Finite Group Databasebeta

← #6460463

Subgroups — #6460463

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35667C211yesmaximal, characteristic
2#35557C211yesmaximal, characteristic
3#29070C214yesmaximal, complemented
4#29066C214yesmaximal, complemented
5#29045C214yesmaximal, complemented
6#29041C214yesmaximal, complemented
7#23745C211yesmaximal, characteristic, complemented
8#23507C211yesmaximal, characteristic, complemented
9#23487C211yesmaximal, characteristic
10#23487C211yesmaximal, characteristic
11C2×#1956C211yesmaximal, characteristic, complemented
12#21151C211yesmaximal, characteristic, complemented
13#21088C211yesmaximal, characteristic, complemented
14#21066C211yesmaximal, characteristic, complemented
15#20974C211yesmaximal, characteristic
16#20974C211yesmaximal, characteristic
17#20880C211yesmaximal, characteristic, complemented
18#20417C211yesmaximal, characteristic, complemented
19#20383C211yesmaximal, characteristic
20C22×#131C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C22×C4⋊Q8C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×#405C2211yescharacteristic
37C22×C4×C8C2211yescharacteristic
38#2546—22no—
39#2545—22no—
40#2512—22no—
41#2511—22no—
42#2497C2214yes—
43#2496C2214yes—
44#2481—24no—
45#2429C2214yes—
46#2429C2214yes—
47#2413—24no—
48#2316C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2211yescharacteristic
51#2310C2211yescharacteristic
52#2297C2211yescharacteristic
53#2296C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#384C2211yescharacteristic
56#2274C2211yescharacteristic
57#2100C2214yes—
58#2099C2214yes—
59#2043C2214yes—
60#2043C2214yes—
61#2036C2214yes—
62#2036C2214yes—
63#2035C2214yes—
64#2035C2214yes—
65#1956C2212yes—
66#1956C2212yes—
67#1956C2212yes—
68#1956C2212yes—
69#1942C2212yes—
70#1942C2212yes—
71#1941C2212yes—
72#1941C2212yes—
73#1935C2214yes—
74#1933C2214yes—
75#1928C2214yes—
76#1927C2214yes—
77#1894C2214yes—
78#1894C2214yes—
79#1892C2214yes—
80#1892C2214yes—
81#1865C2214yes—
82#1863C2214yes—
83#1763C2214yes—
84C4×#103C2214yes—
85#1753C2212yes—
86#1753C2212yes—
87C8×C22⋊C4C2212yes—
88C8×C22⋊C4C2212yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C2×#131—24no—
101C2×#131—42no—
102C2×#131—42no—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2312yes—
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2312yes—
109#536—24no—
110#536—24no—
111#536—42no—
112#536—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—24no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8—42no—
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2312yes—
120C2×#126—24no—
121C2×#124—21no—
122C2×#124—21no—
123C2×#123—22no—
124C2×#123—22no—
125C2×C22⋊Q8C2314yes—
126C2×C22⋊Q8C2314yes—
127#519—24no—
128#519—42no—
129C2×C4×Q8—22no—
130C2×C4×Q8C2314yes—
131C2×#117—22no—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C22×C4⋊C4—22no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#427—24no—
143#427—42no—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#106C2312yes—
147C2×#106C2312yes—
148#419—24no—
149#419—42no—
150#419—42no—
151#419—42no—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168#408—24no—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171#405—42no—
172C2×#97C2312yes—
173C2×C4×C8—42no—
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176C2×C4×C8C2311yescharacteristic
177C2×C4×C8C2311yescharacteristic
178C2×C4×C8C2312yes—
179#400—24no—
180#400—24no—
181#398—24no—
182#398—24no—
183#397—24no—
184#396—24no—
185#396—24no—
186#396—24no—
187#391—21no—
188#391—21no—
189#391—22no—
190#391D412yes—
191#391C2314yes—
192#390—21no—
193#390—21no—
194#390D412yes—
195#389—22no—
196#389—24no—
197#389—24no—
198#389C2314yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#388—24no—
204#388—24no—
205#388—24no—
206#387—24no—
207#387—24no—
208#386—22no—
209#386—22no—
210#386D414yes—
211#385—22no—
212#385—22no—
213#384—22no—
214#384C2312yes—
215#384C2312yes—
216#381—22no—
217#380—22no—
218#380C2314yes—
219#380C2314yes—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2314yes—
224C4×C22⋊C4—22no—
225C4×C22⋊C4C2312yes—
226C4×C22⋊C4C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2312yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8C2×D411yescharacteristic
259C22×Q8#4114yes—
260C22×Q8C2412yes—
261C22×Q8C2412yes—
262C22×D4C2×D412yes—
263C22×D4C2×D411yescharacteristic
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4C2×D412yes—
269C23×C4C2412yes—
270C23×C4C2411yescharacteristic
271#131—416no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—24no—
