Finite Group Databasebeta

← #6625120

Subgroups — #6625120

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33108C211yesmaximal, characteristic, complemented
4#32909C212yesmaximal
5#28306C212yesmaximal
6#28124C212yesmaximal, complemented
7#27067C214yesmaximal, complemented
8#26591C214yesmaximal, complemented
9#26527C212yesmaximal, complemented
10#25722C211yesmaximal, characteristic, complemented
11#25365C212yesmaximal, complemented
12#24209C211yesmaximal, characteristic
13#23184C212yesmaximal, complemented
14#23094C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16C4×#480C211yesmaximal, characteristic
17C4×#451C212yesmaximal, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#494C2212yes—
21C2×#483C2212yes—
22C2×#480C2211yescharacteristic
23C2×#464C2214yes—
24C2×#461C2212yes—
25C2×#451C2212yes—
26C2×#451C2212yes—
27C2×#448C2212yes—
28C2×C4×Q16C2214yes—
29#2857C2211yescharacteristic
30#2850C2212yes—
31#2596—22no—
32#2571C2211yescharacteristic
33#2547C2211yescharacteristic
34#2514C2212yes—
35#2510—22no—
36#2494—21no—
37#2487—21no—
38#2448C2212yes—
39#2365—22no—
40#2365—21no—
41#2365—21no—
42#2365C2212yes—
43#2365C2212yes—
44#2365C2214yes—
45#2365C2211yescharacteristic
46#2354C2211yescharacteristic
47#2352—21no—
48#2347—21no—
49C4×C22⋊Q8C2212yes—
50C4×#121C2211yescharacteristic
51C2×#391C2212yes—
52C2×#389—21no—
53C2×#387—22no—
54C2×C4×C4⋊C4—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#2097C2214yes—
58#2066C2214yes—
59#2056C2214yes—
60#2043C2212yes—
61#2016C2214yes—
62#2015C2214yes—
63#2013C2212yes—
64#1981C2212yes—
65#1981C2212yes—
66#1969C2214yes—
67#1968C2214yes—
68#1956C2212yes—
69#1948C2212yes—
70#1939C2212yes—
71#1927C2214yes—
72#1927C2214yes—
73#1925C2212yes—
74#1918C2212yes—
75#1916C2214yes—
76#1880C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1878C2214yes—
80#1874C2211yescharacteristic
81#1872C2212yes—
82#1863C2214yes—
83#1852C2211yescharacteristic
84#1823C2211yescharacteristic
85#1805C2214yes—
86#1797C2211yescharacteristic
87#1797C2212yes—
88#1787C2212yes—
89C4×#107C2212yes—
90C8×C4⋊C4C2212yes—
91C4×#103C2214yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#124C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×#121C2311yescharacteristic
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—48no—
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—21no—
115C2×C4×D4C2311yescharacteristic
116C2×#117—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C42—22no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#494—28no—
129#483—44no—
130#480—24no—
131#464—48no—
132#461—28no—
133#451—28no—
134#451—44no—
135#448—44no—
136C4×Q16—48no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2311yescharacteristic
154C2×C4×C8C2312yes—
155C2×C4×C8C2314yes—
156#401—22no—
157#400—22no—
158#400—22no—
159#399—22no—
160#397—22no—
161#397—22no—
162#391—22no—
163#391—22no—
164#391C2312yes—
165#391C2314yes—
166#391C2312yes—
167#391C2312yes—
168#391C2314yes—
169#390C2312yes—
170#390C2311yescharacteristic
171#389—21no—
172#389—22no—
173#389—22no—
174#389—24no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389C2312yes—
179#389C2311yescharacteristic
180#388—22no—
181#388—22no—
182#388—22no—
183#388—21no—
184#388—22no—
185#388—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—24no—
190#387—24no—
191#387—24no—
192#387—24no—
193#386—22no—
194#386—22no—
195#386—21no—
196#385—22no—
197#384C2311yescharacteristic
198#383—22no—
199#382—21no—
200#382C2312yes—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—24no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226C2×#95—22no—
227C2×#95—21no—
228C2×#95—22no—
229C2×#95D412yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—28no—
240C22×Q8—22no—
241C22×Q8—24no—
242C22×Q8—24no—
243C22×Q8—44no—
244C22×Q8C2412yes—
245C22×Q8C2414yes—
246C22×D4—21no—
247C22×D4C2411yescharacteristic
