Finite Group Databasebeta

← #7507733

Subgroups — #7507733

Order 512.

1721 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35627C211yesmaximal, characteristic, complemented
2#35621C211yesmaximal, characteristic
3#32100C214yesmaximal, complemented
4#32095C214yesmaximal, complemented
5#32092C214yesmaximal, complemented
6#32087C214yesmaximal, complemented
7C2×#2097C211yesmaximal, characteristic, complemented
8#23611C211yesmaximal, characteristic, complemented
9#23512C211yesmaximal, characteristic, complemented
10#23510C211yesmaximal, characteristic, complemented
11#23510C211yesmaximal, characteristic, complemented
12#23504C211yesmaximal, characteristic
13#21262C211yesmaximal, characteristic, complemented
14#21201C211yesmaximal, characteristic
15#21094C211yesmaximal, characteristic, complemented
16#21068C211yesmaximal, characteristic, complemented
17#21022C211yesmaximal, characteristic, complemented
18#20882C211yesmaximal, characteristic, complemented
19#20881C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#532C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C22×#107C2211yescharacteristic
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#2571—22no—
41#2570—22no—
42#2547C2212yes—
43#2546C2212yes—
44#2545C2212yes—
45#2544C2212yes—
46#2509C2214yes—
47#2508C2214yes—
48#2505—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×#382C2211yescharacteristic
60#2274C2211yescharacteristic
61#2097C2214yes—
62#2097C2214yes—
63#2068C2214yes—
64#2067C2214yes—
65#2043C2214yes—
66#2043C2214yes—
67#2043C2214yes—
68#2043C2212yes—
69#2042C2212yes—
70#2042C2214yes—
71#2042C2212yes—
72#2042C2214yes—
73#2042C2214yes—
74#2041C2212yes—
75#1983C2214yes—
76#1981C2214yes—
77#1961C2212yes—
78#1960C2212yes—
79#1958C2212yes—
80#1958C2212yes—
81#1935C2214yes—
82#1933C2214yes—
83#1927C2214yes—
84#1926C2214yes—
85#1917C2214yes—
86#1916C2214yes—
87#1865C2214yes—
88#1864C2214yes—
89#1863C2214yes—
90#1862C2214yes—
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×#124C2311yescharacteristic
115C2×#124C2311yescharacteristic
116C2×#124C2312yes—
117C2×#124C2312yes—
118C2×#123—22no—
119C2×#123—22no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—22no—
122C2×C22⋊Q8—21no—
123C2×#121—22no—
124#520—42no—
125#520—42no—
126C2×C4×Q8C2312yes—
127C2×C4×D4—22no—
128C2×C4×D4—22no—
129C2×C4×D4C2312yes—
130C2×#117—21no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C2×#107—42no—
141C2×#107C2314yes—
142C2×#106—42no—
143C2×#106C2314yes—
144#424—42no—
145C2×C4⋊C8C2314yes—
146#419—24no—
147#419—42no—
148#417—42no—
149#417—42no—
150#417—42no—
151#417—42no—
152#416—24no—
153#416—24no—
154#416—24no—
155#416—42no—
156C2×#103—42no—
157C2×#103—42no—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2314yes—
169C2×#103C2312yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×#102C2312yes—
175C2×#102C2312yes—
176C2×#102C2312yes—
177C2×#102C2312yes—
178C4×M16—42no—
179C2×#97C2312yes—
180C2×C4×C8C2312yes—
181#400—24no—
182#400—24no—
183#400—24no—
184#398—24no—
185#398—24no—
186#397—24no—
187#396—24no—
188#394—24no—
189#391—22no—
190#391C2312yes—
191#391C2312yes—
192#391C2312yes—
193#390C2312yes—
194#389—22no—
195#389—24no—
196#389—24no—
197#389D412yes—
198#389D412yes—
199#389C2314yes—
200#389C2314yes—
201#389C2314yes—
202#389C2312yes—
203#389C2312yes—
204#388—24no—
205#387—24no—
206#387—24no—
207#386—22no—
208#386—24no—
209#385—22no—
210#385—22no—
211#385—22no—
212#385D414yes—
213#384—21no—
214#384—21no—
215#382—21no—
216#382—21no—
217#382—22no—
218#382—24no—
219#382C2312yes—
220#382C2312yes—
221#380—22no—
222#380—24no—
223#380—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4C2314yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C22⋊C4—22no—
230C2×#95—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#124—22no—
300#124—22no—
301#124—24no—
302#124—41no—
303#124—41no—
304#124—42no—
305#124—42no—
306#124—42no—
307#123—48no—
308#123—48no—
309C22⋊Q8—48no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
312#121—48no—
313C4×Q8—42no—
314C4×Q8—42no—
315C4×D4—42no—
316C4×D4—42no—
317C4×D4—48no—
318C4×D4—48no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—44no—
327#117#4114yes—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4C2×D412yes—
368C2×C22⋊C4C2×D412yes—
369C2×C42—21no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—24no—
375C2×C42C2×D412yes—
376C2×C42C2412yes—
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—44no—
381#106—44no—
382C4⋊C8—44no—
383#103—24no—
384#103—24no—
385#103—24no—
386#103—28no—
