Finite Group Databasebeta

← #7506486

Subgroups — #7506486

Order 512.

2010 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35619C211yesmaximal, characteristic, complemented
2#35617C211yesmaximal, characteristic
3#31859C214yesmaximal, complemented
4#31858C214yesmaximal, complemented
5#31843C214yesmaximal, complemented
6#31842C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23633C211yesmaximal, characteristic, complemented
9#23542C211yesmaximal, characteristic, complemented
10C2×#2015C211yesmaximal, characteristic, complemented
11#23389C211yesmaximal, characteristic, complemented
12#23388C211yesmaximal, characteristic
13#21204C211yesmaximal, characteristic, complemented
14#21094C211yesmaximal, characteristic, complemented
15#21048C211yesmaximal, characteristic, complemented
16#20933C211yesmaximal, characteristic, complemented
17#20914C211yesmaximal, characteristic
18#20913C211yesmaximal, characteristic, complemented
19#20650C211yesmaximal, characteristic, complemented
20C23×Q16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#536C2211yescharacteristic
24C2×#535C2211yescharacteristic
25C22×#124C2211yescharacteristic
26C2×#520C2211yescharacteristic
27C2×#518C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C2×#418C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C2×#408C2211yescharacteristic
38C2×#408C2211yescharacteristic
39C2×C4×M16C2211yescharacteristic
40#2547—22no—
41#2544—22no—
42#2503C2212yescomplemented
43#2502C2212yescomplemented
44#2499C2212yescomplemented
45#2499C2212yescomplemented
46#2447—22no—
47#2446—22no—
48#2427C2214yes—
49#2427C2214yes—
50#2419—24no—
51#2412—24no—
52#2317C2211yescharacteristic
53#2310C2211yescharacteristic
54#2310C2211yescharacteristic
55#2297C2211yescharacteristic
56#2296C2211yescharacteristic
57C2×#391C2211yescharacteristic
58#2285C2211yescharacteristic
59#2101C2212yescomplemented
60#2101C2212yescomplemented
61#2100C2212yescomplemented
62#2099C2212yescomplemented
63#2074C2214yescomplemented
64#2072C2214yescomplemented
65#2056C2214yescomplemented
66#2056C2214yescomplemented
67#2015C2214yes—
68#2015C2214yes—
69#2001C2214yescomplemented
70#2001C2214yescomplemented
71#2000C2214yes—
72#2000C2214yes—
73#1959C2212yes—
74#1959C2212yes—
75#1958C2212yes—
76#1958C2212yes—
77#1935C2214yescomplemented
78#1933C2214yescomplemented
79#1924C2214yescomplemented
80#1923C2214yescomplemented
81#1877C2214yes—
82#1877C2214yes—
83#1875C2214yes—
84#1874C2214yes—
85#1873C2214yescomplemented
86#1872C2214yescomplemented
87#1804C2214yes—
88#1802C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C23×D4C2311yescharacteristic
101C22×Q16—42no—
102C22×Q16—42no—
103C22×Q16—42no—
104C22×Q16C2314yes—
105C22×SD16—24no—
106C22×SD16—42no—
107C22×SD16—42no—
108C22×SD16C2314yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#536—24no—
114#536—42no—
115#536—42no—
116#536—42no—
117#535—24no—
118#535—42no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8—21no—
121C2×#127—21no—
122C2×#124—21no—
123C2×#124—21no—
124C2×#124—21no—
125C2×#124—24no—
126C2×#124—22no—
127C2×#124—42no—
128C2×#124C2312yes—
129C2×#124C2312yes—
130C2×#123—22no—
131C2×#123—22no—
132C2×C22⋊Q8C2314yes—
133C2×C22⋊Q8C2312yes—
134C2×C22⋊Q8C2312yes—
135C2×#121C2314yes—
136C2×#120—22no—
137#520—42no—
