21no—
54C4×#124C2212yes—
55C4×#123—22no—
56C42×Q8C2211yescharacteristic
57C42×Q8C2211yescharacteristic
58C42×D4—21no—
59C42×D4C2212yes—
60#2072C2214yes—
61#2071C2214yes—
62#2071C2214yes—
63#2056C2214yes—
64#2055C2214yes—
65#2055C2214yes—
66#1993C2214yes—
67#1990C2212yes—
68#1988C2214yes—
69#1987C2212yes—
70#1981C2211yescharacteristic
71#1947C2212yes—
72#1947C2214yes—
73#1947C2212yes—
74#1947C2211yescharacteristic
75#1940C2214yes—
76#1939C2212yes—
77#1928C2214yes—
78#1928C2212yes—
79#1878C2212yes—
80#1877C2214yes—
81#1842C2211yescharacteristic
82#1842C2212yes—
83#1805C2214yes—
84#1803C2211yescharacteristic
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2214yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1763C2214yes—
95C4×#103C2212yes—
96C22×Q16C2312yes—
97C22×SD16C2314yes—
98#559—24no—
99#559D418yes—
100Q82—28no—
101Q82—28no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×#127C2311yescharacteristic
112C2×#124—21no—
113C2×#124C2312yes—
114C2×#123—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C22×C42—22no—
132#489—216no—
133#486—28no—
134#477—28no—
135#474—216no—
136#464—216no—
137#462—24no—
138#461—28no—
139#460—28no—
140#446—28no—
141#440—216no—
142C4×Q16—28no—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#97C2314yes—
162C2×C4×C8C2312yes—
163#395—24no—
164#395—24no—
165#395—24no—
166#391—21no—
167#391C2312yes—
168#391C2314yes—
169#391C2312yes—
170#390—21no—
171#390—21no—
172#390—21no—
173#390C2312yes—
174#390C2312yes—
175#389—22no—
176#389—24no—
177#389—24no—
178#389—22no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389—21no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#387—22no—
190#387—22no—
191#386—22no—
192#385—22no—
193#384—22no—
194#384—22no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2314yes—
201#384C2312yes—
202#382—21no—
203#382C2312yes—
204#381—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C43—21no—
233C43C2311yescharacteristic
234C43C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2312yes—
242#336C2312yes—
243#336C2314yes—
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2411yescharacteristic
248C22×Q8C2412yes—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C2×Q16—24no—
256C2×Q16—24no—
257C2×Q16—42no—
258C2×SD16—28no—
259C2×SD16—28no—
260C2×SD16—44no—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—28no—
265C4⋊Q8—24no—
266C4⋊Q8—28no—
267C4⋊Q8—24no—
268C4⋊Q8—22no—
269C4⋊Q8—48no—
270C4⋊Q8—416no—
271C4⋊Q8—84no—
272C4⋊Q8—82no—
273C4⋊Q8—82no—
274C4⋊Q8—84no—
275C4⋊Q8#4114yes—
276#127—22no—
277#127C2×D414yes—
278#124—28no—
279#124—24no—
280#124C2×D418yes—
281#123—216no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—24no—
287C4×Q8—28no—
288C4×Q8—28no—
289C4×Q8—24no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—41no—
293C4×Q8—48no—
294C4×Q8C2×D414yes—
295C4×Q8C2×D414yes—
296C4×Q8#4114yes—
297C4×D4—28no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—42no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—82no—
306C4×D4C2×D418yes—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—84no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—24no—
372#107—28no—
373C4⋊C8—24no—
374C4⋊C8—28no—
375C4⋊C8—24no—
376C4⋊C8—22no—
377C4⋊C8—22no—
378C4⋊C8—44no—
379C4⋊C8—42no—
380#103—24no—
381#103—28no—
382#103—28no—
383#103—24no—
384#103—24no—
385#103—24no—
386#102—28no—
387#102—28no—
388#102—28no—
389#97—28no—
