Finite Group Databasebeta

← #6792514

Subgroups — #6792514

Order 512.

2007 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36126C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#32903C211yesmaximal, characteristic, complemented
4#31638C214yesmaximal, complemented
5#31522C214yesmaximal, complemented
6#29907C212yesmaximal, complemented
7#28404C212yesmaximal, complemented
8#27815C212yesmaximal, complemented
9#25903C211yesmaximal, characteristic, complemented
10#25322C212yesmaximal, complemented
11#24815C212yesmaximal, complemented
12#24503C212yesmaximal, complemented
13#24299C211yesmaximal, characteristic, complemented
14#23931C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16#21859C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C2×#486C2212yes—
23C2×#476C2212yes—
24C2×#460C2212yes—
25C2×#448C2211yescharacteristic
26C2×#448C2212yes—
27C2×C4×SD16C2212yes—
28C2×#431C2212yes—
29#2856C2211yescharacteristic
30#2854C2212yes—
31#2567—21no—
32#2567—22no—
33#2567—21no—
34#2567C2214yes—
35#2567C2212yes—
36#2567C2212yes—
37#2551C2211yescharacteristic
38#2541C2211yescharacteristic
39#2491—21no—
40#2461—22no—
41#2452C2212yes—
42#2445C2212yes—
43#2430—22no—
44#2416—21no—
45C2×#395—22no—
46#2366C2211yescharacteristic
47#2361—21no—
48D4×C4⋊C4C2211yescharacteristic
49C4×#121C2211yescharacteristic
50C2×#391C2212yes—
51C2×#389—21no—
52C2×#389—21no—
53C2×#384C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2100C2214yes—
56#2094C2214yes—
57#2084C2214yes—
58#2071C2214yes—
59#2066C2214yes—
60#2058C2212yes—
61#2055C2214yes—
62#2024C2214yes—
63#2023C2214yes—
64#2023C2212yes—
65#2014C2214yes—
66#1990C2214yes—
67#1990C2212yes—
68#1987C2212yes—
69#1968C2214yes—
70#1957C2214yes—
71#1939C2212yes—
72#1933C2214yes—
73#1916C2214yes—
74#1880C2214yes—
75#1880C2212yes—
76#1877C2214yes—
77#1877C2214yes—
78#1874C2212yes—
79#1874C2212yes—
80#1874C2211yescharacteristic
81#1874C2212yes—
82#1874C2212yes—
83#1865C2212yes—
84#1853C2212yes—
85#1853C2211yescharacteristic
86#1817C2212yes—
87#1803C2212yes—
88#1802C2214yes—
89#1802C2214yes—
90#1797C2211yescharacteristic
91#1787C2211yescharacteristic
92C8×C4⋊C4C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41—22no—
96C23×Q8—21no—
97C23×Q8C2311yescharacteristic
98C22×SD16C2312yes—
99C22×SD16C2314yes—
100#547—48no—
101#532—28no—
102#532—44no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—44no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#127C2311yescharacteristic
111C2×#127C2312yes—
112C2×#124—22no—
113C2×#123—22no—
114C2×#123—22no—
115C2×C22⋊Q8—22no—
116C2×#121—21no—
117C2×#121—22no—
118C2×#121—21no—
119C2×#121—21no—
120C2×#121—21no—
121C2×#121C2312yes—
122C2×#121C2311yescharacteristic
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—44no—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2312yes—
133C2×C4×D4C2312yes—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—22no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4D411yescharacteristic
140C22×C4⋊C4D411yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2312yes—
143C22×C42—21no—
144C22×C42C2311yescharacteristic
145C22×C42C2312yes—
146#486—44no—
147#476—44no—
148#460—44no—
149#448—24no—
150#448—44no—
151C4×SD16—44no—
152#431—44no—
153C2×#106C2314yes—
154C2×#106C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2314yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2311yescharacteristic
168C2×C22⋊C8C2311yescharacteristic
169C2×C4×C8C2312yes—
170C2×C4×C8C2314yes—
171#396—22no—
172#395—24no—
173#395—24no—
174#395—24no—
175#395—24no—
176#395—22no—
177#394—22no—
178#394—22no—
179#393—22no—
180#392—22no—
181#391—22no—
182#391—22no—
183#391C2314yes—
184#391C2314yes—
185#390—21no—
186#390—21no—
187#390—21no—
188#390C2312yes—
189#390C2312yes—
