Finite Group Databasebeta

← #6578599

Subgroups — #6578599

Order 512.

2049 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C212yesmaximal
2#36133C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic
4#33108C211yesmaximal, characteristic
5#33008C212yesmaximal
6#26911C214yesmaximal, complemented
7#26607C214yesmaximal, complemented
8#26462C212yesmaximal, complemented
9#25722C211yesmaximal, characteristic
10#25370C212yesmaximal, complemented
11#24508C212yesmaximal
12#24212C211yesmaximal, characteristic
13#23934C212yesmaximal
14#23187C212yesmaximal, complemented
15#23004C212yesmaximal, complemented
16#22603C211yesmaximal, characteristic
17C4×#483C211yesmaximal, characteristic
18C2×#533C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C2×#494C2212yes—
22#3113—216no—
23C2×#483C2212yes—
24C2×#483C2211yescharacteristic
25C2×#475C2214yes—
26C2×#462C2214yes—
27C2×#462C2212yes—
28C2×#451C2212yes—
29C2×#451C2212yes—
30C2×#451C2212yes—
31C2×#441C2214yes—
32C2×#441C2214yes—
33C2×#432C2212yes—
34#2857C2211yescharacteristic
35#2855C2212yes—
36#2551C2211yescharacteristic
37#2546C2211yescharacteristic
38#2506—21no—
39#2462—22no—
40#2451C2212yes—
41#2446C2212yes—
42#2440—21no—
43#2438—22no—
44#2365—21no—
45#2365—21no—
46#2365—22no—
47#2365—21no—
48#2365C2212yes—
49#2365C2214yes—
50#2365C2211yescharacteristic
51#2365C2212yes—
52Q8×C4⋊C4C2211yescharacteristic
53#2344—21no—
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#2056C2214yes—
63#2043C2212yes—
64#2024C2214yes—
65#2016C2214yes—
66#2016C2214yes—
67#2016C2212yes—
68#1981C2212yes—
69#1981C2212yes—
70#1971C2214yes—
71#1956C2212yes—
72#1948C2212yes—
73#1939C2212yes—
74#1935C2214yes—
75#1927C2214yes—
76#1927C2212yes—
77#1920C2212yes—
78#1917C2214yes—
79#1878C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1872C2211yescharacteristic
84#1872C2212yes—
85#1865C2214yes—
86#1865C2214yes—
87#1853C2211yescharacteristic
88#1823C2211yescharacteristic
89#1817C2212yes—
90#1797C2211yescharacteristic
91#1787C2212yes—
92C4×#106C2212yes—
93#1770C2212yes—
94C4×#103C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×Q8C2312yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×Q16C2314yes—
100#533—28no—
101#533—44no—
102C2×C4⋊Q8—28no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×#125—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2311yescharacteristic
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—28no—
119C2×C4×Q8—44no—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2314yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2314yes—
124C2×C4×Q8C2312yes—
125C2×#117—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C42—22no—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#494—28no—
138#483—28no—
139#483D418yes—
140#475—216no—
141#462—28no—
142#462—216no—
143#451—28no—
144#451—28no—
145#451—44no—
146#441—216no—
147#441—216no—
148#432—28no—
149C2×#106C2314yes—
150C2×#106C2312yes—
151C2×#106C2312yes—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2311yescharacteristic
165C2×C22⋊C8C2311yescharacteristic
166C2×#97C2314yes—
167C2×C4×C8C2312yes—
168#400—22no—
169#398—22no—
170#398—22no—
171#398—22no—
172#395—22no—
173#395—22no—
174#391—22no—
175#391—22no—
176#391—22no—
177#391C2312yes—
178#391C2314yes—
179#391C2312yes—
180#391C2312yes—
181#391C2314yes—
182#391C2311yescharacteristic
183#389—21no—
184#389—21no—
185#389—22no—
186#389—24no—
187#389—22no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#389C2311yescharacteristic
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—21no—
201#387—24no—
202#387—24no—
203#387—24no—
204#387—24no—
205#387—22no—
206#387—22no—
207#385—22no—
208#385—21no—
209#384—21no—
210#384C2311yescharacteristic
211#381—22no—
212#380—21no—
213#380—21no—
214#380C2311yescharacteristic
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—24no—
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—22no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8—44no—
252C22×Q8C2412yes—
253C22×Q8C2411yescharacteristic
254C22×Q8C2414yes—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×Q16—28no—
264C2×Q16—28no—
265C2×Q16—28no—
266C2×Q16—28no—
267C2×Q16—28no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×Q16—44no—
