Finite Group Databasebeta

← #6809659

Subgroups — #6809659

Order 512.

1755 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3236C212yesmaximal
2#36133C211yesmaximal, characteristic
3C2×#2552C211yesmaximal, characteristic
4#33108C211yesmaximal, characteristic
5#30976C214yesmaximal, complemented
6#30834C214yesmaximal, complemented
7#29643C212yesmaximal
8#28816C212yesmaximal, complemented
9#27910C211yesmaximal, characteristic
10#27777C212yesmaximal, complemented
11#27648C211yesmaximal, characteristic
12#26465C212yesmaximal, complemented
13#26047C212yesmaximal, complemented
14#24606C212yesmaximal
15#24064C212yesmaximal
16#22603C211yesmaximal, characteristic
17C4×#483C211yesmaximal, characteristic
18C2×#533C2212yes—
19C22×#125C2212yes—
20#3236—216no—
21C2×#501C2214yes—
22C2×#491C2214yes—
23C2×#484C2212yes—
24C2×#484C2212yes—
25C2×#484C2212yes—
26C2×#483C2212yes—
27C2×#483C2211yescharacteristic
28C2×#479C2214yes—
29C2×#479C2214yes—
30C2×#432C2212yes—
31#2857C2211yescharacteristic
32#2855C2212yes—
33#2552—21no—
34#2552—22no—
35#2552—21no—
36#2552C2214yes—
37#2552C2212yes—
38#2552C2212yes—
39#2551C2211yescharacteristic
40#2546C2211yescharacteristic
41#2506—21no—
42#2462—22no—
43#2451C2212yes—
44#2446C2212yes—
45#2440—21no—
46#2438—22no—
47C2×#398—22no—
48#2365—21no—
49#2365C2211yescharacteristic
50Q8×C4⋊C4C2211yescharacteristic
51#2344—21no—
52C4×C22⋊Q8C2211yescharacteristic
53C2×#389—21no—
54C2×#389C2212yes—
55C2×#384—21no—
56C2×#384C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#2095C2214yes—
60#2086C2214yes—
61#2083C2212yes—
62#2072C2212yes—
63#2066C2214yes—
64#2059C2214yes—
65#2059C2214yes—
66#2057C2214yes—
67#2043C2214yes—
68#2026C2214yes—
69#2026C2214yes—
70#1981C2212yes—
71#1956C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1944C2214yes—
75#1939C2212yes—
76#1935C2214yes—
77#1927C2214yes—
78#1927C2212yes—
79#1920C2212yes—
80#1917C2214yes—
81#1872C2211yescharacteristic
82#1872C2212yes—
83#1863C2214yes—
84#1863C2214yes—
85#1853C2211yescharacteristic
86#1845C2212yes—
87#1824C2212yes—
88#1823C2211yescharacteristic
89#1817C2212yes—
90#1803C2212yes—
91#1797C2211yescharacteristic
92C4×#106C2212yes—
93#1770C2212yes—
94C4×#103C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96#533—28no—
97#533—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×#125—28no—
103C2×#125—22no—
104C2×#125—22no—
105C2×#125—22no—
106C2×#125—44no—
107C2×#125C2314yes—
108C2×#125C2314yes—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2311yescharacteristic
112C2×C4×Q8—22no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×#117—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#501—216no—
131#491—216no—
132#484—28no—
133#484—28no—
134#484—44no—
135#483—28no—
136#483D418yes—
137#479—216no—
138#479—216no—
139#432—28no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2311yescharacteristic
158C2×#97C2314yes—
159C2×C4×C8C2312yes—
160#400—22no—
161#398—22no—
162#398—24no—
163#398—24no—
164#398—24no—
165#398—24no—
166#398—22no—
167#398—22no—
168#395—22no—
169#395—22no—
170#391—22no—
171#391C2312yes—
172#391C2311yescharacteristic
173#391C2312yes—
174#391C2312yes—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—24no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389C2314yes—
188#389C2312yes—
189#389C2312yes—
190#389C2314yes—
191#389C2311yescharacteristic
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—21no—
197#387—22no—
198#387—22no—
199#385—22no—
200#385—21no—
201#384—21no—
202#384—22no—
203#384—22no—
204#384—24no—
205#384—21no—
206#384—21no—
207#384C2312yes—
208#384C2312yes—
209#384C2311yescharacteristic
210#381—22no—
211#380—21no—
212#380—21no—
213#380C2311yescharacteristic