190#390C2312yes—
191#390C2311yescharacteristic
192#389—22no—
193#389—24no—
194#389—22no—
195#389—22no—
196#389—24no—
197#389—22no—
198#389—21no—
199#389C2311yescharacteristic
200#389C2312yes—
201#389C2312yes—
202#388—22no—
203#388—22no—
204#388—21no—
205#386—22no—
206#386—21no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—22no—
211#384—22no—
212#384—22no—
213#384C2314yes—
214#384C2314yes—
215#384C2311yescharacteristic
216#380—21no—
217#380C2311yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D412yes—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—21no—
226C2×#95—22no—
227C2×#95—22no—
228C2×#95D412yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2314yes—
236#336C2312yes—
237#336C2314yes—
238#336C2314yes—
239#336C2312yes—
240C2×#41—48no—
241C2×#41—48no—
242C2×#41—84no—
243C22×Q8—24no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—42no—
251C22×Q8—44no—
252C22×Q8—82no—
253C22×Q8—82no—
254C22×Q8C2412yes—
255C22×D4—21no—
256C22×D4—22no—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4C2412yes—
262C22×D4C2411yescharacteristic
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—22no—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4C2×D412yes—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D412yes—
273C23×C4C2411yescharacteristic
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—28no—
277C2×SD16—42no—
278C2×SD16—44no—
279C2×SD16—44no—
280C22×C8C2412yes—
281C22×C8C2414yes—
282C22×C8C2412yes—
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—44no—
287C4⋊Q8—42no—
288C4⋊Q8—44no—
289C4⋊Q8—44no—
290C4⋊Q8—44no—
291#127—24no—
292#127—24no—
293#127—42no—
294#124—48no—
295#123—48no—
296#123—48no—
297C22⋊Q8—48no—
298#121—22no—
299#121—24no—
300#121—24no—
301#121—48no—
302#121—44no—
303#121—44no—
304#121—44no—
305#121—82no—
306#121#4114yes—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—42no—
310C4×Q8—42no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—48no—
317C4×D4—42no—
318C4×D4—42no—
319C4×D4—44no—
320C4×D4—44no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—44no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4—44no—
342C2×C4⋊C4—44no—
343C2×C4⋊C4—84no—
344C2×C4⋊C4—84no—
345C2×C4⋊C4SD1618yescomplemented
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2414yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—22no—
372C2×C42—24no—
373C2×C42—24no—
374C2×C42—24no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—44no—
379C2×C42C2×D412yes—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D412yes—
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385C2×C42C2412yes—
386C2×C42C2411yescharacteristic
387#106—42no—
388#106—44no—
389C4⋊C8—24no—
390C4⋊C8—44no—
391#103—24no—
392#103—24no—
393#103—42no—
394#103—44no—
395#102—42no—
396#102—44no—
397#102—44no—
398#102—44no—
399#102—44no—
400C22⋊C8—22no—
401C22⋊C8—22no—
402C22⋊C8—24no—
403C4×C8—24no—
404C4×C8—44no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—22no—
413#95—22no—
414#95C2×D412yes—
415#95C2×D412yes—
416#95C2×D412yes—
417#95C2×D412yes—
418C24—21no—
419C24—22no—
420C24C22×D411yescharacteristic
421#41—832no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8—82no—
437C2×Q8—82no—
438C2×Q8—82no—
439C2×Q8—82no—
440C2×Q8—82no—
441C2×Q8—816no—
442C2×Q82−1+414yes—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—24no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4—82no—
465C2×D4—82no—
466C2×D4—81no—
467C2×D4—81no—
468C2×D4C2×#4112yes—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—42no—
488C22×C4—41no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—42no—
496C22×C4—44no—
497C22×C4—42no—
498C22×C4—44no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—42no—
504C22×C4—82no—
505C22×C4—82no—
506C22×C4C2×SD1614yes—
507C22×C4C2×SD1618yes—
