Finite Group Databasebeta

← #6447981

Subgroups — #6447981

Order 512.

1736 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#25003C214yesmaximal, complemented
4#25002C214yesmaximal, complemented
5#24993C212yesmaximal, complemented
6#24992C212yesmaximal, complemented
7#24991C212yesmaximal, complemented
8#24990C212yesmaximal, complemented
9#23748C212yesmaximal, complemented
10#23632C212yesmaximal
11#21302C211yesmaximal, characteristic, complemented
12#21169C211yesmaximal, characteristic, complemented
13#21065C212yesmaximal
14#21039C212yesmaximal, complemented
15#20764C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic
17C4×#418C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#427C2212yes—
21C22×C4⋊C8C2211yescharacteristic
22C2×#418C2211yescharacteristic
23C2×#418C2211yescharacteristic
24C22×#102C2212yes—
25C2×C4×M16C2211yescharacteristic
26#2848—22no—
27#2626—22no—
28#2580C2214yes—
29#2566—22no—
30#2563—21no—
31#2542—21no—
32#2538C2212yes—
33#2538C2212yes—
34#2360—22no—
35#2360—22no—
36#2355C2212yescomplemented
37#2355C2212yescomplemented
38Q8×C4⋊C4C2212yescomplemented
39Q8×C4⋊C4C2212yescomplemented
40#2330—22no—
41#2317C2212yes—
42#2317C2212yes—
43#2309C2212yes—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8—21no—
46#2285C2211yescharacteristic
47C2×#384C2211yescharacteristic
48#2274C2212yes—
49C4×#117C2211yescharacteristic
50#2101C2214yescomplemented
51#2101C2214yescomplemented
52#2100C2214yescomplemented
53#2099C2214yescomplemented
54#2073C2214yes—
55#2073C2214yes—
56#2071C2214yes—
57#2071C2214yes—
58#1985C2214yescomplemented
59#1984C2212yescomplemented
60#1984C2212yescomplemented
61#1949C2212yescomplemented
62#1949C2212yescomplemented
63#1948C2211yescharacteristic, complemented
64#1948C2211yescharacteristic, complemented
65#1947C2211yescharacteristic, complemented
66#1947C2211yescharacteristic, complemented
67#1926C2214yes—
68#1926C2214yes—
69#1925C2214yes—
70#1925C2214yes—
71#1921C2214yescomplemented
72#1920C2214yescomplemented
73#1919C2214yescomplemented
74#1918C2214yescomplemented
75#1836C2214yescomplemented
76#1834C2212yescomplemented
77#1834C2212yescomplemented
78#1802C2212yes—
79#1802C2212yes—
80#1802C2214yes—
81#1762C2212yes—
82#1762C2212yes—
83C4×#102C2212yes—
84C4×#102C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1735C2212yes—
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98#533—24no—
99#533—24no—
100C2×C4⋊Q8D411yescharacteristic
101C2×C4⋊Q8D411yescharacteristic
102C2×#125D412yes—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8D412yes—
106C2×C22⋊Q8D412yes—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4—21no—
126C22×C42C2311yescharacteristic
127#427—28no—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#106C2312yes—
131C2×#106C2312yes—
132C2×C4⋊C8—24no—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136#418—24no—
137#418—24no—
138C2×#102—28no—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C4×M16—24no—
146C2×#97C2312yes—
147C2×C4×C8C2312yes—
148#399—24no—
149#396—24no—
150#396—24no—
151#393—24no—
152#391—22no—
153#391—22no—
154#390—22no—
155#390—22no—
156#390C2314yes—
157#390C2314yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—24no—
163#389—24no—
164#389C2314yes—
165#389C2314yes—
166#387—24no—
167#387—24no—
168#385—24no—
169#385—24no—
170#385—24no—
171#385—24no—
172#384—22no—
173#384—21no—
174#384—21no—
175#384C2312yes—
176#384C2312yes—
177#384C2314yes—
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#382—22no—
181#382C2314yes—
182#382C2312yes—
183#380—21no—
184#380—21no—
185#380—22no—
186#380—22no—
187#379—24no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4C2311yescharacteristic
195C4×C4⋊C4C2311yescharacteristic
196C4×C4⋊C4C2314yes—
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2311yescharacteristic
200C4×C22⋊C4—21no—
201C4×C22⋊C4—21no—
202C4×C22⋊C4—24no—
203#376—21no—
204C43—21no—
205#340C2314yes—
206#340C2314yes—
207#340C2312yes—
208#340C2312yes—
209#340C2314yes—
210#340C2314yes—
211#340C2312yes—
212#340C2312yes—
213#340C2314yes—
214#340C2314yes—
215#340C2312yes—
216#340C2312yes—
217#340C2312yes—
218#340C2312yes—
219#336C2314yes—
220#336C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25—21no—
224C22×Q8C2×D412yes—
225C22×Q8C2×D412yes—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4#4114yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—44no—
239C23×C4—82no—
240C23×C4—82no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—22no—
248C2×M16—22no—
249C2×M16—22no—
250C22×C8—24no—
251C22×C8—24no—
252C22×C8—24no—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259#125—28no—
260C22⋊Q8—28no—
261C22⋊Q8—28no—
262C22⋊Q8—84no—
