Finite Group Databasebeta

← #6578967

Subgroups — #6578967

Order 512.

2047 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C212yesmaximal
2#36133C211yesmaximal, characteristic
3C2×Q8×C4⋊C4C211yesmaximal, characteristic
4#33008C212yesmaximal
5#32906C211yesmaximal, characteristic
6#27905C211yesmaximal, characteristic
7#27647C211yesmaximal, characteristic
8#26389C214yesmaximal, complemented
9#26187C214yesmaximal, complemented
10#26042C212yesmaximal, complemented
11#24732C212yesmaximal, complemented
12#24609C212yesmaximal
13#24063C212yesmaximal
14#23004C212yesmaximal, complemented
15#22674C212yesmaximal, complemented
16#22603C211yesmaximal, characteristic
17C4×#451C211yesmaximal, characteristic
18C2×#533C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#3113—216no—
22C2×#483C2212yes—
23C2×#483C2212yes—
24C2×#483C2212yes—
25C2×#475C2214yes—
26C2×#462C2214yes—
27C2×#462C2212yes—
28C2×#451C2212yes—
29C2×#451C2211yescharacteristic
30C2×#441C2214yes—
31C2×#441C2214yes—
32C2×C4×Q16C2212yes—
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#2365C2211yescharacteristic
46Q8×C4⋊C4—21no—
47Q8×C4⋊C4—22no—
48Q8×C4⋊C4—21no—
49Q8×C4⋊C4C2214yes—
50Q8×C4⋊C4C2211yescharacteristic
51Q8×C4⋊C4C2212yes—
52Q8×C4⋊C4C2212yes—
53#2344—21no—
54C4×C22⋊Q8C2211yescharacteristic
55C2×#389—22no—
56C2×#389C2212yes—
57C2×#384—21no—
58C2×#384C2211yescharacteristic
59C2×C4×C4⋊C4—21no—
60C2×C4×C4⋊C4C2211yescharacteristic
61#2096C2214yes—
62#2088C2214yes—
63#2084C2212yes—
64#2072C2212yes—
65#2068C2214yes—
66#2058C2214yes—
67#2043C2214yes—
68#1956C2212yes—
69#1948C2212yes—
70#1943C2214yes—
71#1939C2212yes—
72#1935C2214yes—
73#1917C2214yes—
74#1878C2214yes—
75#1878C2214yes—
76#1878C2212yes—
77#1878C2214yes—
78#1872C2212yes—
79#1872C2211yescharacteristic
80#1865C2214yes—
81#1865C2214yes—
82#1865C2214yes—
83#1865C2214yes—
84#1863C2212yes—
85#1852C2211yescharacteristic
86#1844C2212yes—
87#1823C2212yes—
88#1817C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2211yescharacteristic
92#1787C2211yescharacteristic
93#1770C2212yes—
94C4×#103C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×Q16C2314yes—
98C22×Q16C2312yes—
99#533—28no—
100#533—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#125—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2311yescharacteristic
109C2×C4×Q8—28no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—28no—
116C2×C4×Q8—44no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2314yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×#117—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C42—21no—
135C22×C42—22no—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#483—28no—
139#483—28no—
140#483—44no—
141#475—216no—
142#462—216no—
143#462—28no—
144#451—28no—
145#451D418yes—
146#441—216no—
147#441—216no—
148C4×Q16—28no—
149#432—28no—
150C2×#107C2312yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154C2×C4⋊C8C2314yes—
155C2×C4⋊C8C2312yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×C22⋊C8C2312yes—
166C2×C22⋊C8C2311yescharacteristic
167C2×C22⋊C8C2311yescharacteristic
168C2×#97C2314yes—
169C2×C4×C8C2312yes—
170#400—22no—
171#398—22no—
172#398—22no—
173#398—22no—
174#395—22no—
175#395—22no—
176#391—22no—
177#391C2312yes—
178#391C2311yescharacteristic
179#391C2312yes—
180#391C2312yes—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389—24no—
190#389—24no—
191#389—24no—
192#389—24no—
193#389—22no—
194#389C2314yes—
195#389C2311yescharacteristic
196#389C2312yes—
197#389C2312yes—
198#389C2314yes—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—21no—
204#387—22no—
205#387—22no—
206#385—22no—
207#385—21no—
208#384—22no—
209#384—24no—
210#384—22no—
211#384—21no—
212#384—21no—
213#384—21no—
214#384C2311yescharacteristic
215#384C2312yes—
216#384C2312yes—
217#381—22no—
218#380—21no—
219#380—21no—
220#380C2311yescharacteristic
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—24no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4D412yes—
229C4×C4⋊C4D411yescharacteristic
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2311yescharacteristic
234C4×C22⋊C4—21no—
235C2×#95—22no—
236C2×#95D412yes—
237#340C2312yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#340C2312yes—
242#336C2312yes—
243#336C2314yes—
244#336C2314yes—
245#336C2312yes—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8C2×D414yes—
252C22×Q8C2412yes—
253C22×Q8C2411yescharacteristic
254C22×Q8C2412yes—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×Q16—24no—
