Finite Group Databasebeta

← #6460141

Subgroups — #6460141

Order 512.

1822 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35674C211yesmaximal, characteristic, complemented
2#35558C211yesmaximal, characteristic
3#28900C214yesmaximal, complemented
4#28896C214yesmaximal, complemented
5#28883C214yesmaximal, complemented
6#28879C214yesmaximal, complemented
7#23656C211yesmaximal, characteristic, complemented
8#23613C211yesmaximal, characteristic, complemented
9#23541C211yesmaximal, characteristic
10C2×#2013C211yesmaximal, characteristic
11#21231C211yesmaximal, characteristic
12#21231C211yesmaximal, characteristic
13#21202C211yesmaximal, characteristic, complemented
14#21095C211yesmaximal, characteristic
15#21090C211yesmaximal, characteristic, complemented
16#21066C211yesmaximal, characteristic, complemented
17#20936C211yesmaximal, characteristic
18#20439C211yesmaximal, characteristic, complemented
19#20417C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×#405C2211yescharacteristic
36C2×#405C2211yescharacteristic
37C2×C4×M16C2211yescharacteristic
38#2513—22no—
39#2513C2214yes—
40#2512—22no—
41#2512C2214yes—
42#2484—24no—
43#2480—24no—
44#2446—22no—
45#2446—22no—
46#2446C2214yescomplemented
47#2446C2214yescomplemented
48#2317C2211yescharacteristic
49#2310C2211yescharacteristic
50#2310C2211yescharacteristic
51#2310C2211yescharacteristic
52#2309C2211yescharacteristic
53#2309C2211yescharacteristic
54#2296C2211yescharacteristic
55#2296C2211yescharacteristic
56C2×#390C2211yescharacteristic
57#2285C2211yescharacteristic
58#2274C2211yescharacteristic
59#2085C2212yescomplemented
60#2085C2212yescomplemented
61#2084C2212yescomplemented
62#2083C2212yescomplemented
63#2068C2214yescomplemented
64#2066C2214yescomplemented
65#2055C2214yes—
66#2055C2214yes—
67#2013C2212yes—
68#2013C2212yes—
69#2013C2212yes—
70#2013C2212yes—
71#1973C2214yes—
72#1973C2214yes—
73#1968C2214yes—
74#1968C2214yes—
75#1960C2214yescomplemented
76#1956C2214yescomplemented
77#1936C2214yes—
78#1935C2214yescomplemented
79#1934C2214yes—
80#1933C2214yescomplemented
81#1928C2214yescomplemented
82#1927C2214yescomplemented
83#1880C2214yes—
84#1880C2214yes—
85#1771C2212yescomplemented
86#1771C2212yescomplemented
87#1770C2212yescomplemented
88C8×C4⋊C4C2212yescomplemented
89#1763C2214yescomplemented
90C4×#103C2214yescomplemented
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—24no—
102C22×SD16—24no—
103C22×SD16—42no—
104C22×SD16—42no—
105C22×SD16—42no—
106C22×SD16C2312yes—
107C22×SD16C2312yes—
108C22×SD16C2312yes—
109C22×SD16C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114C2×C4⋊Q8—24no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8—42no—
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2312yes—
119C2×#126—24no—
120#519—24no—
121#519—42no—
122C2×C4×Q8—22no—
123C2×C4×Q8C2314yes—
124C2×#117—22no—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—22no—
134C22×C42C2311yescharacteristic
135#427—42no—
136#427—42no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141#424—24no—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144#419—24no—
145#419—24no—
146#419—42no—
147#419—42no—
148#418—24no—
149#418—42no—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162#405—24no—
163#405—42no—
164C4×M16—42no—
165C2×#97C2312yes—
166C2×#97C2312yes—
167C2×#97C2312yes—
168C2×C4×C8C2312yes—
169C2×C4×C8C2312yes—
170C2×C4×C8C2312yes—
171#400—24no—
172#400—24no—
173#398—24no—
174#398—24no—
175#397—24no—
176#397—24no—
177#397—24no—
178#396—24no—
179#391—21no—
180#391—21no—
181#391—22no—
182#391—21no—
183#391—21no—
184#391C2314yes—
185#391C2314yes—
186#391C2314yes—
187#390—21no—
188#390—22no—
189#390—21no—
190#390—22no—
191#390C2314yes—
192#390C2314yes—
193#390C2312yes—
194#390C2312yes—
195#389D412yes—
196#389D412yes—
197#389C2314yes—
198#389C2312yes—
199#389C2312yes—
200#388—24no—
201#388—24no—
202#388—24no—
203#387—24no—
204#387—24no—
205#387—24no—
206#387—24no—
207#385—22no—
208#385—22no—
209#385D414yes—
210#384—21no—
211#384C2312yes—
212#382—21no—
213#382C2312yes—
214#381—22no—
215#381—22no—
216#381—22no—
217#380—22no—
218#379—22no—
219#379—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2314yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C22⋊C4—22no—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2311yescharacteristic
236#336C2311yescharacteristic
237#336C2312yes—
238#336C2312yes—
239C25—21no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8#4114yes—
254C22×Q8C2412yes—
255C22×Q8C2412yes—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4C2412yes—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4C2×D412yes—
268C23×C4C2411yescharacteristic
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—42no—
276C2×SD16—42no—
277C2×SD16—42no—
278C2×SD16—42no—
279C2×SD16—42no—
280C2×SD16—42no—
281C2×SD16—416no—
282C2×SD16—416no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16#4114yes—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8#4114yes—