387#103—44no—
388#103—44no—
389#103—42no—
390#103—42no—
391#103—42no—
392#103—42no—
393#103—42no—
394#103—44no—
395#102—24no—
396#102—24no—
397#102—24no—
398#102—42no—
399#102—42no—
400#102—42no—
401#102—42no—
402#102—42no—
403#97—42no—
404C4×C8—42no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95C2×D414yes—
417#95C2×D414yes—
418C24—21no—
419C24—21no—
420C24C22×D411yescharacteristic
421#41—28no—
422#41—28no—
423#41—44no—
424#41—44no—
425#41—44no—
426#41—44no—
427#41—44no—
428#41—44no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—21no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—44no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8C2×#4112yes—
454C2×D4—24no—
455C2×D4—21no—
456C2×D4—21no—
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—44no—
467C2×D4—44no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—82no—
474C2×D4C2×#4112yes—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—42no—
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—84no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D414yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D411yescharacteristic
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4H(5,𝔽2)12yes—
512C22×C4H(5,𝔽2)12yes—
513C22×C4H(5,𝔽2)12yes—
514C22×C42−1+412yes—
515C22×C4C2511yescharacteristic
516M16—44no—
517M16—44no—
518C2×C8—22no—
519C2×C8—22no—
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—44no—
531C2×C8—44no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—48no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—84no—
555C4⋊C4—84no—
556C4⋊C4—82no—
557C4⋊C4—84no—
558C4⋊C4—82no—
559C4⋊C4C2×#4114yes—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—48no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—42no—
570C22⋊C4—84no—
571C42—24no—
572C42—24no—
573C42—21no—
574C42—21no—
575C42—22no—
576C42—21no—
577C42—21no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—44no—
582C42—42no—
583C42—84no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—42no—
588C23—42no—
589C23—41no—
590C23—41no—
591C23—42no—
592C23—42no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23#53512yes—
597C23#54612yes—
598C23#54812yes—
599C23C23×D411yescharacteristic
600Q8—22no—
601Q8—22no—
602Q8—41no—
603Q8—41no—
604Q8—42no—
605Q8—41no—
606Q8—42no—
607Q8—41no—
608Q8—42no—
609Q8—41no—
610Q8—42no—
611Q8—41no—
612Q8—84no—
613D4—22no—
614D4—22no—
615D4—48no—
616D4—41no—
617D4—41no—
618D4—41no—
619D4—41no—
620D4—41no—
621D4—41no—
622D4—88no—
623D4—84no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—48no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—164no—
663C2×C4—162no—
664C2×C4—162no—
665C2×C4—162no—
666C2×C4#53512yes—
667C2×C4#53914yes—
668C2×C4#54014yes—
669C2×C4#54612yes—
670C2×C4#54812yes—
671C2×C4#55314yes—
672C2×C4#56214yes—
673C2×C4C22×#4111yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C2×H(5,𝔽2)11yescharacteristic
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C2×C4C2×H(5,𝔽2)11yescharacteristic
678C2×C4C2×2−1+411yescharacteristic
679C8—42no—
680C8—42no—
681C8—42no—
682C8—42no—
683C22—24no—
684C22—22no—
685C22—22no—
686C22—24no—
687C22—44no—
688C22—42no—
689C22—81no—
690C22—81no—
691C22—82no—
692C22—82no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22—82no—
697C22#271114yes—
698C22C2×#53511yescharacteristic
699C22C2×#54611yescharacteristic
700C22C2×#54811yescharacteristic
701C22#358412yes—
702C22#358612yes—
703C4—44no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—44no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—81no—
715C4—82no—
716C4#271114yes—
717C4C2×#53911yescharacteristic
718C4C2×#54011yescharacteristic
719C4C2×#55311yescharacteristic
720C4C2×#56211yescharacteristic
721C2—24no—
722C2—82no—
723C2—161no—
724C2C2×#271111yescharacteristic
725C2#6081911yescharacteristic
726C2#6082811yescharacteristic
727C2#6119712yes—
728C2#6135012yes—
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
1. #35627C2semidirect
3. #32100C2semidirect
4. #32095C2semidirect
5. #32092C2semidirect
6. #32087C2semidirect
7. C2×#2097C2semidirect
8. #23611C2semidirect
9. #23512C2semidirect
10. #23510C2semidirect
11. #23510C2semidirect
13. #21262C2semidirect
15. #21094C2semidirect
16. #21068C2semidirect
17. #21022C2semidirect
18. #20882C2semidirect
19. #20881C2semidirect

Maximal subgroups

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

subgroupindex
#356272
#356212
#321002
#320952
#320922
#320872
C2×#20972
#236112
#235122
#235102
#235042
#212622
#212012
#210942
#210682
#210222
#208822
#208812