Finite Group Databasebeta

← #6503179

Subgroups — #6503179

Order 512.

2252 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35000C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic, complemented
3#27856C214yesmaximal, complemented
4#26198C214yesmaximal, complemented
5#26126C214yesmaximal, complemented
6#24933C214yesmaximal, complemented
7#24861C214yesmaximal, complemented
8#24550C214yesmaximal, complemented
9#21940C211yesmaximal, characteristic, complemented
10#21758C211yesmaximal, characteristic, complemented
11C2×#1893C211yesmaximal, characteristic, complemented
12C2×#1869C211yesmaximal, characteristic, complemented
13C2×#1808C211yesmaximal, characteristic, complemented
14C24×C8C2211yescharacteristic
15C2×#534—21no—
16C22×#121C2211yescharacteristic
17C2×#518—21no—
18C22×C4×D4C2211yescharacteristic
19C2×#480C2214yescomplemented
20C22×#107C2212yescomplemented
21C22×#102C2212yescomplemented
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2212yescomplemented
24#2854C2212yes—
25#2847C2212yes—
26#2618—21no—
27#2617—21no—
28#2543—22no—
29#2534—22no—
30#2486C2212yes—
31#2416C2214yes—
32#2408C2212yes—
33#2349—22no—
34#2349—22no—
35D4×C4⋊C4C2214yes—
36D4×C4⋊C4C2214yescomplemented
37#2316—21no—
38#2315—21no—
39#2315—21no—
40#2308—21no—
41#2285C2212yescomplemented
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2212yescomplemented
46#2099C2214yescomplemented
47#2093C2214yescomplemented
48#2083C2214yescomplemented
49#2075C2214yescomplemented
50#2075C2214yescomplemented
51#2071C2214yescomplemented
52#2033C2214yescomplemented
53#2033C2214yescomplemented
54#1925C2214yescomplemented
55#1925C2214yescomplemented
56#1918C2214yescomplemented
57#1918C2214yescomplemented
58#1893C2214yes—
59#1893C2214yes—
60#1893C2212yescomplemented
61#1893C2212yescomplemented
62#1874C2214yescomplemented
63#1874C2214yescomplemented
64#1869C2214yes—
65#1869C2212yescomplemented
66#1869C2212yescomplemented
67#1852C2214yescomplemented
68#1852C2214yescomplemented
69#1852C2214yescomplemented
70#1823C2214yescomplemented
71#1823C2212yescomplemented
72#1808C2214yescomplemented
73#1808C2214yescomplemented
74#1808C2214yescomplemented
75#1786C2212yescomplemented
76C2×#340C2212yescomplemented
77C2×#340C2212yescomplemented
78C2×#336C2212yescomplemented
79C23×D4—21no—
80C23×D4D411yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C23×C8—24no—
85C23×C8—44no—
86C23×C8C2311yescharacteristic
87C23×C8C2312yes—
88C23×C8C2312yes—
89C23×C8C2312yes—
90C23×C8C2311yescharacteristic
91#534—48no—
92C2×#123—22no—
93C2×#123—22no—
94C2×#121—24no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121—24no—
98C2×#121—24no—
99C2×#121C2312yes—
100C2×#121C2312yes—
101C2×#121C2314yes—
102C2×#120—22no—
103C2×#120—22no—
104#518—48no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4—24no—
109C2×C4×D4D412yes—
110C2×C4×D4D412yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—22no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4D411yescharacteristic
126C22×C22⋊C4D411yescharacteristic
127C22×C42C2311yescharacteristic
128#480—48no—
129C2×#107—44no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#102—44no—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8—24no—
140C2×C22⋊C8—44no—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2314yes—
145#398—22no—
146#397—22no—
147#396—24no—
148#394—22no—
149#394—24no—
150#392—22no—
151#390C2314yes—
152#389—24no—
153#389—24no—
154#389C2314yes—
155#388—24no—
156#386—22no—
157#386—22no—
158#386—24no—
159#386—24no—
160#385—24no—
161#385—24no—
162#385—24no—
163#385—24no—
164#384C2314yes—
165#382—22no—
166#381—22no—
167#380—22no—
168#380—21no—
169#380—21no—
170#380—22no—
171#380—21no—
172#380—21no—
173#380—22no—
174#380—21no—
175#380—21no—
176#380C2312yes—
177#380C2312yes—
178#380C2312yes—
179#380C2312yes—
180#380C2312yes—
181#380C2314yes—
182#380C2314yes—
183#380C2312yes—
184#379—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4C2314yes—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—22no—
190C2×#95—21no—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25C2×D411yescharacteristic
203C25C2×D411yescharacteristic
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—24no—
209C22×D4—21no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4D818yes—
218C22×D4C2×D412yes—
219C22×D4C2×D412yes—
220C22×D4C2412yes—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4#4114yes—
240C23×C4#4114yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C22×C8—22no—
246C22×C8—24no—
247C22×C8—24no—
248C22×C8—24no—
249C22×C8—22no—
250C22×C8—24no—
251C22×C8—24no—
252C22×C8—24no—
253C22×C8—42no—
254C22×C8—42no—
255C22×C8—42no—
256C22×C8—42no—
257C22×C8—42no—
258C22×C8—416no—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C22×C8C2412yes—
262#123—48no—
263#123—48no—
264#121—24no—
265#121—24no—
266#121—24no—
267#121—24no—
268#121—416no—
269#121—44no—
270#121—44no—
271#121—44no—
272#121—44no—
273#120—48no—
