Finite Group Databasebeta

← #6376018

Subgroups — #6376018

Order 512.

1944 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3106C211yesmaximal, characteristic
2#34226C211yesmaximal, characteristic, complemented
3#34060C211yesmaximal, characteristic, complemented
4#22151C212yesmaximal, complemented
5#22147C212yesmaximal, complemented
6#22119C212yesmaximal, complemented
7#22111C212yesmaximal, complemented
8#22090C214yesmaximal, complemented
9#22053C214yesmaximal, complemented
10#21395C211yesmaximal, characteristic, complemented
11#21359C211yesmaximal, characteristic, complemented
12#21284C212yesmaximal, complemented
13#21280C212yesmaximal, complemented
14#21134C212yesmaximal, complemented
15#21133C212yesmaximal, complemented
16#20743C211yesmaximal, characteristic, complemented
17#20470C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20#3106C4116yescomplemented
21C2×#482C2212yes—
22C2×#477C2211yescharacteristic
23C2×#476C2211yescharacteristic
24C2×#460C2211yescharacteristic
25C2×#459C2211yescharacteristic
26C2×#449C2212yes—
27C2×C4×SD16C2212yes—
28C2×C4×D8C2212yes—
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2367—22no—
32#2363—22no—
33#2357—21no—
34#2349—21no—
35#2341—21no—
36#2339—21no—
37#2324C2214yes—
38#2320C2211yescharacteristic
39#2320C2211yescharacteristic
40#2319C2211yescharacteristic
41#2319C2211yescharacteristic
42#2319C2212yes—
43#2318—21no—
44#2318—21no—
45#2317C2212yes—
46#2314—22no—
47#2311C2212yes—
48C4×C4⋊Q8C2212yes—
49C4×#127C2212yes—
50#2295C2212yes—
51#2293C2212yes—
52#2290—22no—
53#2285—21no—
54#2285C2211yescharacteristic
55#2283—21no—
56#2275C2212yes—
57C4×#117C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#1987C2212yes—
60#1982C2212yes—
61#1972C2214yes—
62#1968C2214yes—
63#1948C2212yes—
64#1948C2212yes—
65#1943C2214yes—
66#1941C2214yes—
67#1936C2214yes—
68#1935C2214yes—
69#1934C2214yes—
70#1934C2214yes—
71#1927C2214yes—
72#1926C2214yes—
73#1919C2214yes—
74#1918C2214yes—
75#1879C2214yes—
76#1877C2214yes—
77#1874C2214yes—
78#1864C2214yes—
79#1864C2214yes—
80#1853C2214yes—
81#1825C2212yes—
82#1803C2212yes—
83#1802C2212yes—
84#1788C2212yes—
85C4×#106C2212yes—
86C4×#106C2212yes—
87#1767C2212yes—
88C4×C4⋊C8C2212yes—
89#1763C2212yes—
90#1762C2212yes—
91#1762C2212yes—
92#1762C2212yes—
93#1750C2214yes—
94#1746C2212yes—
95#1726C2212yes—
96C22×#41—21no—
97C22×#41C2311yescharacteristic
98C22×SD16C2312yes—
99C22×D8C2312yes—
100#532—24no—
101#532C2×C418yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104C2×#124—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C4×#41—24no—
108C4×#41C2×C418yescomplemented
109C2×C4×Q8—22no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2314yes—
121C2×#117—21no—
122C2×#117—21no—
123C2×#117—22no—
124C2×#117C2311yescharacteristic
125C2×#117C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C42—21no—
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130#482C2×C4116yescomplemented
131#477C2×C418yescomplemented
132#476C2×C418yescomplemented
133#460C2×C418yescomplemented
134#459C2×C418yescomplemented
135#449C2×C4116yescomplemented
136C4×SD16C2×C4116yescomplemented
137C4×D8C2×C4116yescomplemented
138#432—24no—
139#431C2×C418yescomplemented
140C2×#107C2312yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×#97C2312yes—
158C2×#97C2314yes—
159C2×C4×C8C2314yes—
160C2×C4×C8C2312yes—
161#391—22no—
162#391C2312yes—
163#391C2312yes—
164#390—24no—
165#390—22no—
166#390—22no—
167#390C2314yes—
168#390C2312yes—
169#390C2312yes—
170#389—22no—
171#389C2314yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#388—24no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—24no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—22no—
188#386—24no—
189#385—24no—
190#385—22no—
191#385—22no—
192#384—22no—
193#384C2312yes—
194#383—21no—
195#383—21no—
196#382—22no—
197#382—21no—
198#382—21no—
199#382—21no—
200#381—21no—
201#381—21no—
202#381—22no—
203#380—21no—
204#380—22no—
205#380—22no—
206#380—21no—
207#380—21no—
208#380—22no—
209#380C2312yes—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—24no—
226C4×C22⋊C4D411yescharacteristic
227C4×C22⋊C4D411yescharacteristic
228C4×C22⋊C4D412yes—
229#376—22no—
230#376C2311yescharacteristic
231#376C2311yescharacteristic
232C2×#95—21no—
233C2×#95—22no—
234C43C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C2×#41—28no—
244C2×#41—22no—
245C2×#41—42no—
246C2×#41—42no—
247C2×#41—84no—
248C2×#41C22×C414yes—
249C22×Q8—21no—
250C22×Q8C2411yescharacteristic
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4C2412yes—
256C22×D4C2411yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—22no—
262C23×C4—24no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2411yescharacteristic
266C2×SD16—44no—
267C2×SD16C22×C418yes—
268C2×D8—44no—
269C2×D8C22×C418yes—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C4⋊Q8—22no—
274C4⋊Q8C22×C414yes—
275#127—22no—
276#127C22×C414yes—
277#124—28no—
278#121—28no—
