Finite Group Databasebeta

← #7507701

Subgroups — #7507701

Order 512.

1787 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35677C211yesmaximal, characteristic
2#35555C211yesmaximal, characteristic, complemented
3#32109C214yesmaximal, complemented
4#32108C214yesmaximal, complemented
5#32107C214yesmaximal, complemented
6#32106C214yesmaximal, complemented
7#23734C211yesmaximal, characteristic, complemented
8#23614C211yesmaximal, characteristic, complemented
9#23512C211yesmaximal, characteristic, complemented
10#23509C211yesmaximal, characteristic
11#23509C211yesmaximal, characteristic
12#23506C211yesmaximal, characteristic, complemented
13#21262C211yesmaximal, characteristic, complemented
14C2×#1959C211yesmaximal, characteristic, complemented
15#21086C211yesmaximal, characteristic, complemented
16#21063C211yesmaximal, characteristic, complemented
17#21021C211yesmaximal, characteristic, complemented
18#20877C211yesmaximal, characteristic, complemented
19#20877C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#535C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#519C2211yescharacteristic
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×#97C2211yescharacteristic
40#2570—22no—
41#2552—22no—
42#2548C2212yes—
43#2548C2212yes—
44#2545C2212yes—
45#2545C2212yes—
46#2507C2214yescomplemented
47#2506C2214yescomplemented
48#2473—24no—
49#2426—22no—
50#2426—22no—
51#2411—24no—
52#2317C2211yescharacteristic
53#2317C2211yescharacteristic
54#2316C2211yescharacteristic
55#2309C2211yescharacteristic
56#2297C2211yescharacteristic
57#2296C2211yescharacteristic
58#2296C2211yescharacteristic
59C2×#384C2211yescharacteristic
60#2274C2211yescharacteristic
61#2096C2212yescomplemented
62#2095C2212yescomplemented
63#2094C2212yescomplemented
64#2093C2212yescomplemented
65#2067C2214yes—
66#2065C2214yes—
67#2043C2214yescomplemented
68#2042C2214yes—
69#2042C2214yes—
70#2042C2214yes—
71#2042C2214yes—
72#2042C2214yescomplemented
73#2041C2214yes—
74#2041C2214yes—
75#1983C2214yescomplemented
76#1981C2214yescomplemented
77#1959C2214yes—
78#1959C2214yes—
79#1936C2212yescomplemented
80#1936C2212yescomplemented
81#1935C2212yescomplemented
82#1935C2212yescomplemented
83#1926C2214yes—
84#1925C2214yes—
85#1917C2214yescomplemented
86#1917C2214yescomplemented
87#1865C2214yescomplemented
88#1865C2214yescomplemented
89#1863C2214yescomplemented
90#1863C2214yescomplemented
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—24no—
105#536—42no—
106#535—42no—
107#535—42no—
108C2×C4⋊Q8C2312yes—
109C2×#127C2312yes—
110C2×#125—24no—
111C2×#124—24no—
112C2×#124—22no—
113C2×#124—42no—
114C2×#124C2312yes—
115C2×#124C2312yes—
116C2×#124C2312yes—
117C2×#124C2312yes—
118C2×#123—22no—
119C2×C22⋊Q8—22no—
120C2×#121—21no—
121C2×#121—21no—
122C2×#120—22no—
123#519—42no—
124#519—42no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×#117—22no—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—22no—
139C22×C42C2311yescharacteristic
140#428—42no—
141#428—42no—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146#424—42no—
147C2×C4⋊C8C2314yes—
148#419—24no—
149#419—24no—
150#418—24no—
151#418—42no—
152#417—42no—
153#417—42no—
154#416—24no—
155#416—42no—
156C2×#103—42no—
157C2×#103—42no—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#102—42no—
167C2×#102—42no—
168C2×#102C2312yes—
169C2×#102C2314yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×#102C2314yes—
175C2×#102C2312yes—
176C2×#97—42no—
177C2×#97C2312yes—
178C2×#97C2312yes—
179#400—24no—
180#398—24no—
181#398—24no—
182#398—24no—
183#396—24no—
184#396—24no—
185#394—24no—
186#394—24no—
187#391C2311yescharacteristic
188#391C2311yescharacteristic
189#390—22no—
190#390—24no—
191#390—24no—
192#390C2312yes—
193#390C2312yes—
194#390C2311yescharacteristic
195#390C2311yescharacteristic
196#389—22no—
197#389—24no—
198#389D412yes—
199#389D412yes—
200#389C2314yes—
201#389C2314yes—
202#389C2314yes—
203#389C2312yes—
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#384—22no—
217#384C2312yes—
218#384C2312yes—
219#382—21no—
220#382—21no—
221#380—22no—
222#379—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4C2314yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—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#4114yes—
253C22×Q8C2411yescharacteristic
254C22×Q8C2412yes—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4C2412yes—
266C22×D4C2411yescharacteristic
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4C2×D412yes—
272C23×C4C2411yescharacteristic
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C4⋊Q8—24no—
290C4⋊Q8—42no—
291#127—42no—
292#127—42no—
293#125—88no—
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—48no—
305#121—44no—
306#121—44no—
307#120—48no—
308C4×Q8—42no—
309C4×Q8—42no—
310C4×Q8—42no—
311C4×Q8—42no—
312C4×Q8—44no—
313C4×Q8—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—48no—
324#117#4114yes—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—84no—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D414yes—
346C2×C4⋊C4C2×D414yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—48no—
