Finite Group Databasebeta

← #6578587

Subgroups — #6578587

Order 512.

2123 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3108C212yesmaximal, complemented
2#36117C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic, complemented
4#33106C211yesmaximal, characteristic
5#33003C212yesmaximal, complemented
6#26903C214yesmaximal, complemented
7#26591C214yesmaximal, complemented
8#26465C212yesmaximal, complemented
9#25718C211yesmaximal, characteristic
10#25369C212yesmaximal, complemented
11#24503C212yesmaximal, complemented
12#24212C211yesmaximal, characteristic, complemented
13#23937C212yesmaximal, complemented
14#23182C212yesmaximal, complemented
15#22998C212yesmaximal, complemented
16#22599C211yesmaximal, characteristic, complemented
17C4×#483C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2212yes—
20C22×C4×Q8C2212yes—
21C2×#492C2212yes—
22#3108—216no—
23C2×#483C2211yescharacteristic
24C2×#480C2212yes—
25C2×#479C2214yes—
26C2×#464C2214yes—
27C2×#462C2212yes—
28C2×#451C2212yes—
29C2×#448C2212yes—
30C2×#448C2212yes—
31C2×C4×Q16C2214yes—
32C2×C4×SD16C2214yes—
33C2×#431C2212yes—
34#2856C2211yescharacteristic
35#2854C2212yes—
36#2552C2211yescharacteristic
37#2548C2211yescharacteristic
38#2512—21no—
39#2501—21no—
40#2479—22no—
41#2454C2212yes—
42#2448C2212yes—
43#2425—22no—
44#2366C2211yescharacteristic
45#2365—21no—
46#2365—22no—
47#2365—21no—
48#2365C2214yes—
49#2365C2212yes—
50#2365C2212yes—
51#2361—21no—
52#2354—21no—
53Q8×C4⋊C4C2211yescharacteristic
54C4×C22⋊Q8C2211yescharacteristic
55C2×#391C2212yes—
56C2×#389—21no—
57C2×#387—22no—
58C2×C4×C4⋊C4—21no—
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61#2066C2214yes—
62#2055C2214yes—
63#2043C2212yes—
64#2026C2214yes—
65#2016C2212yes—
66#2015C2214yes—
67#2013C2214yes—
68#1982C2212yes—
69#1981C2212yes—
70#1968C2214yes—
71#1956C2212yes—
72#1947C2212yes—
73#1939C2212yes—
74#1933C2214yes—
75#1927C2212yes—
76#1925C2214yes—
77#1920C2212yes—
78#1916C2214yes—
79#1880C2214yes—
80#1880C2214yes—
81#1878C2214yes—
82#1877C2214yes—
83#1874C2212yes—
84#1872C2211yescharacteristic
85#1863C2214yes—
86#1863C2214yes—
87#1853C2211yescharacteristic
88#1823C2211yescharacteristic
89#1817C2212yes—
90#1797C2211yescharacteristic
91#1787C2212yes—
92C4×#106C2212yes—
93C8×C4⋊C4C2212yes—
94C4×#103C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×Q8C2312yes—
97C22×Q16C2314yes—
98C22×SD16C2314yes—
99C22×SD16C2314yes—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106#529—28no—
107#529—44no—
108C2×#126—22no—
109C2×#125C2312yes—
110C2×#124C2312yes—
111C2×#123—22no—
112C2×#123—22no—
113C2×C22⋊Q8C2311yescharacteristic
114C2×#121—21no—
115C2×#121—21no—
116C2×#121C2312yes—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—28no—
120C2×C4×Q8—44no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2314yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2314yes—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C2×#117—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4D411yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C42—22no—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140#492—28no—
141#483D418yescomplemented
142#480—28no—
143#479—216no—
144#464—216no—
145#462—28no—
146#451—28no—
147#448—28no—
148#448—44no—
149C4×Q16—216no—
150C4×SD16—216no—
151#431—28no—
152C2×#107C2314yes—
153C2×#106C2312yes—
154C2×#106C2312yes—
155C2×C4⋊C8C2314yes—
156C2×C4⋊C8C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2311yescharacteristic
168C2×C22⋊C8C2311yescharacteristic
169C2×C4×C8C2312yes—
170C2×C4×C8C2314yes—
171#399—22no—
172#399—22no—
173#398—22no—
174#397—22no—
175#396—22no—
176#394—22no—
177#391—22no—
178#391—22no—
179#391C2314yes—
180#391C2314yes—
181#391C2311yescharacteristic
182#390—21no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390C2312yes—
188#390C2312yes—
189#390C2312yes—
190#389—21no—
191#389—22no—
192#389—22no—
193#389—24no—
194#389—22no—
195#389C2312yes—
196#389C2312yes—
197#389C2311yescharacteristic
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#387—21no—
203#387—24no—
204#387—24no—
205#387—24no—
206#387—24no—
207#386—22no—
208#386—22no—
209#385—21no—
210#385—22no—
211#384—21no—
212#384—21no—
213#384C2311yescharacteristic
214#380—22no—
215#380C2311yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4—24no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D412yes—
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2311yescharacteristic
232C4×C22⋊C4—21no—
233C2×#95—22no—
234C2×#95—21no—
235C2×#95—22no—
236C2×#95D412yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#340C2312yes—
241#340C2312yes—
242#336C2312yes—
243#336C2314yes—
244#336C2314yes—
245#336C2312yes—
246C22×Q8—28no—
247C22×Q8—22no—
248C22×Q8—24no—
249C22×Q8—24no—
250C22×Q8—44no—
251C22×Q8C2411yescharacteristic
252C22×Q8C2414yes—
253C22×D4—22no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4C2412yes—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2411yescharacteristic
266C2×Q16—28no—
267C2×Q16—28no—
268C2×Q16—44no—
