Finite Group Databasebeta

← #6502106

Subgroups — #6502106

Order 512.

2122 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2427C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic, complemented
3#29244C212yesmaximal, complemented
4#27391C214yesmaximal, complemented
5#26931C214yesmaximal, complemented
6#26628C214yesmaximal, complemented
7#25157C214yesmaximal, complemented
8#24217C212yesmaximal, complemented
9#24003C211yesmaximal, characteristic, complemented
10#24002C211yesmaximal, characteristic
11#23921C211yesmaximal, characteristic, complemented
12#23919C211yesmaximal, characteristic
13#22044C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic, complemented
15C4×#450C212yesmaximal, complemented
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#483C2212yescomplemented
22C2×#480C2212yes—
23C2×#464C2214yescomplemented
24C2×#451C2212yes—
25C2×#451C2212yescomplemented
26C2×#448C2212yes—
27C2×#450C2212yes—
28C2×C4×SD16C2214yescomplemented
29#2427—22no—
30#2427—22no—
31#2427C2214yes—
32#2427C2214yes—
33C2×#397—21no—
34C2×#394—21no—
35#2368—22no—
36#2360C2212yescomplemented
37#2360C2212yescomplemented
38#2345—22no—
39#2345C2212yescomplemented
40Q8×C22⋊C4C2212yescomplemented
41#2329—21no—
42#2329—21no—
43#2315—21no—
44C4×C22⋊Q8C2212yes—
45C4×C22⋊Q8C2212yes—
46C2×#391C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#386—21no—
49C2×#380—21no—
50C2×#380C2211yescharacteristic
51#2273—21no—
52C2×C4×C22⋊C4C2211yescharacteristic
53#2086C2214yescomplemented
54#2066C2214yescomplemented
55#2042C2214yescomplemented
56#2035C2214yescomplemented
57#2033C2214yes—
58#2015C2214yescomplemented
59#2001C2214yescomplemented
60#2000C2214yes—
61#1983C2214yescomplemented
62#1969C2214yescomplemented
63#1933C2214yescomplemented
64#1928C2214yescomplemented
65#1923C2214yescomplemented
66#1923C2214yescomplemented
67#1920C2214yescomplemented
68#1878C2214yescomplemented
69#1877C2214yescomplemented
70#1874C2212yes—
71#1874C2212yes—
72#1872C2212yes—
73#1872C2212yes—
74#1872C2212yescomplemented
75#1872C2212yescomplemented
76#1864C2214yescomplemented
77#1863C2214yescomplemented
78#1853C2212yes—
79#1852C2212yes—
80#1797C2212yes—
81#1787C2212yes—
82#1786C2212yes—
83#1779C2211yescharacteristic
84#1779C2211yescharacteristic
85C4×#102C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4—21no—
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C24×C4C2311yescharacteristic
93C22×Q16C2314yes—
94C22×SD16C2314yes—
95C2×#124—22no—
96C2×#124—44no—
97C2×#124C2314yes—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—44no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×#121—21no—
109C2×#121—21no—
110C2×#121—44no—
111C2×#121C2312yes—
112C2×#121C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4—44no—
117C2×C4×D4C2314yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4D411yescharacteristic
134C22×C22⋊C4D411yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#483—44no—
138#480—44no—
139#464—48no—
140#451—28no—
141#451—44no—
142#450—28no—
143#448—44no—
144C4×SD16—48no—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C4×C8C2314yes—
163#397—24no—
164#397—24no—
165#394—24no—
166#394—24no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391C2312yes—
171#391C2312yes—
172#391C2314yes—
173#389—22no—
174#389—22no—
175#389—22no—
176#389C2314yes—
177#389C2312yes—
178#389C2312yes—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#386—24no—
186#386—24no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#380—21no—
192#380—21no—
193#380—22no—
194#380—22no—
195#380—24no—
196#380—24no—
197#380C2312yes—
198#380C2312yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4C2312yes—
212C4×C22⋊C4C2312yes—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95—21no—
217C2×#95D411yescharacteristic
218C2×#95D411yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#336C2314yes—
224#336C2314yes—
225C25—21no—
226C25C2×D411yescharacteristic
227C22×Q8—24no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×Q8—24no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8—42no—
234C22×Q8—42no—
235C22×Q8C2412yes—
236C22×Q8C2412yes—
237C22×D4—24no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—24no—
242C22×D4—42no—
243C22×D4—44no—
244C22×D4—44no—
245C22×D4—82no—
246C22×D4C2412yes—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—24no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—24no—
256C23×C4—42no—
257C23×C4—42no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4—82no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271C2×Q16—28no—
272C2×Q16—44no—
273C2×Q16—44no—
274C2×SD16—28no—
275C2×SD16—28no—
276C2×SD16—44no—
277C22×C8C2414yes—
278C22×C8C2414yes—
279#124—28no—
280#124—44no—
281#124—48no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—24no—
288C22⋊Q8—24no—
289C22⋊Q8—42no—
290C22⋊Q8—42no—
291C22⋊Q8—44no—
292C22⋊Q8—44no—
293C22⋊Q8#4118yes—
294#121—24no—
295#121—24no—
296#121—42no—
297#121—42no—
298#121—44no—
299#121—44no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×D4—28no—
303C4×D4—44no—
304C4×D4—48no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—82no—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—42no—
