Finite Group Databasebeta

← #6482068

Subgroups — #6482068

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37639C211yesmaximal, characteristic
2#34207C211yesmaximal, characteristic, complemented
3#32986C214yesmaximal, complemented
4#29898C214yesmaximal, complemented
5#26302C212yesmaximal, complemented
6#26299C212yesmaximal, complemented
7#26235C212yesmaximal, complemented
8#26230C212yesmaximal, complemented
9#25011C212yesmaximal, complemented
10#25010C212yesmaximal, complemented
11#24968C212yesmaximal, complemented
12#24967C212yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21859C211yesmaximal, characteristic, complemented
15#21787C211yesmaximal, characteristic, complemented
16#21390C211yesmaximal, characteristic, complemented
17#21362C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×#520C2212yes—
21C2×#482C2214yes—
22C2×#480C2214yes—
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2819—22no—
26#2619—22no—
27#2615—22no—
28#2615C2214yes—
29#2563—21no—
30#2563C2212yes—
31#2541—21no—
32#2541C2212yes—
33#2537C2212yes—
34#2537C2212yes—
35#2361—22no—
36#2361—22no—
37#2354C2212yes—
38#2354C2212yes—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2212yes—
41#2326—22no—
42#2311C2212yes—
43C4×C4⋊Q8—21no—
44C4×#127—21no—
45#2285C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C4×#117C2211yescharacteristic
48#2077C2214yes—
49#2076C2214yes—
50#2076C2214yes—
51#2075C2214yes—
52#2034C2214yes—
53#2033C2214yes—
54#2033C2214yes—
55#2033C2214yes—
56#1990C2212yes—
57#1987C2211yescharacteristic
58#1987C2211yescharacteristic
59#1982C2211yescharacteristic
60#1982C2211yescharacteristic
61#1957C2212yes—
62#1954C2212yes—
63#1954C2212yes—
64#1954C2212yes—
65#1954C2212yes—
66#1948C2211yescharacteristic
67#1948C2211yescharacteristic
68#1947C2211yescharacteristic
69#1947C2211yescharacteristic
70#1947C2212yes—
71#1947C2212yes—
72#1875C2214yes—
73#1875C2212yes—
74#1875C2212yes—
75#1874C2214yes—
76#1874C2214yes—
77#1854C2212yes—
78#1854C2212yes—
79#1854C2214yes—
80#1852C2214yes—
81#1852C2214yes—
82#1845C2212yes—
83#1834C2212yes—
84#1825C2214yes—
85#1824C2214yes—
86#1802C2212yes—
87#1802C2212yes—
88#1780C2214yes—
89C4×#107C2212yes—
90C4×#106C2212yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×#41—21no—
94C22×#41C2312yes—
95#532—24no—
96#532—24no—
97#532—24no—
98#532—24no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127—22no—
102C2×#127D411yescharacteristic
103C2×#127C2311yescharacteristic
104C2×#124—21no—
105C2×#124D412yes—
106C2×#121—21no—
107C2×#121—22no—
108C2×#121—22no—
109C2×#121D412yes—
110C2×#121D412yes—
111C2×#121C2312yes—
112C2×#121C2314yes—
113C2×#121C2314yes—
114#520—44no—
115#520—44no—
116C2×C4×Q8—22no—
117C2×C4×D4—22no—
118C2×C4×D4—24no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C2×#117C2311yescharacteristic
127C2×#117C2314yes—
128C2×#117C2311yescharacteristic
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134#482—48no—
135#480—48no—
136#431—24no—
137#431—24no—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2312yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2311yescharacteristic
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C4×C8C2312yes—
161C2×C4×C8C2312yes—
162#394—24no—
163#394—24no—
164#393—24no—
165#393—24no—
166#392—24no—
167#392—24no—
168#392—24no—
169#392—24no—
170#391—22no—
171#390—22no—
172#390—22no—
173#390—22no—
174#390—22no—
175#390—22no—
176#389—22no—
177#389—22no—
178#388—24no—
179#388—24no—
180#386—24no—
181#385—24no—
182#385—24no—
183#384—21no—
184#384—21no—
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2311yescharacteristic
189#382—22no—
190#382—21no—
191#380—21no—
192#380—22no—
193#380—22no—
194#380C2314yes—
195#380C2314yes—
196#380C2314yes—
197#380C2314yes—
198#380C2312yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C22⋊C4—21no—
208C4×C22⋊C4—21no—
209#376—21no—
210C43—21no—
211#340C2314yes—
212#340C2314yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223C2×#41—24no—
224C2×#41—24no—
225C2×#41—42no—
226C2×#41—42no—
227C2×#41—42no—
228C2×#41—42no—
229C2×#41—44no—
230C2×#41—82no—
231C2×#41—82no—
232C22×Q8—21no—
233C22×Q8C2×D412yes—
234C22×D4—22no—
235C22×D4—24no—
236C22×D4—24no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2×D412yes—
240C22×D4C2×D412yes—
241C22×D4C2×D414yes—
242C22×D4C2×D412yes—
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4—24no—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2414yes—
250C23×C4C2411yescharacteristic
