Finite Group Databasebeta

← #6447322

Subgroups — #6447322

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35627C211yesmaximal, characteristic, complemented
2#34269C211yesmaximal, characteristic
3#26545C214yesmaximal, complemented
4#26538C214yesmaximal, complemented
5#26537C214yesmaximal, complemented
6#26529C214yesmaximal, complemented
7#23738C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic, complemented
9#23618C211yesmaximal, characteristic, complemented
10#23611C211yesmaximal, characteristic, complemented
11#21263C211yesmaximal, characteristic
12#21206C211yesmaximal, characteristic
13#21173C211yesmaximal, characteristic, complemented
14#21148C211yesmaximal, characteristic, complemented
15#20879C211yesmaximal, characteristic, complemented
16#20872C211yesmaximal, characteristic, complemented
17#20650C211yesmaximal, characteristic, complemented
18#20649C211yesmaximal, characteristic, complemented
19#20454C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C2×#518C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C2×#428C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#417C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C2×#416C2211yescharacteristic
35C2×#416C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C2×#405C2211yescharacteristic
39#2571C2214yescomplemented
40#2570C2214yescomplemented
41#2509—22no—
42#2508—22no—
43#2505—24no—
44#2367C2214yes—
45#2366C2214yes—
46#2358—24no—
47#2324—22no—
48#2317C2211yescharacteristic
49#2317C2211yescharacteristic
50#2316C2211yescharacteristic
51#2311C2211yescharacteristic
52C4×#121—22no—
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2274C2211yescharacteristic
58#2274C2211yescharacteristic
59#2097C2212yescomplemented
60#2096C2212yescomplemented
61#2094C2212yescomplemented
62#2092C2214yes—
63#2092C2214yes—
64#2092C2212yescomplemented
65#2068C2214yescomplemented
66#2067C2214yes—
67#2067C2214yescomplemented
68#2065C2214yes—
69#1983C2214yes—
70#1982C2214yes—
71#1961C2212yes—
72#1959C2212yes—
73#1958C2212yes—
74#1957C2212yes—
75#1949C2214yescomplemented
76#1947C2214yescomplemented
77#1940C2214yescomplemented
78#1939C2214yescomplemented
79#1865C2212yescomplemented
80#1864C2214yes—
81#1864C2212yescomplemented
82#1863C2212yescomplemented
83#1862C2214yes—
84#1862C2212yescomplemented
85#1805C2214yescomplemented
86#1804C2214yes—
87#1804C2214yescomplemented
88#1802C2214yes—
89#1776C2214yescomplemented
90C4×#106C2214yescomplemented
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#535—24no—
104#535Q818yes—
105C2×#127C2312yes—
106C2×#124C2312yes—
107C2×#121—22no—
108C2×#120—22no—
109#518—24no—
110#518—42no—
111C4×#41—24no—
112C4×#41—42no—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—24no—
117C2×C4×D4C2314yes—
118C2×C4×D4C2312yes—
119C2×#117—21no—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—22no—
129C22×C22⋊C4—22no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132#428—42no—
133#427—24no—
134C2×#107—42no—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#106—24no—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143#424—24no—
144#424—42no—
145#423—42no—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149#418—42no—
150#418—42no—
151#417—42no—
152#417—42no—
153#416—24no—
154#416—42no—
155C2×#103C2311yescharacteristic
156C2×#103C2311yescharacteristic
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#102—24no—
161C2×#102—42no—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2311yescharacteristic
165C2×#102C2311yescharacteristic
166C2×#102C2312yes—
167C2×#102C2314yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#102C2314yes—
172C2×#102C2312yes—
173#405—24no—
174C2×#97C2312yes—
175C2×C4×C8C2312yes—
176#400—24no—
177#400—24no—
178#400—24no—
179#398—24no—
180#391C2312yes—
181#390—24no—
182#390—21no—
183#390—21no—
184#390C2312yes—
185#390C2314yes—
186#389—24no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389D412yes—
191#389D412yes—
192#389C2312yes—
193#389C2314yes—
194#389C2312yes—
195#388—24no—
196#388—24no—
197#387—24no—
198#387—24no—
199#386—22no—
200#386—24no—
201#386—24no—
202#385—22no—
203#385—24no—
204#385—22no—
205#385—22no—
206#385D414yes—
207#384C2312yes—
208#384C2312yes—
209#382—24no—
210#382—21no—
211#382—21no—
212#382C2312yes—
213#382C2314yes—
214#382C2312yes—
215#380—22no—
216#380—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4C2314yes—
222C4×C4⋊C4C2314yes—
223C4×C4⋊C4C2314yes—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—24no—
227C2×#95—22no—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#336C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237C25—21no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41C2×Q814yes—
244C22×Q8C2411yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—24no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4C2×Q814yes—
254C22×D4C2412yes—
255C22×D4C2411yescharacteristic
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—24no—
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8#4114yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2411yescharacteristic
