Finite Group Databasebeta

← #7507693

Subgroups — #7507693

Order 512.

1773 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35627C211yesmaximal, characteristic, complemented
2#35617C211yesmaximal, characteristic
3#32089C214yesmaximal, complemented
4#32088C214yesmaximal, complemented
5#32087C214yesmaximal, complemented
6#32086C214yesmaximal, complemented
7#23733C211yesmaximal, characteristic, complemented
8#23615C211yesmaximal, characteristic, complemented
9#23511C211yesmaximal, characteristic
10#23510C211yesmaximal, characteristic, complemented
11#23510C211yesmaximal, characteristic, complemented
12#23506C211yesmaximal, characteristic, complemented
13#21263C211yesmaximal, characteristic
14C2×#1958C211yesmaximal, characteristic, complemented
15#21083C211yesmaximal, characteristic, complemented
16#21064C211yesmaximal, characteristic, complemented
17#21020C211yesmaximal, characteristic, complemented
18#20876C211yesmaximal, characteristic, complemented
19#20876C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#535C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C2×#416C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2571—22no—
41#2570—22no—
42#2547C2212yes—
43#2547C2212yes—
44#2544C2212yes—
45#2544C2212yes—
46#2509C2214yescomplemented
47#2508C2214yescomplemented
48#2505—24no—
49#2427—22no—
50#2427—22no—
51#2412—24no—
52#2317C2211yescharacteristic
53#2317C2211yescharacteristic
54#2316C2211yescharacteristic
55#2310C2211yescharacteristic
56#2297C2211yescharacteristic
57#2296C2211yescharacteristic
58#2296C2211yescharacteristic
59C2×#382C2211yescharacteristic
60#2274C2211yescharacteristic
61#2097C2212yescomplemented
62#2097C2212yescomplemented
63#2092C2212yescomplemented
64#2092C2212yescomplemented
65#2068C2214yescomplemented
66#2066C2214yescomplemented
67#2043C2214yescomplemented
68#2043C2214yescomplemented
69#2042C2214yes—
70#2042C2214yes—
71#2042C2214yes—
72#2042C2214yescomplemented
73#2042C2214yescomplemented
74#2041C2214yes—
75#1983C2214yes—
76#1982C2214yes—
77#1958C2214yes—
78#1958C2214yes—
79#1934C2212yescomplemented
80#1934C2212yescomplemented
81#1933C2212yescomplemented
82#1933C2212yescomplemented
83#1928C2214yescomplemented
84#1927C2214yescomplemented
85#1916C2214yescomplemented
86#1916C2214yescomplemented
87#1864C2214yes—
88#1864C2214yes—
89#1862C2214yes—
90#1862C2214yes—
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C22×M16C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104#536—42no—
105#536—42no—
106#535—24no—
107#535—42no—
108C2×C4⋊Q8C2312yes—
109C2×#127C2312yes—
110C2×#124—24no—
111C2×#124—22no—
112C2×#124—42no—
113C2×#124C2312yes—
114C2×#124C2312yes—
115C2×#124C2312yes—
116C2×#124C2312yes—
117C2×#123—22no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×#121—22no—
121C2×#120—22no—
122#518—42no—
123#518—42no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—22no—
136C22×C22⋊C4—22no—
137C22×C42C2311yescharacteristic
138#428—42no—
139#428—42no—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144#424—42no—
145C2×C4⋊C8C2314yes—
146#419—24no—
147#419—42no—
148#418—24no—
149#418—24no—
150#417—42no—
151#417—42no—
152#416—24no—
153#416—42no—
154C2×#103—42no—
155C2×#103—42no—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#102—42no—
165C2×#102—42no—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#102C2314yes—
172C2×#102C2314yes—
173C2×#102C2312yes—
174C2×C4×C8—42no—
175C2×C4×C8C2312yes—
176C2×C4×C8C2312yes—
177#400—24no—
178#400—24no—
179#400—24no—
180#398—24no—
181#397—24no—
182#397—24no—
183#394—24no—
184#394—24no—
185#391—22no—
186#391C2312yes—
187#391C2312yes—
188#391C2311yescharacteristic
189#391C2311yescharacteristic
190#390C2311yescharacteristic
191#390C2311yescharacteristic
192#389—22no—
193#389—24no—
194#389D412yes—
195#389D412yes—
196#389C2314yes—
197#389C2314yes—
198#389C2314yes—
199#389C2312yes—
200#389C2312yes—
201#387—24no—
202#387—24no—
203#386—22no—
204#386—24no—
205#386—24no—
206#385—22no—
207#385—22no—
208#385—22no—
209#385D414yes—
210#384—21no—
211#384—21no—
212#382—21no—
213#382—21no—
214#382—22no—
215#382—24no—
216#382C2312yes—
217#382C2312yes—
218#381—22no—
219#380—24no—
220#380—24no—
221#380—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4C2314yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C2×#95—22no—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8C2412yes—
254C22×Q8C2411yescharacteristic
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—24no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4#4114yes—
264C22×D4C2411yescharacteristic
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4C2×D412yes—
273C23×C4C2411yescharacteristic
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—42no—
291C4⋊Q8—42no—
292#127—24no—
293#127—42no—
294#124—24no—
295#124—24no—
296#124—24no—
297#124—42no—
298#124—42no—
299#124—42no—
300#124—42no—
301#124—42no—
302#124—48no—
303#123—48no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306#121—48no—