263C22⋊Q8—84no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—84no—
269C4×Q8—84no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—44no—
275C4×D4—44no—
276#117—24no—
277#117—24no—
278#117—22no—
279#117—22no—
280#117—24no—
281#117—24no—
282#117—22no—
283#117#4114yes—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4C2×D414yes—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42#4114yes—
341C2×C42#4114yes—
342C2×C42C2414yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—24no—
346#107—24no—
347#106—24no—
348#106—24no—
349C4⋊C8—22no—
350C4⋊C8—22no—
351C4⋊C8—24no—
352#102—28no—
353#102—28no—
354#102—24no—
355#102—24no—
356#102—24no—
357#102—24no—
358#97—24no—
359C4×C8—24no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—24no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372C24—22no—
373C24—24no—
374C24—42no—
375C24—42no—
376C24—81no—
377C24—81no—
378C24—81no—
379C24—81no—
380C24C22×D411yescharacteristic
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×D4—28no—
388C2×D4—24no—
389C2×D4—28no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—42no—
398C2×D4—44no—
399C2×D4C2×#4114yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4—816no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4#12314yes—
426C22×C4#12314yes—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C42−1+412yes—
439C22×C42−1+412yes—
440C22×C4C2511yescharacteristic
441M16—44no—
442M16—44no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—48no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4C2×#4112yes—
491C4⋊C4C2×#4112yes—
492C22⋊C4—28no—
493C22⋊C4—24no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C42—24no—
503C42—24no—
504C42—24no—
505C42—42no—
506C42—42no—
507C42—44no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—84no—
512C42—84no—
513C42#12314yes—
514C42#12314yes—
515C42C2×#4112yes—
516C42C2×#4112yes—
517C42C2×#4112yes—
518C42C2×#4112yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—24no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—48no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23C2×#12312yes—
538C23#54812yes—
539C23#54812yes—
540C23C23×D411yescharacteristic
541Q8—48no—
542Q8—48no—
543D4—24no—
544D4—24no—
545D4—48no—
546D4—42no—
547D4—42no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—88no—
587C2×C4—84no—
588C2×C4—88no—
589C2×C4—164no—
590C2×C4C2×#12312yes—
591C2×C4C2×#12312yes—
592C2×C4C2×#12312yes—
593C2×C4C2×#12312yes—
594C2×C4C2×#12312yes—
595C2×C4#52914yes—
596C2×C4#54114yes—
597C2×C4#54812yes—
598C2×C4#54812yes—
599C2×C4#55214yes—
600C2×C4#55614yes—
601C2×C4#57612yes—
602C2×C4#57712yes—
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×2−1+411yescharacteristic
608C2×C4C2×2−1+411yescharacteristic
609C8—42no—
610C8—42no—
611C8—42no—
612C8—42no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—816no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22#242114yes—
628C22C22×#12311yescharacteristic
629C22C2×#54811yescharacteristic
630C22C2×#54811yescharacteristic
631C22C2×#57611yescharacteristic
632C22C2×#57711yescharacteristic
633C22#358511yescharacteristic
634C22#358611yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—44no—
639C4—44no—
640C4—42no—
641C4—42no—
642C4—82no—
643C4—81no—
644C4—84no—
645C4—81no—
646C4—82no—
647C4—82no—
648C4#242114yes—
649C4#309614yes—
650C4C2×#52911yescharacteristic
651C4C2×#54111yescharacteristic
652C4C2×#55211yescharacteristic
653C4C2×#55611yescharacteristic
654C2—24no—
655C2—84no—
656C2C2×#242111yescharacteristic
657C2C2×#309611yescharacteristic
658C2#6051711yescharacteristic
659C2#6133911yescharacteristic
660C2#6134011yescharacteristic
661C2#6134511yescharacteristic
662C2#6134711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #25003C2semidirect
4. #25002C2semidirect
5. #24993C2semidirect
6. #24992C2semidirect
7. #24991C2semidirect
8. #24990C2semidirect
9. #23748C2semidirect
11. #21302C2semidirect
12. #21169C2semidirect
14. #21039C2semidirect
15. #20764C2semidirect
36. #2355C22semidirect
37. #2355C22semidirect
38. Q8×C4⋊C4C22semidirect
39. Q8×C4⋊C4C22semidirect
50. #2101C22semidirect
51. #2101C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
58. #1985C22semidirect
59. #1984C22semidirect
60. #1984C22semidirect
61. #1949C22semidirect
62. #1949C22semidirect
63. #1948C22semidirect
64. #1948C22semidirect
65. #1947C22semidirect
66. #1947C22semidirect
71. #1921C22semidirect
72. #1920C22semidirect
73. #1919C22semidirect
74. #1918C22semidirect
75. #1836C22semidirect
76. #1834C22semidirect
77. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#351722
#342142
#250032
#250022
#249932
#249922
#249912
#249902
#237482
#236322
#213022
#211692
#210652
#210392
#207642
C2×#18022
C4×#4182