Finite Group Databasebeta

← #6458574

Subgroups — #6458574

Order 512.

1803 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35667C211yesmaximal, characteristic
2#35555C211yesmaximal, characteristic, complemented
3#27765C214yesmaximal, complemented
4#27762C214yesmaximal, complemented
5#27752C214yesmaximal, complemented
6#27749C214yesmaximal, complemented
7#23633C211yesmaximal, characteristic, complemented
8#23630C211yesmaximal, characteristic
9#23512C211yesmaximal, characteristic
10#23507C211yesmaximal, characteristic, complemented
11#21262C211yesmaximal, characteristic
12#21205C211yesmaximal, characteristic
13#21205C211yesmaximal, characteristic
14C2×#1956C211yesmaximal, characteristic, complemented
15#21088C211yesmaximal, characteristic, complemented
16#21083C211yesmaximal, characteristic
17#20651C211yesmaximal, characteristic, complemented
18#20435C211yesmaximal, characteristic, complemented
19#20417C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#417C2211yescharacteristic
26C2×#417C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C2×#405C2211yescharacteristic
38C2×#405C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2570C2214yes—
41#2552C2214yes—
42#2546C2214yes—
43#2545C2214yes—
44#2507—22no—
45#2506—22no—
46#2473—24no—
47#2429—22no—
48#2429—22no—
49#2413—24no—
50#2317C2211yescharacteristic
51#2317C2211yescharacteristic
52#2316C2211yescharacteristic
53#2310C2211yescharacteristic
54#2297C2211yescharacteristic
55#2296C2211yescharacteristic
56#2296C2211yescharacteristic
57C2×#384C2211yescharacteristic
58#2274C2211yescharacteristic
59#2074C2214yes—
60#2072C2214yes—
61#2072C2214yes—
62#2071C2214yes—
63#2043C2214yes—
64#2043C2214yes—
65#2043C2214yes—
66#2042C2214yes—
67#1983C2214yes—
68#1981C2214yes—
69#1960C2212yes—
70#1960C2212yes—
71#1959C2212yes—
72#1959C2212yes—
73#1958C2212yes—
74#1958C2212yes—
75#1956C2212yes—
76#1956C2212yes—
77#1956C2214yes—
78#1956C2214yes—
79#1935C2214yes—
80#1934C2212yes—
81#1934C2212yes—
82#1933C2212yes—
83#1933C2212yes—
84#1933C2214yes—
85#1805C2214yes—
86#1803C2214yes—
87#1771C2214yes—
88C8×C4⋊C4C2214yes—
89#1763C2214yes—
90C4×#103C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C22×M16C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104#536—24no—
105#536—24no—
106#536—24no—
107#536—42no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—42no—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2314yes—
114C2×C4⋊Q8C2312yes—
115C2×#125—24no—
116C2×#124C2312yes—
117C2×#124C2312yes—
118C2×#123—22no—
119C2×#123—22no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122#519—24no—
123#519—24no—
124C2×C4×Q8—22no—
125C2×C4×Q8C2314yes—
126C2×#117—22no—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—22no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#424—24no—
138C2×C4⋊C8C2314yes—
139#419—24no—
140#419—42no—
141#419—42no—
142#419—42no—
143#417—42no—
144#417—42no—
145#416—24no—
146#416—42no—
147C2×#103—24no—
148C2×#103—24no—
149C2×#103—42no—
150C2×#103—42no—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2311yescharacteristic
156C2×#103C2311yescharacteristic
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#102C2311yescharacteristic
169C2×#102C2311yescharacteristic
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173#405—24no—
174#405—42no—
175C2×#97C2312yes—
176C2×#97C2312yes—
177C2×C4×C8—42no—
178C2×C4×C8C2312yes—
179C2×C4×C8C2312yes—
180C2×C4×C8C2312yes—
181C2×C4×C8C2312yes—
182#400—24no—
183#398—24no—
184#398—24no—
185#398—24no—
186#398—24no—
187#398—24no—
188#396—24no—
189#396—24no—
190#391—22no—
191#391C2314yes—
192#391C2312yes—
193#390C2312yes—
194#389—22no—
195#389—22no—
196#389—22no—
197#389—24no—
198#389—24no—
199#389—24no—
200#389D412yes—
201#389D412yes—
202#389C2312yes—
203#389C2314yes—
204#389C2312yes—
205#388—24no—
206#388—24no—
207#387—24no—
208#387—24no—
209#386—22no—
210#385—22no—
211#385—22no—
212#385—22no—
213#385D414yes—
214#384—21no—
215#384—21no—
216#384C2314yes—
217#384C2312yes—
218#384C2312yes—
219#382C2312yes—
220#382C2312yes—
221#381—22no—
222#380—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4C2314yes—
227C4×C4⋊C4C2314yes—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—21no—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#336C2314yes—
238#336C2314yes—
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×Q8#4114yes—
259C22×Q8C2412yes—
260C22×Q8C2411yescharacteristic
261C22×Q8C2412yes—
262C22×D4—21no—
263C22×D4C2411yescharacteristic