269C2×SD16—28no—
270C2×SD16—28no—
271C2×SD16—28no—
272C2×SD16—44no—
273C2×SD16—44no—
274C2×SD16—44no—
275C22×C8C2412yes—
276C22×C8C2414yes—
277C22×C8C2412yes—
278C4⋊Q8—28no—
279C4⋊Q8—28no—
280C4⋊Q8—28no—
281C4⋊Q8—24no—
282C4⋊Q8#4118yes—
283#127—24no—
284#126—48no—
285#125—24no—
286#125—42no—
287#124—24no—
288#124—42no—
289#123—48no—
290#123—84no—
291C22⋊Q8—22no—
292C22⋊Q8C2×D414yes—
293#121—24no—
294#121—24no—
295#121—44no—
296#121—82no—
297C4×Q8—24no—
298C4×Q8—28no—
299C4×Q8—28no—
300C4×Q8—24no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×Q8—42no—
304C4×Q8—48no—
305C4×Q8—48no—
306C4×D4—24no—
307C4×D4—42no—
308C4×D4—48no—
309C4×D4—44no—
310#117—48no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—48no—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D414yes—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—48no—
372C2×C42—44no—
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2412yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—28no—
382#106—42no—
383#106C2×D418yes—
384C4⋊C8—28no—
385C4⋊C8—24no—
386#103—28no—
387#103—28no—
388#103—28no—
389#103—44no—
390#103—42no—
391#103—44no—
392#103C2×D418yes—
393#102—28no—
394#102—44no—
395C22⋊C8—24no—
396C22⋊C8—22no—
397C22⋊C8C2×D414yes—
398C4×C8—28no—
399C4×C8—24no—
400#95—24no—
401#95—24no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—21no—
407#95—21no—
408#95—22no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—22no—
413#95C2×D412yes—
414#95C2×D412yes—
415#95C2×D412yes—
416#95C2×D412yes—
417C24—22no—
418C24C22×D411yescharacteristic
419C2×Q8—28no—
420C2×Q8—24no—
421C2×Q8—21no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—416no—
429C2×Q8—416no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8C22×D412yes—
433C2×Q8C2×#4118yes—
434C2×D4—22no—
435C2×D4—24no—
436C2×D4—22no—
437C2×D4—24no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—81no—
446C2×D4—81no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—41no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—82no—
480C22×C4#13514yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4H(5,𝔽2)12yes—
492C22×C4C2511yescharacteristic
493Q16—416no—
494SD16—416no—
495SD16—88no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8C22×D414yes—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—81no—
533C4⋊C4—81no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4#13714yes—
537C4⋊C4C22×D414yes—
538C4⋊C4C22×D412yes—
539C22⋊C4—24no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—42no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C42—22no—
552C42—22no—
553C42—24no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—48no—
561C42—48no—
562C42—44no—
563C42—42no—
564C42—44no—
565C42—44no—
566C42—42no—
567C42C2×#4114yes—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23D4212yes—
580C23C2×#13512yes—
581C23#57712yes—
582C23C23×D411yescharacteristic
583Q8—28no—
584Q8—24no—
585Q8—28no—
586Q8—416no—
587Q8—44no—
588Q8—42no—
589Q8—44no—
590D4—44no—
591D4—44no—
592D4—88no—
593D4—84no—
594D4—162no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—28no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—48no—
623C2×C4—44no—
624C2×C4—416no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—88no—
648C2×C4—84no—
649C2×C4#52512yes—
650C2×C4#52512yes—
651C2×C4D4212yes—
652C2×C4C2×#13512yes—
653C2×C4C2×#13512yes—
654C2×C4C2×#13714yes—
655C2×C4C2×#13712yes—
656C2×C4C23×D411yescharacteristic
657C2×C4C22×#4112yes—
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4#58512yes—
660C8—24no—
661C8—44no—
662C8—42no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22#328414yes—
677C22C2×#52511yescharacteristic
678C22C2×D4211yescharacteristic
679C22#348711yescharacteristic
680C22C22×#13511yescharacteristic
681C22C22×#13711yescharacteristic
682C22C2×#57711yescharacteristic
683C22#358611yescharacteristic
684C4—22no—
685C4—22no—
686C4—24no—
687C4—22no—
688C4—48no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—44no—
693C4—42no—
694C4—42no—
695C4—41no—
696C4—82no—
697C4—81no—
698C4—82no—
699C4—81no—
700C4#310814yescomplemented
701C4D4×#4112yes—
702C4#351312yes—
703C2—24no—
704C2—162no—
705C2#3610211yescharacteristic
706C2C2×#310811yescharacteristic
707C2#6058211yescharacteristic
708C2C2×#328411yescharacteristic
709C2D4×#13711yescharacteristic
710C2#6188211yescharacteristic
711C2#6190611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
1. C2×#3108C2semidirect
3. C2×#2365C2semidirect
5. #33003C2semidirect
6. #26903C2semidirect
7. #26591C2semidirect
8. #26465C2semidirect
10. #25369C2semidirect
11. #24503C2semidirect
12. #24212C2semidirect
13. #23937C2semidirect
14. #23182C2semidirect
15. #22998C2semidirect
16. #22599C2semidirect
17. C4×#483C2semidirect
141. #483D4semidirect
700. C4#3108semidirect

Maximal subgroups

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

subgroupindex
C2×#31082
#361172
C2×#23652
#331062
#330032
#269032
#265912
#264652
#257182
#253692
#245032
#242122
#239372
#231822
#229982
#225992
C4×#4832