274#120—48no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—416no—
282C4×D4—416no—
283#117—48no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338#107—28no—
339#107—44no—
340#107—44no—
341#107—44no—
342#102—28no—
343#102—44no—
344#102—44no—
345#102—44no—
346C22⋊C8—24no—
347C22⋊C8—24no—
348C22⋊C8—28no—
349C22⋊C8—44no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—82no—
371C24—82no—
372C24#12314yes—
373C24C2×D814yes—
374C24C2×D814yes—
375C24#13714yes—
376C24C22×D412yes—
377C24C22×D411yescharacteristic
378C24C22×D411yescharacteristic
379C24C22×D411yescharacteristic
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—232no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—48no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—42no—
398C2×D4—82no—
399C2×D4—82no—
400C2×D4—82no—
401C2×D4—84no—
402C2×D4—84no—
403C2×D4—816no—
404C2×D4C2×D818yes—
405C2×D4C2×D818yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—84no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—84no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—84no—
452C22×C4#12314yes—
453C22×C4#12314yes—
454C22×C4#12314yes—
455C22×C4C2×D814yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2511yescharacteristic
469C2×C8—28no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—48no—
479C2×C8—48no—
480C2×C8—44no—
481C2×C8—84no—
482C2×C8H(5,𝔽2)14yes—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—42no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—84no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—84no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—48no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—22no—
532C23—28no—
533C23—44no—
534C23—44no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—44no—
539C23—42no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—82no—
544C23—84no—
545C23—82no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23—164no—
550C23—164no—
551C23#480116yescomplemented
552C23C2×#12312yes—
553C23C2×#12312yes—
554C23C2×#12312yes—
555C23C22×D814yes—
556C23C22×D812yes—
557C23C22×D812yes—
558C23C22×D812yes—
559C23C2×#13712yes—
560C23C2×#13712yes—
561C23C2×#13712yes—
562C23C23×D411yescharacteristic
563D4—216no—
564D4—216no—
565D4—42no—
566D4—42no—
567D4—44no—
568D4—84no—
569D4—88no—
570D4—88no—
571D4—168no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—84no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—84no—
613C2×C4—164no—
614C2×C4—164no—
615C2×C4—168no—
616C2×C4—164no—
617C2×C4C2×#12312yes—
618C2×C4C2×#12312yes—
619C2×C4C2×#12312yes—
620C2×C4C22×D814yes—
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4#58512yes—
625C8—24no—
626C8—42no—
627C8—44no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—28no—
641C22—24no—
642C22—416no—
643C22—42no—
644C22—44no—
645C22—42no—
646C22—88no—
647C22—82no—
648C22—84no—
649C22—84no—
650C22—82no—
651C22—82no—
652C22—168no—
653C22C2×#48014yes—
654C22C2×#48014yes—
655C22C2×#48014yes—
656C22C22×#12311yescharacteristic
657C22#349612yes—
658C22C23×D811yescharacteristic
659C22C22×#13711yescharacteristic
660C22#358412yes—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—24no—
665C4—42no—
666C4—42no—
667C4—84no—
668C4—84no—
669C4—84no—
670C4—82no—
671C4#349011yescharacteristic
672C4#349111yescharacteristic
673C4#350512yes—
674C2—24no—
675C2—24no—
676C2—24no—
677C2—44no—
678C2—162no—
679C2#3536011yescharacteristic
680C2C22×#48011yescharacteristic
681C2#6051611yescharacteristic
682C2#6224512yes—
683C2#6225412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35000C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34181C2semidirect
3. #27856C2semidirect
4. #26198C2semidirect
5. #26126C2semidirect
6. #24933C2semidirect
7. #24861C2semidirect
8. #24550C2semidirect
9. #21940C2semidirect
10. #21758C2semidirect
11. C2×#1893C2semidirect
12. C2×#1869C2semidirect
13. C2×#1808C2semidirect
19. C2×#480C22semidirect
20. C22×#107C22semidirect
21. C22×#102C22semidirect
23. C22×C22⋊C8C22semidirect
36. D4×C4⋊C4C22semidirect
41. #2285C22semidirect
45. C2×#380C22semidirect
46. #2099C22semidirect
47. #2093C22semidirect
48. #2083C22semidirect
49. #2075C22semidirect
50. #2075C22semidirect
51. #2071C22semidirect
52. #2033C22semidirect
53. #2033C22semidirect
54. #1925C22semidirect
55. #1925C22semidirect
56. #1918C22semidirect
57. #1918C22semidirect
60. #1893C22semidirect
61. #1893C22semidirect
62. #1874C22semidirect
63. #1874C22semidirect
65. #1869C22semidirect
66. #1869C22semidirect
67. #1852C22semidirect
68. #1852C22semidirect
69. #1852C22semidirect
70. #1823C22semidirect
71. #1823C22semidirect
72. #1808C22semidirect
73. #1808C22semidirect
74. #1808C22semidirect
75. #1786C22semidirect
76. C2×#340C22semidirect
77. C2×#340C22semidirect
78. C2×#336C22semidirect
551. C23#480semidirect

Maximal subgroups

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

subgroupindex
#350002
#341812
#278562
#261982
#261262
#249332
#248612
#245502
#219402
#217582
C2×#18932
C2×#18692
C2×#18082