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258#127—24no—
259#127—22no—
260#127—22no—
261#127—84no—
262#124—28no—
263#124—44no—
264#121—28no—
265#121—28no—
266#121—24no—
267#121—24no—
268#121—44no—
269#121—44no—
270#121—44no—
271#121—44no—
272#121—44no—
273#121—84no—
274#121—84no—
275C4×Q8—48no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—48no—
283C4×D4—416no—
284C4×D4—44no—
285C4×D4—84no—
286C4×D4—84no—
287#117—22no—
288#117—22no—
289#117—28no—
290#117—41no—
291#117—41no—
292#117—44no—
293#117—82no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4C2×D414yes—
334C2×C22⋊C4C2×D414yes—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42#4114yes—
346C2×C42#4114yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—24no—
350#107—42no—
351#107—44no—
352#107—44no—
353#106—24no—
354#106—42no—
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358C4⋊C8—22no—
359#102—24no—
360#102—24no—
361#102—42no—
362#102—42no—
363#102—44no—
364#102—44no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367C22⋊C8—44no—
368C22⋊C8—44no—
369C22⋊C8—44no—
370C22⋊C8—44no—
371C4×C8—24no—
372C4×C8—24no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—24no—
382#95—24no—
383C24—22no—
384C24C22×D414yes—
385C24C22×D411yescharacteristic
386#41—48no—
387#41—48no—
388#41—48no—
389#41—48no—
390#41—816no—
391C2×Q8—24no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—81no—
396C2×Q8—81no—
397C2×D4—28no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4—84no—
428C2×D4—84no—
429C2×D4—84no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4#12314yes—
462C22×C4#12314yes—
463C22×C4C22×D414yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2511yescharacteristic
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4H(5,𝔽2)12yes—
512C4⋊C4H(5,𝔽2)12yes—
513C22⋊C4—24no—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—48no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—82no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C42—24no—
528C42—48no—
529C42—42no—
530C42—44no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—42no—
535C42#12314yes—
536C42#12314yes—
537C42C2×#4112yes—
538C42C2×#4112yes—
539C42C2×#4112yes—
540C42C2×#4112yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23C2×#12312yes—
562C23C23×D411yescharacteristic
563Q8—48no—
564Q8—84no—
565D4—48no—
566D4—416no—
567D4—48no—
568D4—48no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—84no—
574D4—88no—
575D4—84no—
576D4—168no—
577D4—168no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—88no—
622C2×C4—168no—
623C2×C4—164no—
624C2×C4C2×#12312yes—
625C2×C4C2×#12312yes—
626C2×C4C2×#12312yes—
627C2×C4C2×#12312yes—
628C2×C4C2×#12312yes—
629C2×C4#57612yes—
630C2×C4#57612yes—
631C2×C4#57712yes—
632C2×C4#57812yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4#58511yescharacteristic
638C2×C4#58511yescharacteristic
639C8—42no—
640C8—42no—
641C8—42no—
642C8—42no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22C22×#12311yescharacteristic
665C22#348411yescharacteristic
666C22#348511yescharacteristic
667C22C2×#57611yescharacteristic
668C22C2×#57611yescharacteristic
669C22C2×#57711yescharacteristic
670C22C2×#57811yescharacteristic
671C4—22no—
672C4—22no—
673C4—22no—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—81no—
678C4—82no—
679C4—81no—
680C4—82no—
681C4—84no—
682C4—82no—
683C4#309614yes—
684C4#309814yes—
685C4#346011yescharacteristic
686C4#346011yescharacteristic
687C4#350611yescharacteristic
688C4#350611yescharacteristic
689C2—42no—
690C2—84no—
691C2—162no—
692C2#3515211yescharacteristic
693C2C2×#309611yescharacteristic
694C2C2×#309811yescharacteristic
695C2#6166111yescharacteristic
696C2#6166711yescharacteristic
697C2#6178811yescharacteristic
698C2#6179111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37639C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34207C2semidirect
3. #32986C2semidirect
4. #29898C2semidirect
5. #26302C2semidirect
6. #26299C2semidirect
7. #26235C2semidirect
8. #26230C2semidirect
9. #25011C2semidirect
10. #25010C2semidirect
11. #24968C2semidirect
12. #24967C2semidirect
13. #21982C2semidirect
14. #21859C2semidirect
15. #21787C2semidirect
16. #21390C2semidirect
17. #21362C2semidirect

Maximal subgroups

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

subgroupindex
#376392
#342072
#329862
#298982
#263022
#262992
#262352
#262302
#250112
#250102
#249682
#249672
#219822
#218592
#217872
#213902
#213622