264C2×Q16—24no—
265C2×Q16—28no—
266C2×Q16—28no—
267C2×Q16—28no—
268C2×Q16C2×D418yes—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—42no—
276C4⋊Q8—42no—
277C4⋊Q8—44no—
278#125—48no—
279C22⋊Q8—22no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—48no—
283C22⋊Q8C2×D414yes—
284C4×Q8—28no—
285C4×Q8—28no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—28no—
290C4×Q8—28no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×Q8—42no—
297C4×Q8—42no—
298C4×Q8—48no—
299C4×Q8—48no—
300C4×Q8—48no—
301C4×Q8#4118yes—
302#117—48no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—48no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—84no—
339C2×C4⋊C4—82no—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2414yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C42—28no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—48no—
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2412yes—
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—24no—
377#106—28no—
378#106—28no—
379#106—44no—
380C4⋊C8—28no—
381C4⋊C8—24no—
382#103—28no—
383#103—28no—
384#103—24no—
385#103—24no—
386#103—28no—
387#103—24no—
388#103—28no—
389#103—44no—
390#103C2×D418yes—
391C22⋊C8—24no—
392C22⋊C8—22no—
393C22⋊C8C2×D414yes—
394#97—28no—
395C4×C8—24no—
396#95—24no—
397#95—24no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—21no—
404#95—21no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D412yes—
414#95C2×D412yes—
415C24C22×D411yescharacteristic
416C2×Q8—24no—
417C2×Q8—216no—
418C2×Q8—24no—
419C2×Q8—216no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—82no—
434C2×Q8C22×D412yes—
435C2×Q8C22×D414yes—
436C2×Q8C2×#4114yes—
437C2×Q82−1+414yes—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4#13714yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4H(5,𝔽2)12yes—
481C22×C4C2511yescharacteristic
482Q16—216no—
483Q16—416no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8C22×D414yes—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—28no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—81no—
527C4⋊C4—81no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4—82no—
533C4⋊C4#13714yes—
534C4⋊C4C22×D412yes—
535C4⋊C4C2×#4114yes—
536C22⋊C4—24no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C42—28no—
540C42—28no—
541C42—22no—
542C42—22no—
543C42—24no—
544C42—24no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—48no—
551C42—48no—
552C42—44no—
553C42—44no—
554C42—42no—
555C42—42no—
556C42C2×#4114yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23D4212yes—
562C23C2×#13712yes—
563C23#57712yes—
564C23C23×D411yescharacteristic
565Q8—216no—
566Q8—24no—
567Q8—24no—
568Q8—24no—
569Q8—24no—
570Q8—42no—
571Q8—44no—
572Q8—44no—
573Q8—88no—
574Q8#54718yes—
575C2×C4—216no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—22no—
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—22no—
604C2×C4—21no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—48no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—88no—
631C2×C4#52512yes—
632C2×C4#52512yes—
633C2×C4D4212yes—
634C2×C4C2×#13712yes—
635C2×C4C2×#13714yes—
636C2×C4C2×#13712yes—
637C2×C4C2×#13712yes—
638C2×C4C23×D411yescharacteristic
639C2×C4C22×#4112yes—
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×2−1+412yes—
642C8—24no—
643C8—44no—
644C8—42no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22#327814yes—
652C22C2×#52511yescharacteristic
653C22C2×D4211yescharacteristic
654C22#348611yescharacteristic
655C22C22×#13711yescharacteristic
656C22C22×#13711yescharacteristic
657C22C2×#57711yescharacteristic
658C22#358611yescharacteristic
659C4—28no—
660C4—22no—
661C4—22no—
662C4—24no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—44no—
671C4—82no—
672C4—84no—
673C4—82no—
674C4—81no—
675C4—81no—
676C4#311314yescomplemented
677C4C2×#54712yes—
678C4#349212yes—
679C2—24no—
680C2#3610311yescharacteristic
681C2C2×#311311yescharacteristic
682C2#6058511yescharacteristic
683C2C2×#327811yescharacteristic
684C2D4×#13711yescharacteristic
685C2#6182911yescharacteristic
686C2#6183811yescharacteristic
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
8. #26389C2semidirect
9. #26187C2semidirect
10. #26042C2semidirect
11. #24732C2semidirect
14. #23004C2semidirect
15. #22674C2semidirect
676. 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×Q8×C4⋊C42
#330082
#329062
#279052
#276472
#263892
#261872
#260422
#247322
#246092
#240632
#230042
#226742
#226032
C4×#4512