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C2×Q16—28no—
258C2×Q16—28no—
259C2×Q16—28no—
260C2×Q16—44no—
261C2×Q16—44no—
262C2×Q16—44no—
263C2×SD16—28no—
264C2×SD16—44no—
265C2×SD16—44no—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—28no—
270C4⋊Q8—28no—
271C4⋊Q8—28no—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8#4118yes—
275#124—44no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278#121—22no—
279#121—22no—
280C4×Q8—28no—
281C4×Q8—28no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—48no—
286C4×Q8—48no—
287C4×D4—24no—
288C4×D4—44no—
289#117—44no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42—48no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—24no—
356#107—24no—
357#106—44no—
358C4⋊C8—24no—
359C4⋊C8—44no—
360#103—28no—
361#103—28no—
362#103—28no—
363#103—44no—
364#103—44no—
365#103—44no—
366#103—44no—
367#102—24no—
368#102—24no—
369C22⋊C8—24no—
370C22⋊C8—22no—
371C22⋊C8#4114yes—
372C4×C8—24no—
373C4×C8—44no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
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—
397C24—21no—
398C24C22×D411yescharacteristic
399C2×Q8—28no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—416no—
410C2×Q8—416no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8C2×#4118yes—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4#13514yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C2×#4112yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4C2511yescharacteristic
467Q16—416no—
468Q16—88no—
469SD16—88no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8C2×#4114yes—
479C4⋊C4—24no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4#13714yes—
510C4⋊C4H(5,𝔽2)14yes—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—81no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—24no—
525C42—24no—
526C42—42no—
527C42—42no—
528C42—48no—
529C42—48no—
530C42—44no—
531C42—44no—
532C42—42no—
533C42—44no—
534C42—82no—
535C42C2×#4114yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23#55012yes—
548C23C2×#13512yes—
549C23#57812yes—
550C23C23×D411yescharacteristic
551Q8—28no—
552Q8—28no—
553Q8—416no—
554Q8—44no—
555Q8—44no—
556Q8—44no—
557Q8—44no—
558D4—42no—
559D4—42no—
560D4—84no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—416no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4#52512yes—
616C2×C4#52512yes—
617C2×C4#55012yes—
618C2×C4C2×#13512yes—
619C2×C4C2×#13512yes—
620C2×C4C2×#13714yes—
621C2×C4C2×#13712yes—
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4112yes—
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C8—24no—
627C8—44no—
628C8—42no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22#339314yes—
643C22C2×#52511yescharacteristic
644C22C2×#55011yescharacteristic
645C22#348411yescharacteristic
646C22C22×#13511yescharacteristic
647C22C22×#13711yescharacteristic
648C22C2×#57811yescharacteristic
649C22#358611yescharacteristic
650C4—22no—
651C4—24no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—42no—
658C4—41no—
659C4—48no—
660C4—81no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4—81no—
665C4—82no—
666C4#310814yes—
667C4#348112yes—
668C4#351912yes—
669C2—24no—
670C2—161no—
671C2#3652911yescharacteristic
672C2C2×#310811yescharacteristic
673C2#6058311yescharacteristic
674C2C2×#339311yescharacteristic
675C2#6140411yescharacteristic
676C2#6156511yescharacteristic
677C2#6160211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2365C2semidirect
3. #33108C2semidirect
6. #28124C2semidirect
7. #27067C2semidirect
8. #26591C2semidirect
9. #26527C2semidirect
10. #25722C2semidirect
11. #25365C2semidirect
13. #23184C2semidirect
14. #23094C2semidirect
15. #22625C2semidirect
17. C4×#451C2semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#23652
#331082
#329092
#283062
#281242
#270672
#265912
#265272
#257222
#253652
#242092
#231842
#230942
#226252
C4×#4802
C4×#4512