366C2×C22⋊C4C2×D412yes—
367C2×C22⋊C4C2×D412yes—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—24no—
376C2×C42C2×D412yes—
377C2×C42C2412yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—42no—
382#107—42no—
383#106—42no—
384#106—42no—
385C4⋊C8—44no—
386#103—28no—
387#103—24no—
388#103—28no—
389#103—44no—
390#103—44no—
391#103—42no—
392#103—42no—
393#103—42no—
394#102—24no—
395#102—28no—
396#102—44no—
397#102—44no—
398#102—42no—
399#102—42no—
400#102—42no—
401#102—44no—
402#97—42no—
403#97—42no—
404#95—24no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95C2×D414yes—
413#95C2×D414yes—
414C24—22no—
415C24—21no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—82no—
420C24—82no—
421C24C22×D411yescharacteristic
422#41—44no—
423#41—44no—
424#41—44no—
425#41—44no—
426C2×Q8—24no—
427C2×Q8—22no—
428C2×Q8—28no—
429C2×Q8—28no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8C2×#4114yes—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—84no—
470C2×D4—84no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
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—42no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4C22×D412yes—
493C22×C4C22×D414yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C42−1+412yes—
504C22×C4C2511yescharacteristic
505M16—44no—
506M16—44no—
507C2×C8—22no—
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—44no—
520C2×C8—44no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—48no—
538C4⋊C4—44no—
539C4⋊C4—48no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—84no—
546C4⋊C4—84no—
547C4⋊C4—84no—
548C4⋊C4—84no—
549C4⋊C4C2×#4114yes—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—48no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—84no—
561C22⋊C4—82no—
562C22⋊C4—84no—
563C22⋊C4—82no—
564C42—24no—
565C42—24no—
566C42—21no—
567C42—21no—
568C42—22no—
569C42—21no—
570C42—21no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—44no—
575C42—44no—
576C42—42no—
577C42—42no—
578C42—84no—
579C23—22no—
580C23—22no—
581C23—42no—
582C23—44no—
583C23—42no—
584C23—44no—
585C23—42no—
586C23—41no—
587C23—41no—
588C23—42no—
589C23—41no—
590C23—42no—
591C23—88no—
592C23—84no—
593C23—164no—
594C23—162no—
595C23—162no—
596C23#53512yes—
597C23#54612yes—
598C23#54812yes—
599C23C23×D411yescharacteristic
600Q8—24no—
601Q8—24no—
602Q8—42no—
603Q8—48no—
604Q8—42no—
605Q8—41no—
606Q8—41no—
607Q8—41no—
608Q8—41no—
609D4—42no—
610D4—41no—
611D4—42no—
612D4—41no—
613D4—42no—
614D4—41no—
615D4—42no—
616D4—41no—
617D4—84no—
618D4—84no—
619D4—164no—
620D4—164no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—48no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—48no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—84no—
657C2×C4—162no—
658C2×C4—162no—
659C2×C4#53512yes—
660C2×C4#53914yes—
661C2×C4#54014yes—
662C2×C4#54612yes—
663C2×C4#54812yes—
664C2×C4#55314yes—
665C2×C4#56214yes—
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×H(5,𝔽2)11yescharacteristic
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C2×C4C2×2−1+411yescharacteristic
672C8—42no—
673C8—42no—
674C8—42no—
675C8—42no—
676C22—24no—
677C22—22no—
678C22—22no—
679C22—24no—
680C22—82no—
681C22—84no—
682C22—81no—
683C22—82no—
684C22—81no—
685C22—82no—
686C22—81no—
687C22—82no—
688C22—82no—
689C22—84no—
690C22—168no—
691C22#271114yes—
692C22C2×#53511yescharacteristic
693C22C2×#54611yescharacteristic
694C22C2×#54811yescharacteristic
695C22#358512yes—
696C22#358612yes—
697C4—44no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—44no—
702C4—42no—
703C4—42no—
704C4—81no—
705C4—82no—
706C4—82no—
707C4—81no—
708C4—81no—
709C4—82no—
710C4#271114yes—
711C4C2×#53911yescharacteristic
712C4C2×#54011yescharacteristic
713C4C2×#55311yescharacteristic
714C4C2×#56211yescharacteristic
715C2—24no—
716C2—161no—
717C2—162no—
718C2C2×#271111yescharacteristic
719C2#6082511yescharacteristic
720C2#6082611yescharacteristic
721C2#6117812yes—
722C2#6133612yes—
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
2. #35555C2semidirect
3. #32109C2semidirect
4. #32108C2semidirect
5. #32107C2semidirect
6. #32106C2semidirect
7. #23734C2semidirect
8. #23614C2semidirect
9. #23512C2semidirect
12. #23506C2semidirect
13. #21262C2semidirect
14. C2×#1959C2semidirect
15. #21086C2semidirect
16. #21063C2semidirect
17. #21021C2semidirect
18. #20877C2semidirect
19. #20877C2semidirect
46. #2507C22semidirect
47. #2506C22semidirect
61. #2096C22semidirect
62. #2095C22semidirect
63. #2094C22semidirect
64. #2093C22semidirect
67. #2043C22semidirect
72. #2042C22semidirect
75. #1983C22semidirect
76. #1981C22semidirect
79. #1936C22semidirect
80. #1936C22semidirect
81. #1935C22semidirect
82. #1935C22semidirect
85. #1917C22semidirect
86. #1917C22semidirect
87. #1865C22semidirect
88. #1865C22semidirect
89. #1863C22semidirect
90. #1863C22semidirect

Maximal subgroups

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

subgroupindex
#356772
#355552
#321092
#321082
#321072
#321062
#237342
#236142
#235122
#235092
#235062
#212622
C2×#19592
#210862
#210632
#210212
#208772