Finite Group Databasebeta

← #6458265

Subgroups — #6458265

Order 512.

2062 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34184C211yesmaximal, characteristic, complemented
2C4×#535C211yesmaximal, characteristic
3#27688C214yesmaximal, complemented
4#27687C214yesmaximal, complemented
5#27684C214yesmaximal, complemented
6#27683C214yesmaximal, complemented
7#23748C212yesmaximal, complemented
8#23735C212yesmaximal, complemented
9#21172C211yesmaximal, characteristic, complemented
10#21169C211yesmaximal, characteristic, complemented
11#20914C212yesmaximal
12#20849C212yesmaximal, complemented
13#20646C211yesmaximal, characteristic, complemented
14C2×#1804C211yesmaximal, characteristic, complemented
15#20629C211yesmaximal, characteristic, complemented
16C2×#535C2211yescharacteristic
17C2×#520C2212yes—
18C2×C4×#41C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#428C2212yes—
21C2×#427C2212yes—
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×#418C2212yes—
25C22×#102C2212yes—
26C2×#408C2212yes—
27#2351—21no—
28#2351—21no—
29#2349—21no—
30#2349—22no—
31#2349—21no—
32#2349—22no—
33#2328—21no—
34#2326—21no—
35#2324C2214yes—
36#2321—22no—
37#2317C2211yescharacteristic
38#2317C2212yes—
39#2317C2214yescomplemented
40#2311C2211yescharacteristic
41#2309C2211yescharacteristic
42C4×#127—21no—
43C4×#124—21no—
44C4×#121C2214yes—
45C4×#120—22no—
46#2285C2211yescharacteristic
47#2285C2212yescomplemented
48C2×#382C2211yescharacteristic
49C2×#382C2212yescomplemented
50C4×#117C2211yescharacteristic
51#2101C2214yescomplemented
52#2101C2214yescomplemented
53#2100C2214yescomplemented
54#2099C2214yescomplemented
55#2094C2214yes—
56#2093C2214yes—
57#2092C2214yes—
58#2092C2214yes—
59#1949C2212yescomplemented
60#1949C2214yescomplemented
61#1949C2212yescomplemented
62#1949C2214yescomplemented
63#1948C2212yescomplemented
64#1947C2212yescomplemented
65#1875C2214yes—
66#1875C2214yes—
67#1874C2214yes—
68#1874C2214yes—
69#1854C2214yescomplemented
70#1854C2214yescomplemented
71#1853C2214yescomplemented
72#1852C2214yescomplemented
73#1804C2212yes—
74#1804C2214yes—
75#1804C2212yes—
76#1804C2212yes—
77#1804C2212yes—
78#1802C2214yes—
79#1798C2212yescomplemented
80#1798C2212yescomplemented
81#1797C2214yescomplemented
82#1736C2212yes—
83#1736C2212yes—
84#1735C2212yes—
85C2×#340C2212yes—
86#1729C2212yes—
87C22×#41C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×M16C2312yes—
91C23×C8C2312yes—
92#535—24no—
93#535Q818yes—
94C2×#127—21no—
95C2×#124—21no—
96C2×#121C2314yes—
97C2×#120—22no—
98#520—44no—
99#520—44no—
100C4×#41—24no—
101C4×#41—24no—
102C2×C4×Q8—22no—
103C2×C4×Q8D411yescharacteristic
104C2×C4×Q8D411yescharacteristic
105C2×C4×D4—24no—
106C2×C4×D4—22no—
107C2×C4×D4—24no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—24no—
111C2×C4×D4—24no—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D412yes—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D412yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C2×#117C2314yes—
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C22⋊C4—21no—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#428—44no—
131#427—44no—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#107C2312yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138#423—24no—
139C2×C4⋊C8—24no—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145#418—44no—
146C2×#102—44no—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151#408—28no—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154#391—21no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390—24no—
159#390C2314yes—
160#389—22no—
161#389—22no—
162#389—24no—
163#389—24no—
164#389C2314yes—
165#388—24no—
166#388—24no—
167#386—22no—
168#386—24no—
169#386—24no—
170#385—24no—
171#385—22no—
172#385—24no—
173#385—24no—
174#385—24no—
175#384—21no—
176#384C2312yes—
177#384C2314yes—
178#382—21no—
179#382—21no—
180#382—21no—
181#382—22no—
182#382—22no—
183#382C2312yes—
184#382C2312yes—
185#382C2311yescharacteristic
186#382C2311yescharacteristic
187#382C2314yes—
188#380—21no—
189#380—21no—
190#380—22no—
191#380—22no—
192#380C2314yes—
193#379—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—21no—
202C4×C22⋊C4—24no—
203#376—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—22no—
207C2×#95—22no—
208C43—21no—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2312yes—
214#340C2314yes—
215#340C2312yes—
216#340C2314yes—
217#340C2312yes—
218#340C2312yes—
219#336C2314yes—
220#336C2314yes—
221C25—21no—
222C2×#41—22no—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41C2×Q814yes—
227C2×#41#4114yes—
228C22×Q8C2×D411yescharacteristic
229C22×D4—24no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4C2×D412yes—
237C22×D4C2×D411yescharacteristic
238C22×D4C2×Q814yes—
239C22×D4C2412yes—
240C23×C4—21no—
241C23×C4—24no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—44no—
246C23×C4—84no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C2×M16—24no—
254C2×M16—24no—
255C2×M16—24no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8—24no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2414yes—