279#121—24no—
280#121C22×C418yes—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—84no—
284C4×Q8C22×C414yes—
285C4×Q8C22×C414yes—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—24no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—84no—
295C4×D4—84no—
296C4×D4C22×C414yes—
297C4×D4C22×C414yes—
298C4×D4C22×C418yes—
299#117—22no—
300#117—28no—
301#117—42no—
302#117—42no—
303#117—84no—
304#117—82no—
305#117C22×C414yes—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—28no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—24no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—82no—
352C2×C42C22×C414yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42#4114yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107C22×C418yes—
365#106—44no—
366#106—44no—
367#106C22×C418yes—
368C4⋊C8—24no—
369C4⋊C8C22×C414yes—
370C4⋊C8C22×C414yes—
371#103—44no—
372#103C22×C418yes—
373#102—44no—
374#102—44no—
375#102—44no—
376#102C22×C418yes—
377#102C22×C418yes—
378#102C22×C418yes—
379C22⋊C8—24no—
380C22⋊C8C22×C418yes—
381#97—24no—
382#97—44no—
383C4×C8—44no—
384C4×C8C22×C418yes—
385#95—22no—
386#95—24no—
387#95—21no—
388#95—21no—
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#95—22no—
398#95—22no—
399#95—24no—
400#95C2×D411yescharacteristic
401#95C2×D411yescharacteristic
402#95C2×D412yes—
403#95C2×D411yescharacteristic
404#95C2×D411yescharacteristic
405#95C2×D412yes—
406C24—21no—
407C24—22no—
408C24C22×D411yescharacteristic
409#41—44no—
410#41—44no—
411#41—48no—
412#41—816no—
413C2×Q8—24no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—44no—
418C2×Q8C23×C412yes—
419C2×D4—24no—
420C2×D4—28no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—84no—
434C2×D4C23×C412yes—
435C2×D4C23×C414yes—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—82no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4C4×D414yes—
464C22×C4C4×D414yes—
465C22×C4C23×C412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4112yes—
476C22×C4C2511yescharacteristic
477SD16—48no—
478D8—48no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8C23×C414yes—
485C2×C8C23×C414yes—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—84no—
508C4⋊C4C23×C412yes—
509C4⋊C4C23×C412yes—
510C4⋊C4C23×C414yes—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—48no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—82no—
526C42—24no—
527C42—24no—
528C42—24no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—42no—
536C42—42no—
537C42—44no—
538C42—81no—
539C42—81no—
540C42—82no—
541C42—84no—
542C42C4×D414yes—
543C42C4×D414yes—
544C42C23×C412yes—
545C42C23×C412yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—24no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—41no—
559C23—42no—
560C23—41no—
561C23—42no—
562C23C2×C4×D412yes—
563C23C23×D411yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—48no—
567Q8—42no—
568D4—22no—
569D4—24no—
570D4—24no—
571D4—22no—
572D4—48no—
573D4—48no—
574D4—44no—
575D4—42no—
576D4—48no—
577D4—88no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—48no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—84no—
621C2×C4—88no—
622C2×C4—88no—
623C2×C4—168no—
624C2×C4C2×C4×D412yes—
625C2×C4C2×C4×D412yes—
626C2×C4C2×C4×D412yes—
627C2×C4C2×C4×D412yes—
628C2×C4C2×C4×D412yes—
629C2×C4#57712yes—
630C2×C4#57712yes—
631C2×C4C24×C411yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4#58512yes—
634C2×C4#58512yes—
635C8—24no—
636C8—24no—
637C8—44no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—82no—
650C22—84no—
651C22—84no—
652C22—82no—
653C22C22×C4×D411yescharacteristic
654C22#344912yes—
655C22C2×#57711yescharacteristic
656C22C2×#57711yescharacteristic
657C22#358512yes—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—82no—
672C4—84no—
673C4#295214yes—
674C4#342811yescharacteristic
675C4#342811yescharacteristic
676C4#345512yes—
677C2—42no—
678C2—82no—
679C2—84no—
680C2#3406511yescharacteristic
681C2C2×#295211yescharacteristic
682C2#6016811yescharacteristic
683C2#6098912yes—
684C2#6099412yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3106C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34226C2semidirect
3. #34060C2semidirect
4. #22151C2semidirect
5. #22147C2semidirect
6. #22119C2semidirect
7. #22111C2semidirect
8. #22090C2semidirect
9. #22053C2semidirect
10. #21395C2semidirect
11. #21359C2semidirect
12. #21284C2semidirect
13. #21280C2semidirect
14. #21134C2semidirect
15. #21133C2semidirect
16. #20743C2semidirect
17. #20470C2semidirect
20. #3106C4semidirect
108. C4×#41C2×C4semidirect
130. #482C2×C4semidirect
131. #477C2×C4semidirect
132. #476C2×C4semidirect
133. #460C2×C4semidirect
134. #459C2×C4semidirect
135. #449C2×C4semidirect
136. C4×SD16C2×C4semidirect
137. C4×D8C2×C4semidirect
139. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31062
#342262
#340602
#221512
#221472
#221192
#221112
#220902
#220532
#213952
#213592
#212842
#212802
#211342
#211332
#207432
#204702