271C2×Q16—44no—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C4⋊Q8—28no—
276C4⋊Q8—28no—
277C4⋊Q8—28no—
278C4⋊Q8—24no—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—42no—
283C4⋊Q8—42no—
284C4⋊Q8—44no—
285C4⋊Q8#4118yes—
286#125—48no—
287C22⋊Q8—22no—
288C22⋊Q8—24no—
289C22⋊Q8—24no—
290C22⋊Q8—48no—
291C22⋊Q8C2×D414yes—
292C4×Q8—24no—
293C4×Q8—28no—
294C4×Q8—28no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×Q8—42no—
300C4×Q8—42no—
301C4×Q8—48no—
302C4×Q8—48no—
303C4×Q8—48no—
304#117—48no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—48no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D414yes—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—48no—
363C2×C42—44no—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2412yes—
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#106—28no—
373#106—42no—
374#106C2×D418yes—
375C4⋊C8—28no—
376C4⋊C8—24no—
377#103—28no—
378#103—28no—
379#103—28no—
380#103—28no—
381#103—44no—
382#103—44no—
383#103—42no—
384#103—44no—
385#103C2×D418yes—
386C22⋊C8—24no—
387C22⋊C8—22no—
388C22⋊C8C2×D414yes—
389#97—28no—
390C4×C8—24no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—21no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95C2×D412yes—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D412yes—
414C24C22×D411yescharacteristic
415C2×Q8—28no—
416C2×Q8—24no—
417C2×Q8—21no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—44no—
429C2×Q8—416no—
430C2×Q8—416no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—82no—
435C2×Q8C22×D412yes—
436C2×Q8C2×#4118yes—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—28no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4#13714yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4H(5,𝔽2)12yes—
480C22×C4C2511yescharacteristic
481Q16—416no—
482Q16—416no—
483Q16—88no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8C22×D414yes—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—81no—
527C4⋊C4—81no—
528C4⋊C4—82no—
529C4⋊C4#13714yes—
530C4⋊C4C22×D412yes—
531C4⋊C4C22×D414yes—
532C22⋊C4—24no—
533C22⋊C4—44no—
534C22⋊C4—42no—
535C42—22no—
536C42—22no—
537C42—24no—
538C42—24no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—48no—
545C42—48no—
546C42—44no—
547C42—44no—
548C42—42no—
549C42—44no—
550C42—44no—
551C42—42no—
552C42C2×#4114yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23D4212yes—
558C23C2×#13712yes—
559C23#57812yes—
560C23C23×D411yescharacteristic
561Q8—28no—
562Q8—24no—
563Q8—28no—
564Q8—416no—
565Q8—44no—
566Q8—42no—
567Q8—44no—
568Q8—44no—
569Q8—44no—
570Q8—88no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—416no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—84no—
627C2×C4#52512yes—
628C2×C4#52512yes—
629C2×C4D4212yes—
630C2×C4C2×#13712yes—
631C2×C4C2×#13714yes—
632C2×C4C2×#13712yes—
633C2×C4C2×#13712yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C22×#4112yes—
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4#58512yes—
638C8—24no—
639C8—44no—
640C8—42no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22#327614yes—
648C22C2×#52511yescharacteristic
649C22C2×D4211yescharacteristic
650C22#348711yescharacteristic
651C22C22×#13711yescharacteristic
652C22C22×#13711yescharacteristic
653C22C2×#57811yescharacteristic
654C22#358611yescharacteristic
655C4—22no—
656C4—22no—
657C4—24no—
658C4—22no—
659C4—48no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—42no—
665C4—42no—
666C4—41no—
667C4—82no—
668C4—82no—
669C4—81no—
670C4—82no—
671C4—81no—
672C4#311314yescomplemented
673C4D4×#4112yes—
674C4#351312yes—
675C2—24no—
676C2#3610211yescharacteristic
677C2C2×#311311yescharacteristic
678C2#6058311yescharacteristic
679C2C2×#327611yescharacteristic
680C2D4×#13711yescharacteristic
681C2#6188911yescharacteristic
682C2#6190011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
6. #26911C2semidirect
7. #26607C2semidirect
8. #26462C2semidirect
10. #25370C2semidirect
14. #23187C2semidirect
15. #23004C2semidirect
672. C4#3113semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
#361332
C2×#23652
#331082
#330082
#269112
#266072
#264622
#257222
#253702
#245082
#242122
#239342
#231872
#230042
#226032
C4×#4832