Finite Group Databasebeta

← #7507725

Subgroups — #7507725

Order 512.

1763 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35621C211yesmaximal, characteristic
2#35555C211yesmaximal, characteristic, complemented
3#32106C214yesmaximal, complemented
4#32105C214yesmaximal, complemented
5#32082C214yesmaximal, complemented
6#32081C214yesmaximal, complemented
7C2×#2096C211yesmaximal, characteristic, complemented
8#23610C211yesmaximal, characteristic, complemented
9#23512C211yesmaximal, characteristic, complemented
10#23510C211yesmaximal, characteristic, complemented
11#23510C211yesmaximal, characteristic, complemented
12#23504C211yesmaximal, characteristic
13#21262C211yesmaximal, characteristic, complemented
14#21200C211yesmaximal, characteristic
15#21094C211yesmaximal, characteristic, complemented
16#21068C211yesmaximal, characteristic, complemented
17#21022C211yesmaximal, characteristic, complemented
18#20882C211yesmaximal, characteristic, complemented
19#20882C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#532C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#417C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C2×#416C2211yescharacteristic
35C2×#416C2211yescharacteristic
36C2×#416C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C2×C4×M16C2211yescharacteristic
40#2570—22no—
41#2552—22no—
42#2547C2212yes—
43#2546C2212yes—
44#2545C2212yes—
45#2544C2212yes—
46#2507C2214yes—
47#2506C2214yes—
48#2473—24no—
49#2425—24no—
50#2417—22no—
51#2415—22no—
52#2317C2211yescharacteristic
53#2317C2211yescharacteristic
54#2316C2211yescharacteristic
55#2311C2211yescharacteristic
56#2296C2211yescharacteristic
57#2296C2211yescharacteristic
58C2×#391C2211yescharacteristic
59C2×#384C2211yescharacteristic
60#2274C2211yescharacteristic
61#2096C2214yes—
62#2096C2214yes—
63#2066C2214yes—
64#2065C2214yes—
65#2043C2212yes—
66#2043C2214yes—
67#2043C2214yes—
68#2043C2214yes—
69#2042C2214yes—
70#2042C2214yes—
71#2042C2214yes—
72#2042C2212yes—
73#2042C2212yes—
74#2041C2212yes—
75#1983C2214yes—
76#1981C2214yes—
77#1959C2212yes—
78#1959C2212yes—
79#1957C2212yes—
80#1956C2212yes—
81#1935C2214yes—
82#1933C2214yes—
83#1927C2214yes—
84#1926C2214yes—
85#1917C2214yes—
86#1916C2214yes—
87#1865C2214yes—
88#1865C2214yes—
89#1864C2214yes—
90#1864C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C22×#41C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C22×M16C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104#536—24no—
105#536—42no—
106#536—42no—
107#536—42no—
108#532—24no—
109#532—42no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×#127C2311yescharacteristic
114C2×#125—24no—
115C2×#124C2311yescharacteristic
116C2×#124C2311yescharacteristic
117C2×#124C2312yes—
118C2×#124C2312yes—
119C2×#123—22no—
120C2×#123—22no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—21no—
124C2×#121—22no—
125#520—42no—
126#520—42no—
127C2×C4×Q8C2312yes—
128C2×C4×D4—22no—
129C2×C4×D4—22no—
130C2×C4×D4C2312yes—
131C2×#117—21no—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4—22no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C2×#106—42no—
143C2×#106—42no—
144C2×#106C2314yes—
145C2×#106C2314yes—
146#424—42no—
147C2×C4⋊C8C2314yes—
148#419—24no—
149#419—42no—
150#417—42no—
151#417—42no—
152#417—42no—
153#417—42no—
154#416—24no—
155#416—24no—
156#416—24no—
157#416—42no—
158C2×#103—42no—
159C2×#103—42no—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2314yes—
171C2×#103C2312yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×#102C2312yes—
175C2×#102C2312yes—
176C2×#102C2312yes—
177C2×#102C2312yes—
178C2×#102C2312yes—
179C2×#102C2312yes—
180C4×M16—42no—
181C2×#97C2312yes—
182C2×C4×C8C2312yes—
183#400—24no—
184#398—24no—
185#398—24no—
186#398—24no—
187#398—24no—
188#397—24no—
189#396—24no—
190#394—24no—
191#391—22no—
192#391C2312yes—
193#391C2312yes—
194#391C2312yes—
195#390C2312yes—
196#389—22no—
197#389—24no—
198#389—24no—
199#389D412yes—
200#389D412yes—
201#389C2314yes—
202#389C2314yes—
203#389C2314yes—
204#389C2312yes—
205#389C2312yes—
206#388—24no—
207#387—24no—
208#387—24no—
209#386—22no—
210#386—24no—
211#385—22no—
212#385—22no—
213#385—22no—
214#385D414yes—
215#384—21no—
216#384—21no—
217#384—22no—
218#384C2312yes—
219#384C2312yes—
220#382—21no—
221#382—21no—
222#380—22no—
223#380—24no—
224#380—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4C2314yes—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C22⋊C4—22no—
231C2×#95—22no—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41#4114yes—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8C2412yes—
259C22×Q8C2411yescharacteristic
260C22×Q8C2411yescharacteristic
261C22×D4—21no—
262C22×D4—21no—
263C22×D4—22no—
264C22×D4C2411yescharacteristic
265C22×D4C2411yescharacteristic
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—22no—
273C23×C4C2×D412yes—
274C23×C4C2411yescharacteristic
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C4⋊Q8—22no—
292C4⋊Q8—24no—
293C4⋊Q8—41no—
294C4⋊Q8—42no—
295C4⋊Q8—42no—
296C4⋊Q8—42no—
297#127—22no—
298#127—41no—
299#125—88no—
300#124—22no—