294C22×C8#4114yes—
295C22×C8C2412yes—
296C22×C8C2411yescharacteristic
297C22×C8C2411yescharacteristic
298C22×C8C2411yescharacteristic
299C22×C8C2411yescharacteristic
300C22×C8C2412yes—
301C4⋊Q8—24no—
302C4⋊Q8—24no—
303C4⋊Q8—42no—
304C4⋊Q8—42no—
305C4⋊Q8—48no—
306#126—88no—
307C4×Q8—28no—
308C4×Q8—44no—
309C4×Q8—48no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—48no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—48no—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D414yes—
339C2×C4⋊C4C2×D414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—84no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—24no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—24no—
366C2×C42C2×D412yes—
367C2×C42C2412yes—
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—42no—
372#107—42no—
373#106—42no—
374#106—42no—
375C4⋊C8—24no—
376C4⋊C8—24no—
377#103—24no—
378#103—24no—
379#103—24no—
380#103—24no—
381#103—28no—
382#103—28no—
383#103—44no—
384#103—44no—
385#102—28no—
386#102—44no—
387#97—24no—
388#97—42no—
389#97—42no—
390C4×C8—24no—
391C4×C8—42no—
392C4×C8—42no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95C2×D414yes—
406#95C2×D414yes—
407C24—22no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24—82no—
412C24—82no—
413C24C22×D411yescharacteristic
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—28no—
417C2×Q8—28no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—44no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8C2×#4114yes—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—44no—
452C2×D4—44no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—82no—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4C2511yescharacteristic
484SD16—48no—
485SD16—48no—
486SD16—48no—
487SD16—48no—
488M16—28no—
489M16—44no—
490C2×C8—28no—
491C2×C8—28no—
492C2×C8—22no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—22no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8C2×#4112yes—
506C2×C8C2×#4112yes—
507C2×C8C2×#4114yes—
508C2×C8C2×#4112yes—
509C2×C8C2×#4112yes—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—28no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—82no—
533C4⋊C4—84no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—48no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C42—24no—
547C42—22no—
548C42—22no—
549C42—21no—
550C42—21no—
551C42—22no—
552C42—21no—
553C42—22no—
554C42—21no—
555C42—48no—
556C42—42no—
557C42—42no—
558C42—42no—
559C42—42no—
560C42—84no—
561C23—22no—
562C23—22no—
563C23—44no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—88no—
569C23—84no—
570C23—164no—
571C23—162no—
572C23—162no—
573C23#53212yes—
574C23#54612yes—
575C23#55012yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—24no—
579Q8—42no—
580Q8—42no—
581Q8—48no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—84no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—84no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—84no—
627C2×C4—162no—
628C2×C4#52914yes—
629C2×C4#53212yes—
630C2×C4#53814yes—
631C2×C4#54612yes—
632C2×C4#55012yes—
633C2×C4#55114yes—
634C2×C4#55314yes—
635C2×C4#57714yes—
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C8—22no—
643C8—22no—
644C8—22no—
645C8—22no—
646C8—24no—
647C8—42no—
648C22—24no—
649C22—22no—
650C22—22no—
651C22—24no—
652C22—84no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—84no—
657C22—168no—
658C22#255414yes—
659C22C2×#53211yescharacteristic
660C22C2×#54611yescharacteristic
661C22C2×#55011yescharacteristic
662C22C2×#57712yes—
663C22#358612yes—
664C4—24no—
665C4—44no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#255414yes—
677C4C2×#52911yescharacteristic
678C4C2×#53811yescharacteristic
679C4C2×#55111yescharacteristic
680C4C2×#55311yescharacteristic
681C2—24no—
682C2—162no—
683C2C2×#255411yescharacteristic
684C2#6071211yescharacteristic
685C2#6072011yescharacteristic
686C2#6118512yes—
687C2#6142012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #35674C2semidirect
3. #28900C2semidirect
4. #28896C2semidirect
5. #28883C2semidirect
6. #28879C2semidirect
7. #23656C2semidirect
8. #23613C2semidirect
13. #21202C2semidirect
15. #21090C2semidirect
16. #21066C2semidirect
18. #20439C2semidirect
19. #20417C2semidirect
46. #2446C22semidirect
47. #2446C22semidirect
59. #2085C22semidirect
60. #2085C22semidirect
61. #2084C22semidirect
62. #2083C22semidirect
63. #2068C22semidirect
64. #2066C22semidirect
75. #1960C22semidirect
76. #1956C22semidirect
78. #1935C22semidirect
80. #1933C22semidirect
81. #1928C22semidirect
82. #1927C22semidirect
85. #1771C22semidirect
86. #1771C22semidirect
87. #1770C22semidirect
88. C8×C4⋊C4C22semidirect
89. #1763C22semidirect
90. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#356742
#355582
#289002
#288962
#288832
#288792
#236562
#236132
#235412
C2×#20132
#212312
#212022
#210952
#210902
#210662
#209362
#204392
#204172