280C22×C8C2411yescharacteristic
281#127—24no—
282#127C2×Q818yes—
283#124—24no—
284#124C2×Q818yes—
285#121—216no—
286#120—216no—
287C4×Q8—24no—
288C4×Q8—42no—
289C4×D4—24no—
290C4×D4—28no—
291C4×D4—42no—
292C4×D4—44no—
293C4×D4—48no—
294C4×D4—48no—
295C4×D4—88no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—44no—
304#117C2×Q814yes—
305#117#4114yes—
306#117#4114yes—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—48no—
343C2×C22⋊C4—84no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—44no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360#107—24no—
361#107—44no—
362#107—42no—
363#106—24no—
364#106—24no—
365#106—24no—
366#106—42no—
367C4⋊C8—28no—
368C4⋊C8—44no—
369C4⋊C8—44no—
370#103—22no—
371#103—22no—
372#103—42no—
373#103—42no—
374#103—42no—
375#102—24no—
376#102—24no—
377#102—22no—
378#102—22no—
379#102—44no—
380#102—42no—
381#102—42no—
382#102—42no—
383#102—44no—
384#102—42no—
385#102—42no—
386#97—24no—
387C4×C8—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95C2×D414yes—
399#95C2×D414yes—
400C24—22no—
401C24—21no—
402C24—24no—
403C24—42no—
404C24—42no—
405C24—82no—
406C24—82no—
407C24C22×D411yescharacteristic
408#41—28no—
409#41—28no—
410#41—44no—
411#41—44no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8C22×Q812yes—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—216no—
421C2×D4—28no—
422C2×D4—28no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4C22×Q814yes—
442C2×D4C22×Q812yes—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—82no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×Q812yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4C2511yescharacteristic
481M16—44no—
482M16—44no—
483C2×C8—28no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—44no—
498C2×C8C2×#4112yes—
499C2×C8C2×#4112yes—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—84no—
525C4⋊C4C22⋊Q818yes—
526C4⋊C4C2×#4114yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—216no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4C22⋊Q818yes—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—44no—
546C42—42no—
547C42—42no—
548C42—84no—
549C42—84no—
550C42C22×Q814yes—
551C42C2×#4114yes—
552C23—22no—
553C23—22no—
554C23—24no—
555C23—22no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—41no—
561C23—42no—
562C23—41no—
563C23—42no—
564C23—48no—
565C23—88no—
566C23—164no—
567C23—162no—
568C23—162no—
569C23C2×C22⋊Q812yes—
570C23#54612yes—
571C23#55012yes—
572C23C23×D411yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—41no—
576Q8—41no—
577D4—24no—
578D4—24no—
579D4—22no—
580D4—22no—
581D4—48no—
582D4—42no—
583D4—41no—
584D4—42no—
585D4—41no—
586D4—48no—
587D4—816no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4—168no—
622C2×C4—162no—
623C2×C4—162no—
624C2×C4—162no—
625C2×C4C2×C22⋊Q818yes—
626C2×C4C2×C22⋊Q812yes—
627C2×C4#52914yes—
628C2×C4#54014yes—
629C2×C4#54612yes—
630C2×C4#55012yes—
631C2×C4#55414yes—
632C2×C4#55414yes—
633C2×C4C23×Q811yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C8—22no—
640C8—22no—
641C8—42no—
642C8—42no—
643C8—42no—
644C22—24no—
645C22—24no—
646C22—22no—
647C22—22no—
648C22—44no—
649C22—48no—
650C22—42no—
651C22—44no—
652C22—82no—
653C22—84no—
654C22—81no—
655C22—82no—
656C22—81no—
657C22—82no—
658C22—168no—
659C22#245414yes—
660C22C22×C22⋊Q811yescharacteristic
661C22C2×#54611yescharacteristic
662C22C2×#55011yescharacteristic
663C22#358412yes—
664C22#358512yes—
665C4—28no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—44no—
670C4—42no—
671C4—82no—
672C4—81no—
673C4—82no—
674C4—81no—
675C4—82no—
676C4—82no—
677C4#245414yes—
678C4C2×#52911yescharacteristic
679C4C2×#54011yescharacteristic
680C4C2×#55411yescharacteristic
681C4C2×#55411yescharacteristic
682C2—24no—
683C2—82no—
684C2—84no—
685C2C2×#245411yescharacteristic
686C2#6049612yes—
687C2#6119612yes—
688C2#6141211yescharacteristic
689C2#6141711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35627C2semidirect
3. #26545C2semidirect
4. #26538C2semidirect
5. #26537C2semidirect
6. #26529C2semidirect
7. #23738C2semidirect
8. C2×#2092C2semidirect
9. #23618C2semidirect
10. #23611C2semidirect
13. #21173C2semidirect
14. #21148C2semidirect
15. #20879C2semidirect
16. #20872C2semidirect
17. #20650C2semidirect
18. #20649C2semidirect
19. #20454C2semidirect
39. #2571C22semidirect
40. #2570C22semidirect
59. #2097C22semidirect
60. #2096C22semidirect
61. #2094C22semidirect
64. #2092C22semidirect
65. #2068C22semidirect
67. #2067C22semidirect
75. #1949C22semidirect
76. #1947C22semidirect
77. #1940C22semidirect
78. #1939C22semidirect
79. #1865C22semidirect
81. #1864C22semidirect
82. #1863C22semidirect
84. #1862C22semidirect
85. #1805C22semidirect
87. #1804C22semidirect
89. #1776C22semidirect
90. C4×#106C22semidirect

Maximal subgroups

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

subgroupindex
#356272
#342692
#265452
#265382
#265372
#265292
#237382
C2×#20922
#236182
#236112
#212632
#212062
#211732
#211482
#208792
#208722
#206502
#206492
#204542