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D412yes—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—21no—
229C2×#95—21no—
230C2×#95—22no—
231C2×#95D412yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#340C2314yes—
238#340C2312yes—
239#336C2312yes—
240#336C2314yes—
241#336C2314yes—
242#336C2312yes—
243C22×Q8—22no—
244C22×Q8C2412yes—
245C22×Q8C2411yescharacteristic
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2411yescharacteristic
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263#125—28no—
264#125—28no—
265#125—44no—
266#125—44no—
267#125—48no—
268#125—48no—
269#125—48no—
270C22⋊Q8—22no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—48no—
274C22⋊Q8C2×D414yes—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—42no—
279C4×Q8—42no—
280C4×Q8—48no—
281#117—48no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4—84no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—44no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—28no—
357#107—28no—
358#107—44no—
359#106—28no—
360#106—24no—
361#106C2×D418yes—
362C4⋊C8—28no—
363C4⋊C8—24no—
364#103—28no—
365#103—28no—
366#103—28no—
367#103—24no—
368#103—28no—
369#103—44no—
370#103C2×D418yes—
371C22⋊C8—24no—
372C22⋊C8—22no—
373C22⋊C8C2×D414yes—
374#97—28no—
375C4×C8—24no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24C22×D411yescharacteristic
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—82no—
408C2×Q8C22×D412yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4#13714yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4H(5,𝔽2)12yes—
452C22×C4C2511yescharacteristic
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8C22×D414yes—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—42no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—84no—
503C4⋊C4—82no—
504C4⋊C4—84no—
505C4⋊C4C22×D414yes—
506C4⋊C4C22×D412yes—
507C4⋊C42−1+414yes—
508C22⋊C4—24no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—44no—
521C42—44no—
522C42—42no—
523C42—44no—
524C42—44no—
525C42—42no—
526C422−1+414yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23D4212yes—
532C23C2×#13712yes—
533C23#57812yes—
534C23C23×D411yescharacteristic
535Q8—24no—
536Q8—42no—
537Q8—44no—
538Q8—44no—
539Q8—88no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—88no—
597C2×C4#53612yes—
598C2×C4#53612yes—
599C2×C4D4212yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4#57512yes—
603C2×C4C23×D411yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+412yes—
606C2×C4C2×2−1+412yes—
607C8—24no—
608C8—44no—
609C8—42no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22#327614yes—
617C22C2×#53611yescharacteristic
618C22C2×D4211yescharacteristic
619C22#348611yescharacteristic
620C22C22×#13711yescharacteristic
621C22C2×#57511yescharacteristic
622C22C2×#57811yescharacteristic
623C22#358511yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—42no—
629C4—44no—
630C4—44no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—41no—
636C4—84no—
637C4—82no—
638C4—81no—
639C4—82no—
640C4—81no—
641C4#323614yescomplemented
642C4#349912yes—
643C4#354412yes—
644C2—24no—
645C2#3933711yescharacteristic
646C2C2×#323611yescharacteristic
647C2#6087411yescharacteristic
648C2C2×#327611yescharacteristic
649C2#6107011yescharacteristic
650C2#6182611yescharacteristic
651C2#6184211yescharacteristic
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
5. #30976C2semidirect
6. #30834C2semidirect
8. #28816C2semidirect
10. #27777C2semidirect
12. #26465C2semidirect
13. #26047C2semidirect
641. C4#3236semidirect

Maximal subgroups

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

subgroupindex
C2×#32362
#361332
C2×#25522
#331082
#309762
#308342
#296432
#288162
#279102
#277772
#276482
#264652
#260472
#246062
#240642
#226032
C4×#4832