279C2×M16—42no—
280C2×M16—42no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—24no—
286C22×C8—42no—
287C22×C8—42no—
288C22×C8#4114yes—
289C22×C8#4114yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2411yescharacteristic
293C22×C8C2411yescharacteristic
294C22×C8C2411yescharacteristic
295C22×C8C2411yescharacteristic
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—42no—
299C4⋊Q8—42no—
300C4⋊Q8—48no—
301C4⋊Q8—44no—
302C4⋊Q8—44no—
303#126—88no—
304#124—44no—
305#124—44no—
306#123—48no—
307#123—48no—
308C22⋊Q8—28no—
309C22⋊Q8—28no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
312C4×Q8—28no—
313C4×Q8—44no—
314C4×Q8—48no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—48no—
325#117#4114yes—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—48no—
345C2×C4⋊C4—84no—
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D414yes—
363C2×C22⋊C4C2×D414yes—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—24no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—24no—
372C2×C42C2×D412yes—
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—24no—
377#107—42no—
378#106—24no—
379#106—42no—
380#103—28no—
381#103—24no—
382#103—24no—
383#103—24no—
384#103—24no—
385#103—24no—
386#103—24no—
387#103—24no—
388#103—24no—
389#103—44no—
390#103—44no—
391#103—44no—
392C22⋊C8—24no—
393C22⋊C8—24no—
394#97—42no—
395C4×C8—41no—
396C4×C8—41no—
397C4×C8—41no—
398C4×C8—41no—
399C4×C8—42no—
400#95—24no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95C2×D414yes—
411#95C2×D414yes—
412C24C22×D412yes—
413C24C22×D411yescharacteristic
414#41—44no—
415#41—44no—
416#41—44no—
417#41—44no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—28no—
424C2×Q8—28no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8C2×#4114yes—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4C22×D412yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4C2511yescharacteristic
492M16—48no—
493M16—44no—
494M16—44no—
495C2×C8—28no—
496C2×C8—28no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—48no—
512C2×C8C2×#4112yes—
513C2×C8C2×#4112yes—
514C2×C8C2×#4112yes—
515C2×C8C2×#4112yes—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—48no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4C2×#4114yes—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C42—24no—
555C42—24no—
556C42—21no—
557C42—21no—
558C42—22no—
559C42—21no—
560C42—21no—
561C42—22no—
562C42—48no—
563C42—42no—
564C42—42no—
565C42—42no—
566C42—84no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23—24no—
574C23#53212yes—
575C23#54612yes—
576C23#54612yes—
577C23C23×D411yescharacteristic
578Q8—24no—
579Q8—24no—
580Q8—42no—
581Q8—42no—
582Q8—48no—
583Q8—44no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—84no—
589D4—48no—
590D4—44no—
591C2×C4—216no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—48no—
608C2×C4—42no—
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—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—84no—
630C2×C4—162no—
631C2×C4—162no—
632C2×C4#52914yes—
633C2×C4#53212yes—
634C2×C4#53814yes—
635C2×C4#54612yes—
636C2×C4#54612yes—
637C2×C4#55314yes—
638C2×C4#55314yes—
639C2×C4#57714yes—
640C2×C4#57814yes—
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C8—22no—
648C8—22no—
649C8—22no—
650C8—22no—
651C8—42no—
652C8—42no—
653C22—24no—
654C22—22no—
655C22—22no—
656C22—24no—
657C22—42no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22#255914yes—
663C22C2×#53211yescharacteristic
664C22C2×#54611yescharacteristic
665C22C2×#54611yescharacteristic
666C22C2×#57712yes—
667C22C2×#57812yes—
668C4—24no—
669C4—24no—
670C4—42no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#255914yes—
681C4C2×#52911yescharacteristic
682C4C2×#53811yescharacteristic
683C4C2×#55311yescharacteristic
684C4C2×#55311yescharacteristic
685C2—24no—
686C2—82no—
687C2C2×#255911yescharacteristic
688C2#6071211yescharacteristic
689C2#6071511yescharacteristic
690C2#6116812yes—
691C2#6121712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29070C2semidirect
4. #29066C2semidirect
5. #29045C2semidirect
6. #29041C2semidirect
7. #23745C2semidirect
8. #23507C2semidirect
11. C2×#1956C2semidirect
12. #21151C2semidirect
13. #21088C2semidirect
14. #21066C2semidirect
17. #20880C2semidirect
18. #20417C2semidirect

Maximal subgroups

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

subgroupindex
#356672
#355572
#290702
#290662
#290452
#290412
#237452
#235072
#234872
C2×#19562
#211512
#210882
#210662
#209742
#208802
#204172
#203832