301#124—22no—
302#124—24no—
303#124—41no—
304#124—41no—
305#124—42no—
306#124—42no—
307#124—42no—
308#123—48no—
309#123—48no—
310C22⋊Q8—48no—
311C22⋊Q8—44no—
312C22⋊Q8—44no—
313#121—48no—
314C4×Q8—42no—
315C4×Q8—42no—
316C4×D4—42no—
317C4×D4—42no—
318C4×D4—48no—
319C4×D4—48no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—44no—
328#117#4114yes—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4—84no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—24no—
379C2×C42C2×D412yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#106—44no—
385#106—44no—
386C4⋊C8—44no—
387#103—24no—
388#103—24no—
389#103—24no—
390#103—28no—
391#103—44no—
392#103—44no—
393#103—42no—
394#103—42no—
395#103—42no—
396#103—42no—
397#103—42no—
398#103—44no—
399#102—24no—
400#102—24no—
401#102—24no—
402#102—42no—
403#102—42no—
404#102—42no—
405#102—42no—
406#102—42no—
407#97—42no—
408C4×C8—42no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95C2×D414yes—
419#95C2×D414yes—
420C24—21no—
421C24—21no—
422C24C22×D411yescharacteristic
423#41—28no—
424#41—28no—
425#41—44no—
426#41—44no—
427#41—44no—
428#41—44no—
429#41—44no—
430#41—44no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8C2×#4112yes—
456C2×D4—24no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—44no—
475C2×D4—82no—
476C2×D4C2×#4112yes—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—84no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4C22×D412yes—
506C22×C4C22×D414yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4H(5,𝔽2)12yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C42−1+412yes—
517C22×C4C2511yescharacteristic
518M16—44no—
519M16—44no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—44no—
533C2×C8—44no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—28no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—48no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—48no—
558C4⋊C4—44no—
559C4⋊C4—84no—
560C4⋊C4—84no—
561C4⋊C4—84no—
562C4⋊C4—84no—
563C4⋊C4C2×#4114yes—
564C22⋊C4—24no—
565C22⋊C4—24no—
566C22⋊C4—24no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—48no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—42no—
574C22⋊C4—84no—
575C42—24no—
576C42—24no—
577C42—21no—
578C42—21no—
579C42—22no—
580C42—21no—
581C42—21no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—44no—
586C42—42no—
587C42—84no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—42no—
592C23—41no—
593C23—41no—
594C23—42no—
595C23—42no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—42no—
600C23#53512yes—
601C23#54612yes—
602C23#54812yes—
603C23C23×D411yescharacteristic
604Q8—22no—
605Q8—22no—
606Q8—41no—
607Q8—41no—
608Q8—42no—
609Q8—41no—
610Q8—42no—
611Q8—41no—
612Q8—42no—
613Q8—41no—
614Q8—42no—
615Q8—41no—
616Q8—84no—
617D4—22no—
618D4—22no—
619D4—48no—
620D4—41no—
621D4—41no—
622D4—41no—
623D4—41no—
624D4—41no—
625D4—41no—
626D4—88no—
627D4—84no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—24no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—22no—
642C2×C4—48no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—48no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4—164no—
666C2×C4—162no—
667C2×C4—162no—
668C2×C4—162no—
669C2×C4#53512yes—
670C2×C4#53914yes—
671C2×C4#54014yes—
672C2×C4#54612yes—
673C2×C4#54812yes—
674C2×C4#55314yes—
675C2×C4#56214yes—
676C2×C4#57714yes—
677C2×C4C22×#4111yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C2×C4C2×H(5,𝔽2)11yescharacteristic
680C2×C4C2×H(5,𝔽2)11yescharacteristic
681C2×C4C2×H(5,𝔽2)11yescharacteristic
682C2×C4C2×2−1+411yescharacteristic
683C8—42no—
684C8—42no—
685C8—42no—
686C8—42no—
687C22—24no—
688C22—22no—
689C22—22no—
690C22—24no—
691C22—44no—
692C22—42no—
693C22—81no—
694C22—81no—
695C22—82no—
696C22—82no—
697C22—81no—
698C22—81no—
699C22—82no—
700C22—81no—
701C22#271114yes—
702C22C2×#53511yescharacteristic
703C22C2×#54611yescharacteristic
704C22C2×#54811yescharacteristic
705C22C2×#57712yes—
706C22#358612yes—
707C4—24no—
708C4—44no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—44no—
713C4—42no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—81no—
719C4—82no—
720C4#271114yes—
721C4C2×#53911yescharacteristic
722C4C2×#54011yescharacteristic
723C4C2×#55311yescharacteristic
724C4C2×#56211yescharacteristic
725C2—24no—
726C2—82no—
727C2—161no—
728C2C2×#271111yescharacteristic
729C2#6082011yescharacteristic
730C2#6082611yescharacteristic
731C2#6119312yes—
732C2#6134612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #32106C2semidirect
4. #32105C2semidirect
5. #32082C2semidirect
6. #32081C2semidirect
7. C2×#2096C2semidirect
8. #23610C2semidirect
9. #23512C2semidirect
10. #23510C2semidirect
11. #23510C2semidirect
13. #21262C2semidirect
15. #21094C2semidirect
16. #21068C2semidirect
17. #21022C2semidirect
18. #20882C2semidirect
19. #20882C2semidirect

Maximal subgroups

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

subgroupindex
#356212
#355552
#321062
#321052
#320822
#320812
C2×#20962
#236102
#235122
#235102
#235042
#212622
#212002
#210942
#210682
#210222
#208822