138#520—42no—
139#518—24no—
140#518—42no—
141C2×C4×Q8—21no—
142C2×C4×D4—22no—
143C2×C4×D4—21no—
144C2×C4×D4—22no—
145C2×C4×D4C2314yes—
146C2×#117C2311yescharacteristic
147C2×#117C2311yescharacteristic
148C2×#117C2311yescharacteristic
149C2×#117C2311yescharacteristic
150C2×#117C2312yes—
151C2×#117C2311yescharacteristic
152C22×C4⋊C4C2311yescharacteristic
153C22×C4⋊C4C2312yes—
154C22×C22⋊C4—22no—
155C22×C22⋊C4—22no—
156C22×C42C2311yescharacteristic
157#424—42no—
158C2×C4⋊C8C2314yes—
159#419—24no—
160#419—42no—
161#419—42no—
162#419—42no—
163#418—42no—
164#418—42no—
165C2×#103—42no—
166C2×#103C2314yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2314yes—
170C2×#103C2312yes—
171C2×#103C2314yes—
172C2×#103C2312yes—
173C2×#102—42no—
174C2×#102C2312yes—
175C2×#102C2314yes—
176C2×#102C2312yes—
177C2×#102C2312yes—
178C2×#102C2312yes—
179#408—42no—
180#408—42no—
181C2×C22⋊C8C2314yes—
182C2×C22⋊C8C2314yes—
183C4×M16—42no—
184C2×#97C2312yes—
185C2×C4×C8C2312yes—
186#400—24no—
187#398—24no—
188#397—24no—
189#397—24no—
190#397—24no—
191#397—24no—
192#394—24no—
193#394—24no—
194#391—22no—
195#391—22no—
196#391—22no—
197#391D412yes—
198#391C2314yes—
199#391C2312yes—
200#391C2312yes—
201#391C2312yes—
202#391C2312yes—
203#390D412yes—
204#389—22no—
205#389C2312yes—
206#389C2312yes—
207#387—24no—
208#387—24no—
209#386—24no—
210#386—24no—
211#386D414yes—
212#385—22no—
213#385—22no—
214#385—24no—
215#384C2312yes—
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#382C2312yes—
219#382C2311yescharacteristic
220#382C2311yescharacteristic
221#381—22no—
222#381—22no—
223#381—24no—
224#380—22no—
225#380—24no—
226#380—24no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C22⋊C4C2312yes—
230C4×C22⋊C4C2312yes—
231C2×#95—22no—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2314yes—
236#340C2312yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#340C2312yes—
241#340C2312yes—
242#336C2314yes—
243#336C2312yes—
244#336C2312yes—
245C25—21no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41—22no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2412yes—
268C22×Q8C2412yes—
269C22×D4—24no—
270C22×D4—21no—
271C22×D4—22no—
272C22×D4—22no—
273C22×D4—22no—
274C22×D4—22no—
275C22×D4—22no—
276C22×D4C2×D412yes—
277C22×D4C2×D411yescharacteristic
278C22×D4#4114yes—
279C22×D4C2412yes—
280C23×C4—21no—
281C23×C4—21no—
282C23×C4—21no—
283C23×C4—22no—
284C23×C4—24no—
285C23×C4C2×D412yes—
286C23×C4C2412yes—
287C23×C4C2411yescharacteristic
288C2×Q16—44no—
289C2×Q16—44no—
290C2×Q16—44no—
291C2×Q16—416no—
292C2×SD16—28no—
293C2×SD16—44no—
294C2×SD16—44no—
295C2×SD16—416no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16—22no—
300C2×M16—22no—
301C2×M16—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8—22no—
307C22×C8—22no—
308C22×C8C2412yes—
309C22×C8C2412yes—
310C22×C8C2412yes—
311C22×C8C2412yes—
312C4⋊Q8—44no—
313C4⋊Q8—44no—
314#127—44no—
315#124—24no—
316#124—24no—
317#124—42no—
318#124—42no—
319#124—48no—
320#124—44no—
321#124—44no—
322#124—44no—
323#123—48no—
324#123—48no—
325C22⋊Q8—28no—
326C22⋊Q8—44no—
327C22⋊Q8—42no—
328C22⋊Q8—42no—
329C22⋊Q8—42no—
330C22⋊Q8—42no—
331#121—28no—
332#121—44no—
333#120—48no—
334C4×Q8—44no—
335C4×D4—28no—
336C4×D4—44no—
337C4×D4—48no—
338C4×D4—48no—
339C4×D4—44no—