361C2×C42—42no—
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2412yes—
365C2×C42C2412yes—
366#107—44no—
367#106—28no—
368C4⋊C8—44no—
369#103—28no—
370#103—28no—
371#103—44no—
372#103—44no—
373#103—44no—
374#102—28no—
375#102—28no—
376#102—44no—
377C22⋊C8—24no—
378C22⋊C8—24no—
379C22⋊C8—42no—
380C22⋊C8—42no—
381C22⋊C8—42no—
382C22⋊C8—42no—
383C4×C8—44no—
384#95—22no—
385#95—24no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—22no—
402C24—22no—
403C24—24no—
404C24—22no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24—41no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24#13514yes—
416C24C22×D411yescharacteristic
417C24C22×D411yescharacteristic
418C24C22×D411yescharacteristic
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—81no—
431C2×Q8—81no—
432C2×Q8—88no—
433C2×Q8—88no—
434C2×Q8C2×#4114yes—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—416no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—42no—
446C2×D4—82no—
447C2×D4—82no—
448C2×D4—816no—
449C2×D4H(5,𝔽2)14yes—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—48no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—48no—
500C22×C4—88no—
501C22×C4—84no—
502C22×C4—81no—
503C22×C4—81no—
504C22×C4—81no—
505C22×C4—81no—
506C22×C4—816no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—81no—
510C22×C4—81no—
511C22×C4#13714yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D412yes—
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D412yes—
519C22×C4C2×#4114yes—
520C22×C4C2511yescharacteristic
521Q16—88no—
522SD16—416no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—28no—
532C4⋊C4—28no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—81no—
545C4⋊C4—81no—
546C4⋊C4—84no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4C2×#4114yes—
550C4⋊C4H(5,𝔽2)14yes—
551C22⋊C4—28no—
552C22⋊C4—28no—
553C22⋊C4—24no—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—48no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—48no—
563C22⋊C4—48no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C42—24no—
567C42—24no—
568C42—42no—
569C42—42no—
570C42—82no—
571C42—82no—
572C42—82no—
573C42—82no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—28no—
591C23—24no—
592C23—44no—
593C23—44no—
594C23—48no—
595C23—48no—
596C23—48no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—82no—
604C23—82no—
605C23—82no—
606C23—82no—
607C23—84no—
608C23—84no—
609C23—84no—
610C23—84no—
611C23#52512yes—
612C23C2×#13512yes—
613C23C2×#13512yes—
614C23C2×#13512yes—
615C23C2×#13712yes—
616C23C23×D411yescharacteristic
617Q8—28no—
618Q8—44no—
619Q8—44no—
620Q8—44no—
621Q8—84no—
622Q8—84no—
623D4—28no—
624D4—48no—
625D4—44no—
626D4—88no—
627D4—88no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—28no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—44no—
652C2×C4—48no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—82no—
674C2×C4—82no—
675C2×C4—816no—
676C2×C4—82no—
677C2×C4—82no—
678C2×C4—82no—
679C2×C4—82no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—84no—
684C2×C4—162no—
685C2×C4—162no—
686C2×C4#52512yes—
687C2×C4C2×#13714yes—
688C2×C4C22×#4112yes—
689C2×C4C2×H(5,𝔽2)12yes—
690C2×C4C2×H(5,𝔽2)12yes—
691C8—44no—
692C8—44no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—88no—
717C22—816no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22#310814yes—
723C22C2×#52511yescharacteristic
724C22#348411yescharacteristic
725C22#348411yescharacteristic
726C22C22×#13511yescharacteristic
727C22C22×#13711yescharacteristic
728C22#358511yescharacteristic
729C22#358611yescharacteristic
730C4—22no—
731C4—24no—
732C4—41no—
733C4—41no—
734C4—44no—
735C4—42no—
736C4—42no—
737C4—44no—
738C4—42no—
739C4—44no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4#351914yes—
745C2—24no—
746C2—42no—
747C2—42no—
748C2—84no—
749C2#3535511yescharacteristic
750C2C2×#310811yescharacteristic
751C2#6056711yescharacteristic
752C2#6156111yescharacteristic
753C2#6156411yescharacteristic
754C2#6157011yescharacteristic
755C2#6158611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34212C2semidirect
3. #29244C2semidirect
4. #27391C2semidirect
5. #26931C2semidirect
6. #26628C2semidirect
7. #25157C2semidirect
8. #24217C2semidirect
9. #24003C2semidirect
11. #23921C2semidirect
13. #22044C2semidirect
14. #21902C2semidirect
15. C4×#450C2semidirect
21. C2×#483C22semidirect
23. C2×#464C22semidirect
25. C2×#451C22semidirect
28. C2×C4×SD16C22semidirect
36. #2360C22semidirect
37. #2360C22semidirect
39. #2345C22semidirect
40. Q8×C22⋊C4C22semidirect
53. #2086C22semidirect
54. #2066C22semidirect
55. #2042C22semidirect
56. #2035C22semidirect
58. #2015C22semidirect
59. #2001C22semidirect
61. #1983C22semidirect
62. #1969C22semidirect
63. #1933C22semidirect
64. #1928C22semidirect
65. #1923C22semidirect
66. #1923C22semidirect
67. #1920C22semidirect
68. #1878C22semidirect
69. #1877C22semidirect
74. #1872C22semidirect
75. #1872C22semidirect
76. #1864C22semidirect
77. #1863C22semidirect
85. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24272
#342122
#292442
#273912
#269312
#266282
#251572
#242172
#240032
#240022
#239212
#239192
#220442
#219022
C4×#4502