390C4×C8—24no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95—21no—
400#95—24no—
401#95—22no—
402#95—24no—
403#95—22no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407C24—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—48no—
421C2×Q8—44no—
422C2×Q8—84no—
423C2×Q8—82no—
424C2×Q8—84no—
425C2×Q8—82no—
426C2×Q8#12114yes—
427C2×Q8C22×D412yes—
428C2×Q8C2×#4114yes—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—81no—
439C2×D4—81no—
440C2×D4C22×D414yes—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—28no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2511yescharacteristic
469Q16—48no—
470SD16—416no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—82no—
512C4⋊C4#12114yes—
513C4⋊C4#12118yes—
514C4⋊C4#13714yes—
515C4⋊C4C22×D412yes—
516C4⋊C42−1+412yes—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—22no—
535C42—21no—
536C42—41no—
537C42—42no—
538C42—42no—
539C42—48no—
540C42—44no—
541C42—42no—
542C42—42no—
543C42—42no—
544C42—81no—
545C42—82no—
546C42—82no—
547C42#13714yes—
548C42C22×D412yes—
549C42C22×D414yes—
550C42C2×#4114yes—
551C42C2×#4112yes—
552C42C2×#4112yes—
553C422−1+412yes—
554C23—24no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23C23×D411yescharacteristic
562Q8—28no—
563Q8—24no—
564Q8—22no—
565Q8—24no—
566Q8—22no—
567Q8—41no—
568Q8—42no—
569Q8—41no—
570Q8—42no—
571Q8—168no—
572Q8—164no—
573D4—24no—
574D4—24no—
575D4—42no—
576D4—42no—
577D4—48no—
578D4—84no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—88no—
622C2×C4—88no—
623C2×C4—88no—
624C2×C4C2×#12114yes—
625C2×C4C2×#12114yes—
626C2×C4C2×#12112yes—
627C2×C4C2×#12112yes—
628C2×C4#54812yes—
629C2×C4#54812yes—
630C2×C4C2×#13714yes—
631C2×C4C2×#13712yes—
632C2×C4C2×#13712yes—
633C2×C4C2×#13712yes—
634C2×C4C2×#13712yes—
635C2×C4#57612yes—
636C2×C4C23×D411yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×2−1+411yescharacteristic
640C2×C4C2×2−1+411yescharacteristic
641C2×C4#58511yescharacteristic
642C8—44no—
643C8—42no—
644C8—42no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22C22×#12111yescharacteristic
653C22C2×#54811yescharacteristic
654C22#348911yescharacteristic
655C22C22×#13711yescharacteristic
656C22C22×#13711yescharacteristic
657C22C2×#57611yescharacteristic
658C22#358511yescharacteristic
659C4—24no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—24no—
664C4—24no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—82no—
672C4—84no—
673C4—82no—
674C4—81no—
675C4#303914yes—
676C4#337514yescomplemented
677C4D4×#4111yescharacteristic
678C4#348211yescharacteristic
679C4#349911yescharacteristic
680C4#351811yescharacteristic
681C2—84no—
682C2#3644411yescharacteristic
683C2C2×#303911yescharacteristic
684C2#6042811yescharacteristic
685C2C2×#337511yescharacteristic
686C2#6131911yescharacteristic
687C2#6211211yescharacteristic
688C2#6212711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Q8C2semidirect
4. #33198C2semidirect
5. #33195C2semidirect
6. #28726C2semidirect
8. #28254C2semidirect
10. #25336C2semidirect
12. #24696C2semidirect
13. #24682C2semidirect
14. #22722C2semidirect
15. #22525C2semidirect
16. #21504C2semidirect
17. C4×#462C2semidirect
676. C4#3375semidirect

Maximal subgroups

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

subgroupindex
C2×#33752
Q8×C4⋊Q82
C4×#5592
#331982
#331952
#287262
#287162
#282542
#282522
#253362
#253222
#246962
#246822
#227222
#225252
#215042
C4×#4622