340#117—22no—
341#117—22no—
342#117—22no—
343#117—22no—
344#117—22no—
345#117—22no—
346#117—22no—
347#117—22no—
348#117—22no—
349#117—42no—
350#117—42no—
351#117#4114yes—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—42no—
365C2×C4⋊C4—42no—
366C2×C4⋊C4C2×D412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2411yescharacteristic
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2411yescharacteristic
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2412yes—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—48no—
387C2×C22⋊C4—48no—
388C2×C22⋊C4C2×D412yes—
389C2×C22⋊C4C2×D414yes—
390C2×C22⋊C4C2×D414yes—
391C2×C42—21no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42—21no—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42C2×D412yes—
399C2×C42C2412yes—
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C4⋊C8—44no—
403#103—28no—
404#103—44no—
405#103—42no—
406#103—42no—
407#103—44no—
408#103—42no—
409#103—42no—
410#102—44no—
411#102—42no—
412#102—42no—
413#102—42no—
414#102—42no—
415C22⋊C8—44no—
416C22⋊C8—44no—
417#97—42no—
418C4×C8—42no—
419#95—24no—
420#95—24no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—24no—
430#95C2×D414yes—
431#95C2×D414yes—
432C24—22no—
433C24—24no—
434C24—42no—
435C24—42no—
436C24—82no—
437C24—82no—
438C24C22×D412yes—
439C24C22×D411yescharacteristic
440#41—44no—
441#41—44no—
442#41—44no—
443#41—44no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—44no—
466C2×Q8—44no—
467C2×Q8—44no—
468C2×Q8—44no—
469C2×Q8—44no—
470C2×Q8—44no—
471C2×Q8—44no—
472C2×Q8—44no—
473C2×D4—24no—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—22no—
477C2×D4—28no—
478C2×D4—24no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—28no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—48no—
488C2×D4—42no—
489C2×D4—44no—
490C2×D4—42no—
491C2×D4—44no—
492C2×D4—84no—
493C2×D4—84no—
494C2×D4C2×#4114yes—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—48no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—42no—
517C22×C4—42no—
518C22×C4—42no—
519C22×C4—42no—
520C22×C4—44no—
521C22×C4—42no—
522C22×C4—84no—
523C22×C4—84no—
524C22×C4—82no—
525C22×C4—84no—
526C22×C4C22×D412yes—
527C22×C4C22×D414yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C22×D411yescharacteristic
531C22×C4C2×#4112yes—
532C22×C4C2×#4112yes—
533C22×C4H(5,𝔽2)12yes—
534C22×C4H(5,𝔽2)12yes—
535C22×C4H(5,𝔽2)12yes—
536C22×C4H(5,𝔽2)12yes—
537C22×C4C2511yescharacteristic
538Q16—416no—
539SD16—416no—
540M16—44no—
541M16—44no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—22no—
554C2×C8—44no—
555C2×C8—44no—
556C4⋊C4—24no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—22no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—22no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—48no—
571C4⋊C4—44no—
572C4⋊C4—44no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—42no—
576C4⋊C4—84no—
577C4⋊C4—82no—
578C4⋊C4—82no—
579C4⋊C4—82no—
580C4⋊C4C2×#4114yes—
581C22⋊C4—24no—
582C22⋊C4—24no—
583C22⋊C4—44no—
584C22⋊C4—44no—
585C22⋊C4—48no—
586C22⋊C4—48no—
587C22⋊C4—48no—
588C22⋊C4—48no—
589C22⋊C4—48no—
590C22⋊C4—44no—
591C22⋊C4—44no—
592C22⋊C4—44no—
593C22⋊C4—42no—
594C22⋊C4—42no—
595C22⋊C4—42no—
596C42—24no—
597C42—24no—
598C42—21no—
599C42—21no—
600C42—22no—
601C42—21no—
602C42—21no—