262#127—28no—
263#124—28no—
264#121—28no—
265#121C2×Q8116yes—
266#120—216no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—48no—
270C4×D4—28no—
271C4×D4—28no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—416no—
279C4×D4—48no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—88no—
283#117—22no—
284#117—22no—
285#117—22no—
286#117—24no—
287#117—24no—
288#117—24no—
289#117—24no—
290#117—22no—
291#117—22no—
292#117—44no—
293#117—44no—
294#117C2×Q814yes—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—21no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42#4114yes—
346C2×C42#4114yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—42no—
351#107—42no—
352#107—44no—
353#106—42no—
354#106—42no—
355#106—44no—
356C4⋊C8—24no—
357C4⋊C8—22no—
358C4⋊C8—22no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361#102—44no—
362#102—44no—
363#102—44no—
364#102—44no—
365C22⋊C8—28no—
366C22⋊C8—28no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—21no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—21no—
380#95—21no—
381C24—22no—
382C24—24no—
383C24—42no—
384C24—42no—
385C24—82no—
386C24—82no—
387C24C22×D412yes—
388C24C22×D411yescharacteristic
389#41—28no—
390#41—28no—
391#41—28no—
392#41—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8C22⋊Q814yes—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—216no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—84no—
414C2×D4—84no—
415C2×D4C22⋊Q818yes—
416C2×D4C22⋊Q814yes—
417C2×D4C22×Q814yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—84no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—84no—
444C22×C4—82no—
445C22×C4—84no—
446C22×C4—816no—
447C22×C4#12314yes—
448C22×C4#12314yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×Q814yes—
458C22×C4C22×Q812yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4C2511yescharacteristic
466M16—48no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—48no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—48no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4C22×Q814yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—216no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—48no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C42—24no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—48no—
520C42—44no—
521C42—42no—
522C42—44no—
523C42—82no—
524C42#12314yes—
525C42#12314yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C42C2×#4112yes—
529C42C2×#4112yes—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—48no—
542C23—88no—
543C23—84no—
544C23—84no—
545C23—88no—
546C23C2×C22⋊Q814yes—
547C23C2×C22⋊Q812yes—
548C23C2×#12312yes—
549C23#53512yes—
550C23C23×D411yescharacteristic
551Q8—28no—
552D4—216no—
553D4—24no—
554D4—24no—
555D4—28no—
556D4—42no—
557D4—42no—
558D4—48no—
559D4—816no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—88no—
596C2×C4—88no—
597C2×C4—84no—
598C2×C4—164no—
599C2×C4—168no—
600C2×C4C2×C22⋊Q814yes—
601C2×C4C2×C22⋊Q812yes—
602C2×C4C2×#12312yes—
603C2×C4C2×#12312yes—
604C2×C4C2×#12312yes—
605C2×C4C2×#12312yes—
606C2×C4C2×#12312yes—
607C2×C4#53512yes—
608C2×C4#55314yes—
609C2×C4#55314yes—
610C2×C4#55414yes—
611C2×C4#55414yes—
612C2×C4C23×Q811yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C8—42no—
619C8—42no—
620C8—44no—
621C22—24no—
622C22—22no—
623C22—22no—
624C22—24no—
625C22—28no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—44no—
630C22—48no—
631C22—44no—
632C22—816no—
633C22—82no—
634C22—82no—
635C22—84no—
636C22#248714yes—
637C22C22×C22⋊Q811yescharacteristic
638C22C22×#12311yescharacteristic
639C22C2×#53511yescharacteristic
640C22#358412yes—
641C22#358512yes—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—24no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—84no—
654C4—82no—
655C4#248714yes—
656C4C2×#55311yescharacteristic
657C4C2×#55311yescharacteristic
658C4C2×#55411yescharacteristic
659C4C2×#55411yescharacteristic
660C2—24no—
661C2—44no—
662C2—84no—
663C2C2×#248711yescharacteristic
664C2#6049612yes—
665C2#6051611yescharacteristic
666C2#6051811yescharacteristic
667C2#6083412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34184C2semidirect
3. #27688C2semidirect
4. #27687C2semidirect
5. #27684C2semidirect
6. #27683C2semidirect
7. #23748C2semidirect
8. #23735C2semidirect
9. #21172C2semidirect
10. #21169C2semidirect
12. #20849C2semidirect
13. #20646C2semidirect
14. C2×#1804C2semidirect
15. #20629C2semidirect
39. #2317C22semidirect
47. #2285C22semidirect
49. C2×#382C22semidirect
51. #2101C22semidirect
52. #2101C22semidirect
53. #2100C22semidirect
54. #2099C22semidirect
59. #1949C22semidirect
60. #1949C22semidirect
61. #1949C22semidirect
62. #1949C22semidirect
63. #1948C22semidirect
64. #1947C22semidirect
69. #1854C22semidirect
70. #1854C22semidirect
71. #1853C22semidirect
72. #1852C22semidirect
79. #1798C22semidirect
80. #1798C22semidirect
81. #1797C22semidirect

Maximal subgroups

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

subgroupindex
#341842
C4×#5352
#276882
#276872
#276842
#276832
#237482
#237352
#211722
#211692
#209142
#208492
#206462
C2×#18042
#206292