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8#4114yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2411yescharacteristic
287C22×C8C2411yescharacteristic
288C4⋊Q8—28no—
289C4⋊Q8—24no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—44no—
293C4⋊Q8—42no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296#125—88no—
297#124—24no—
298#124—24no—
299#124—24no—
300#124—42no—
301#123—48no—
302#123—48no—
303C22⋊Q8—48no—
304C22⋊Q8—48no—
305C4×Q8—28no—
306C4×Q8—28no—
307C4×Q8—48no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—48no—
317#117#4114yes—
318#117#4114yes—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—48no—
338C2×C4⋊C4—84no—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D414yes—
341C2×C4⋊C4C2×D414yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
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—24no—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2412yes—
369C2×C42C2412yes—
370C4⋊C8—28no—
371#103—28no—
372#103—24no—
373#103—24no—
374#103—24no—
375#103—24no—
376#103—22no—
377#103—22no—
378#103—44no—
379#103—44no—
380#103—42no—
381#103—42no—
382#103—42no—
383#103—44no—
384#103—42no—
385#103—42no—
386#103—42no—
387#103—42no—
388#102—22no—
389#102—22no—
390#102—42no—
391#102—42no—
392#102—42no—
393#97—24no—
394#97—42no—
395C4×C8—24no—
396C4×C8—42no—
397C4×C8—42no—
398C4×C8—42no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95C2×D414yes—
410#95C2×D414yes—
411C24—21no—
412C24C22×D411yescharacteristic
413#41—44no—
414#41—44no—
415#41—44no—
416#41—44no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8C2×#4114yes—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—82no—
482C22×C4—82no—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C4C2511yescharacteristic
495M16—44no—
496M16—44no—
497C2×C8—28no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—44no—
512C2×C8C2×#4112yes—
513C2×C8C2×#4112yes—
514C4⋊C4—24no—
515C4⋊C4—28no—
516C4⋊C4—28no—
517C4⋊C4—21no—
518C4⋊C4—21no—
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—21no—
528C4⋊C4—21no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—48no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4#12418yes—
544C22⋊C4—24no—
545C22⋊C4—48no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4#12418yes—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—42no—
560C42—48no—
561C42—42no—
562C42—42no—
563C42—84no—
564C42C2×#4114yes—
565C42C2×#4114yes—
566C23—22no—
567C23—22no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23C2×#12412yes—
574C23#54612yes—
575C23#55012yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—24no—
579Q8—42no—
580Q8—42no—
581Q8—48no—
582Q8—42no—
583Q8—41no—
584Q8—42no—
585Q8—41no—
586Q8—42no—
587Q8—42no—
588Q8—41no—
589Q8—41no—
590Q8—84no—
591D4—41no—
592D4—41no—
593D4—41no—
594D4—41no—
595D4—84no—
596C2×C4—216no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—162no—
633C2×C4—162no—
634C2×C4C2×#12418yes—
635C2×C4C2×#12412yes—
636C2×C4#52914yes—
637C2×C4#54014yes—
638C2×C4#54612yes—
639C2×C4#55012yes—
640C2×C4#55314yes—
641C2×C4#55314yes—
642C2×C4#57714yes—
643C2×C4#57814yes—
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C8—22no—
651C8—22no—
652C8—42no—
653C8—42no—
654C8—42no—
655C22—24no—
656C22—22no—
657C22—22no—
658C22—24no—
659C22—82no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—82no—
664C22#249014yes—
665C22C22×#12411yescharacteristic
666C22C2×#54611yescharacteristic
667C22C2×#55011yescharacteristic
668C22C2×#57712yes—
669C22C2×#57812yes—
670C4—28no—
671C4—24no—
672C4—24no—
673C4—44no—
674C4—44no—
675C4—82no—
676C4—81no—
677C4—82no—
678C4—81no—
679C4—82no—
680C4—82no—
681C4—81no—
682C4—82no—
683C4#249014yes—
684C4C2×#52911yescharacteristic
685C4C2×#54011yescharacteristic
686C4C2×#55311yescharacteristic
687C4C2×#55311yescharacteristic
688C2—24no—
689C2—161no—
690C2C2×#249011yescharacteristic
691C2#6062312yes—
692C2#6116812yes—
693C2#6141011yescharacteristic
694C2#6141111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
2. #35555C2semidirect
3. #27765C2semidirect
4. #27762C2semidirect
5. #27752C2semidirect
6. #27749C2semidirect
7. #23633C2semidirect
10. #23507C2semidirect
14. C2×#1956C2semidirect
15. #21088C2semidirect
17. #20651C2semidirect
18. #20435C2semidirect
19. #20417C2semidirect

Maximal subgroups

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

subgroupindex
#356672
#355552
#277652
#277622
#277522
#277492
#236332
#236302
#235122
#235072
#212622
#212052
C2×#19562
#210882
#210832
#206512
#204352
#204172