603C42—22no—
604C42—44no—
605C42—42no—
606C42—42no—
607C42—42no—
608C23—22no—
609C23—22no—
610C23—24no—
611C23—24no—
612C23—22no—
613C23—24no—
614C23—22no—
615C23—22no—
616C23—44no—
617C23—44no—
618C23—42no—
619C23—42no—
620C23—48no—
621C23—88no—
622C23—164no—
623C23—162no—
624C23—162no—
625C23#53512yes—
626C23#54612yes—
627C23#54612yes—
628C23C23×D411yescharacteristic
629Q8—44no—
630Q8—42no—
631Q8—42no—
632Q8—42no—
633Q8—42no—
634Q8—42no—
635Q8—42no—
636Q8—42no—
637Q8—42no—
638Q8—84no—
639Q8—84no—
640D4—24no—
641D4—24no—
642D4—48no—
643D4—42no—
644D4—48no—
645D4—44no—
646D4—42no—
647D4—168no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—24no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—41no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—84no—
687C2×C4—84no—
688C2×C4—84no—
689C2×C4—168no—
690C2×C4—162no—
691C2×C4—164no—
692C2×C4#53512yes—
693C2×C4#53814yes—
694C2×C4#54014yes—
695C2×C4#54612yes—
696C2×C4#54612yes—
697C2×C4#55314yes—
698C2×C4#56014yes—
699C2×C4C22×#4111yescharacteristic
700C2×C4C22×#4111yescharacteristic
701C2×C4C2×H(5,𝔽2)11yescharacteristic
702C2×C4C2×H(5,𝔽2)11yescharacteristic
703C2×C4C2×H(5,𝔽2)11yescharacteristic
704C2×C4C2×H(5,𝔽2)11yescharacteristic
705C8—42no—
706C8—42no—
707C8—42no—
708C8—42no—
709C22—24no—
710C22—24no—
711C22—22no—
712C22—22no—
713C22—44no—
714C22—48no—
715C22—44no—
716C22—42no—
717C22—42no—
718C22—44no—
719C22—42no—
720C22—84no—
721C22—82no—
722C22—82no—
723C22—168no—
724C22#270314yes—
725C22C2×#53511yescharacteristic
726C22C2×#54611yescharacteristic
727C22C2×#54611yescharacteristic
728C22#358412yes—
729C22#358612yes—
730C4—42no—
731C4—42no—
732C4—44no—
733C4—44no—
734C4—42no—
735C4—82no—
736C4—81no—
737C4—82no—
738C4—81no—
739C4—82no—
740C4—82no—
741C4—81no—
742C4—81no—
743C4#270314yes—
744C4C2×#53811yescharacteristic
745C4C2×#54011yescharacteristic
746C4C2×#55311yescharacteristic
747C4C2×#56011yescharacteristic
748C2—24no—
749C2—82no—
750C2—84no—
751C2C2×#270311yescharacteristic
752C2#6081911yescharacteristic
753C2#6082811yescharacteristic
754C2#6118812yes—
755C2#6120712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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
1. #35619C2semidirect
3. #31859C2semidirect
4. #31858C2semidirect
5. #31843C2semidirect
6. #31842C2semidirect
7. #23744C2semidirect
8. #23633C2semidirect
9. #23542C2semidirect
10. C2×#2015C2semidirect
11. #23389C2semidirect
13. #21204C2semidirect
14. #21094C2semidirect
15. #21048C2semidirect
16. #20933C2semidirect
18. #20913C2semidirect
19. #20650C2semidirect
42. #2503C22semidirect
43. #2502C22semidirect
44. #2499C22semidirect
45. #2499C22semidirect
59. #2101C22semidirect
60. #2101C22semidirect
61. #2100C22semidirect
62. #2099C22semidirect
63. #2074C22semidirect
64. #2072C22semidirect
65. #2056C22semidirect
66. #2056C22semidirect
69. #2001C22semidirect
70. #2001C22semidirect
77. #1935C22semidirect
78. #1933C22semidirect
79. #1924C22semidirect
80. #1923C22semidirect
85. #1873C22semidirect
86. #1872C22semidirect

Maximal subgroups

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

subgroupindex
#356192
#356172
#318592
#318582
#318432
#318422
#237442
#236332
#235422
C2×#20152
#233892
#233882
#212042
#210942
#210482
#209332
#209142
#209132
#206502