508C22×C4C22×D412yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D412yes—
516C22×C4C22×D411yescharacteristic
517C22×C4C2×#4112yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C4C2511yescharacteristic
520SD16—48no—
521SD16—88no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—24no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—22no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—84no—
554C4⋊C4—84no—
555C4⋊C4—84no—
556C4⋊C4—84no—
557C4⋊C4—81no—
558C4⋊C4C2×SD1614yes—
559C4⋊C4C2×SD1614yes—
560C4⋊C4C2×SD1618yes—
561C4⋊C4#13714yes—
562C4⋊C4C2×#4112yes—
563C22⋊C4—24no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—42no—
567C22⋊C4—42no—
568C22⋊C4—44no—
569C22⋊C4—42no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—82no—
575C22⋊C4—81no—
576C22⋊C4—82no—
577C42—22no—
578C42—22no—
579C42—24no—
580C42—24no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—44no—
587C42—44no—
588C42—42no—
589C42H(5,𝔽2)14yes—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—22no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23#54612yes—
609C23C22×SD1612yes—
610C23#57612yes—
611C23C23×D411yescharacteristic
612Q8—24no—
613Q8—24no—
614Q8—48no—
615Q8—42no—
616Q8—42no—
617Q8—88no—
618Q8—88no—
619Q8—88no—
620D4—24no—
621D4—48no—
622D4—44no—
623D4—42no—
624D4—44no—
625D4—88no—
626D4—88no—
627D4—88no—
628D4—84no—
629D4—162no—
630C2×C4—22no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—21no—
638C2×C4—216no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—21no—
643C2×C4—28no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—24no—
652C2×C4—22no—
653C2×C4—21no—
654C2×C4—48no—
655C2×C4—44no—
656C2×C4—48no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—42no—
672C2×C4—41no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—88no—
677C2×C4—88no—
678C2×C4—84no—
679C2×C4—88no—
680C2×C4—84no—
681C2×C4#53512yes—
682C2×C4#53512yes—
683C2×C4#54612yes—
684C2×C4C22×SD1612yes—
685C2×C4C22×SD1614yes—
686C2×C4C22×SD1614yes—
687C2×C4C22×SD1612yes—
688C2×C4C2×#13714yes—
689C2×C4C2×#13712yes—
690C2×C4C22×#4111yescharacteristic
691C2×C4C2×H(5,𝔽2)12yes—
692C2×C4C2×H(5,𝔽2)11yescharacteristic
693C2×C4C2×2−1+412yes—
694C8—42no—
695C8—42no—
696C8—44no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—81no—
707C22—81no—
708C22—81no—
709C22—81no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22#329514yes—
718C22C2×#53511yescharacteristic
719C22C2×#54611yescharacteristic
720C22#348611yescharacteristic
721C22C23×SD1611yescharacteristic
722C22C22×#13711yescharacteristic
723C22C2×#57611yescharacteristic
724C22#358511yescharacteristic
725C4—22no—
726C4—22no—
727C4—24no—
728C4—24no—
729C4—24no—
730C4—22no—
731C4—41no—
732C4—42no—
733C4—42no—
734C4—44no—
735C4—81no—
736C4—81no—
737C4—82no—
738C4—84no—
739C4—82no—
740C4#320614yes—
741C4#352012yes—
742C4#353612yes—
743C2—24no—
744C2—82no—
745C2—162no—
746C2#3890511yescharacteristic
747C2C2×#320611yescharacteristic
748C2#6081111yescharacteristic
749C2C2×#329511yescharacteristic
750C2#6116211yescharacteristic
751C2#6182011yescharacteristic
752C2#6184411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36126C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2567C2semidirect
3. #32903C2semidirect
4. #31638C2semidirect
5. #31522C2semidirect
6. #29907C2semidirect
7. #28404C2semidirect
8. #27815C2semidirect
9. #25903C2semidirect
10. #25322C2semidirect
11. #24815C2semidirect
12. #24503C2semidirect
13. #24299C2semidirect
14. #23931C2semidirect
15. #22599C2semidirect
16. #21859C2semidirect
17. C4×#448C2semidirect
345. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#361262
C2×#25672
#329032
#316382
#315222
#299072
#284042
#278152
#259032
#253222
#248152
#245032
#242992
#239312
#225992
#218592
C4×#4482