307#120—48no—
308C4×D4—42no—
309C4×D4—42no—
310C4×D4—42no—
311C4×D4—42no—
312C4×D4—44no—
313C4×D4—44no—
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—22no—
323#117#4114yes—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D414yes—
341C2×C4⋊C4C2×D414yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—48no—
361C2×C22⋊C4—48no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—24no—
370C2×C42C2×D412yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—42no—
376#107—42no—
377#106—42no—
378#106—42no—
379C4⋊C8—44no—
380#103—24no—
381#103—28no—
382#103—44no—
383#103—44no—
384#103—42no—
385#103—42no—
386#103—42no—
387#103—44no—
388#102—28no—
389#102—24no—
390#102—28no—
391#102—44no—
392#102—44no—
393#102—42no—
394#102—42no—
395#102—42no—
396C4×C8—42no—
397C4×C8—42no—
398#95—24no—
399#95—24no—
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—
412C24—22no—
413C24—24no—
414C24—21no—
415C24—42no—
416C24—42no—
417C24—82no—
418C24—82no—
419C24C22×D411yescharacteristic
420#41—44no—
421#41—44no—
422#41—44no—
423#41—44no—
424C2×Q8—22no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×D4—28no—
444C2×D4—24no—
445C2×D4—28no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—48no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4—44no—
465C2×D4—84no—
466C2×D4—84no—
467C2×D4C2×#4114yes—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—84no—
490C22×C4—84no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4C22×D414yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4H(5,𝔽2)12yes—
504C22×C42−1+412yes—
505C22×C4C2511yescharacteristic
506M16—44no—
507M16—44no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—44no—
521C2×C8—44no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—84no—
545C4⋊C4—84no—
546C4⋊C4—84no—
547C4⋊C4C2×#4114yes—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C42—24no—
563C42—24no—
564C42—21no—
565C42—21no—
566C42—22no—
567C42—21no—
568C42—21no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—42no—
573C42—42no—
574C42—84no—
575C23—22no—
576C23—22no—
577C23—24no—
578C23—44no—
579C23—44no—
580C23—42no—
581C23—42no—
582C23—48no—
583C23—42no—
584C23—42no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—88no—
589C23—162no—
590C23—162no—
591C23—164no—
592C23#53512yes—
593C23#54612yes—
594C23#54812yes—
595C23C23×D411yescharacteristic
596Q8—42no—
597Q8—41no—
598Q8—42no—
599Q8—41no—
600Q8—42no—
601Q8—41no—
602Q8—42no—
603Q8—41no—
604Q8—84no—
605D4—24no—
606D4—24no—
607D4—41no—
608D4—42no—
609D4—41no—
610D4—41no—
611D4—48no—
612D4—42no—
613D4—41no—
614D4—84no—
615D4—168no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—164no—
653C2×C4—164no—
654C2×C4—162no—
655C2×C4—162no—
656C2×C4#53512yes—
657C2×C4#53914yes—
658C2×C4#54014yes—
659C2×C4#54612yes—
660C2×C4#54812yes—
661C2×C4#55314yes—
662C2×C4#56214yes—
663C2×C4C22×#4111yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C2×C4C2×H(5,𝔽2)11yescharacteristic
668C2×C4C2×2−1+411yescharacteristic
669C8—42no—
670C8—42no—
671C8—42no—
672C8—42no—
673C22—22no—
674C22—22no—
675C22—24no—
676C22—24no—
677C22—44no—
678C22—48no—
679C22—82no—
680C22—82no—
681C22—84no—
682C22—81no—
683C22—81no—
684C22—82no—
685C22—81no—
686C22—82no—
687C22—168no—
688C22#271114yes—
689C22C2×#53511yescharacteristic
690C22C2×#54611yescharacteristic
691C22C2×#54811yescharacteristic
692C22#358412yes—
693C22#358412yes—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—44no—
700C4—82no—
701C4—82no—
702C4—81no—
703C4—81no—
704C4—81no—
705C4—82no—
706C4—82no—
707C4#271114yes—
708C4C2×#53911yescharacteristic
709C4C2×#54011yescharacteristic
710C4C2×#55311yescharacteristic
711C4C2×#56211yescharacteristic
712C2—24no—
713C2—84no—
714C2—161no—
715C2C2×#271111yescharacteristic
716C2#6081911yescharacteristic
717C2#6081911yescharacteristic
718C2#6116912yes—
719C2#6133112yes—
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. #32089C2semidirect
4. #32088C2semidirect
5. #32087C2semidirect
6. #32086C2semidirect
7. #23733C2semidirect
8. #23615C2semidirect
10. #23510C2semidirect
11. #23510C2semidirect
12. #23506C2semidirect
14. C2×#1958C2semidirect
15. #21083C2semidirect
16. #21064C2semidirect
17. #21020C2semidirect
18. #20876C2semidirect
19. #20876C2semidirect
46. #2509C22semidirect
47. #2508C22semidirect
61. #2097C22semidirect
62. #2097C22semidirect
63. #2092C22semidirect
64. #2092C22semidirect
65. #2068C22semidirect
66. #2066C22semidirect
67. #2043C22semidirect
68. #2043C22semidirect
72. #2042C22semidirect
73. #2042C22semidirect
79. #1934C22semidirect
80. #1934C22semidirect
81. #1933C22semidirect
82. #1933C22semidirect
83. #1928C22semidirect
84. #1927C22semidirect
85. #1916C22semidirect
86. #1916C22semidirect

Maximal subgroups

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

subgroupindex
#356272
#356172
#320892
#320882
#320872
#320862
#237332
#236152
#235112
#235102
#235062
#212632
C2×#19582
#210832
#210642
#210202
#208762