Finite Group Databasebeta

← #6395284

Subgroups — #6395284

Order 512.

2226 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3253C211yesmaximal, characteristic
2#34207C211yesmaximal, characteristic, complemented
3#34155C211yesmaximal, characteristic, complemented
4#23140C214yesmaximal, complemented
5#23099C212yesmaximal, complemented
6#23067C212yesmaximal, complemented
7#23037C214yesmaximal, complemented
8#22995C212yesmaximal, complemented
9#22963C212yesmaximal, complemented
10#22151C212yesmaximal, complemented
11#22111C212yesmaximal, complemented
12#21859C211yesmaximal, characteristic, complemented
13#21614C212yesmaximal, complemented
14C4×#459C212yesmaximal, complemented
15#21390C211yesmaximal, characteristic, complemented
16#21171C211yesmaximal, characteristic, complemented
17#20550C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×#520—21no—
21#3253C4116yescomplemented
22C2×#476C2212yes—
23C2×#476C2212yes—
24C2×#469C2214yes—
25C2×#459C2212yes—
26C2×#459C2212yes—
27C2×C4×D8C2214yes—
28C2×#431C2211yescharacteristic
29C2×#428C2212yes—
30C2×#423C2211yescharacteristic
31#2372—22no—
32#2369C2211yescharacteristic
33#2369C2211yescharacteristic
34#2366C2211yescharacteristic
35#2366C2211yescharacteristic
36#2363—21no—
37#2363—21no—
38#2363C2212yes—
39#2363C2212yes—
40#2361—22no—
41#2361C2214yes—
42#2354—21no—
43#2354C2212yes—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46#2326—22no—
47#2311—21no—
48C4×C4⋊Q8C2212yes—
49C4×#127C2212yes—
50#2297—22no—
51#2293—21no—
52#2293C2212yes—
53C2×#384C2211yescharacteristic
54C42×D4—21no—
55C42×D4C2212yes—
56C4×#117C2211yescharacteristic
57C4×#117C2211yescharacteristic
58#1990C2212yes—
59#1987C2211yescharacteristic
60#1987C2211yescharacteristic
61#1982C2211yescharacteristic
62#1982C2211yescharacteristic
63#1972C2214yes—
64#1957C2212yes—
65#1948C2212yes—
66#1948C2212yes—
67#1947C2212yes—
68#1947C2212yes—
69#1941C2214yes—
70#1936C2214yes—
71#1934C2214yes—
72#1926C2214yes—
73#1925C2214yes—
74#1919C2214yes—
75#1919C2214yes—
76#1879C2214yes—
77#1879C2214yes—
78#1875C2212yes—
79#1875C2212yes—
80#1874C2214yes—
81#1864C2214yes—
82#1862C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1776C2214yes—
86#1770C2212yes—
87C8×C4⋊C4C2212yes—
88#1767C2212yes—
89C4×C4⋊C8C2212yes—
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93C4×#102C2214yes—
94#1753C2214yes—
95#1736C2212yes—
96C22×#41—21no—
97C22×#41—21no—
98C22×D8C2314yes—
99C22×M16C2312yes—
100#532—24no—
101#532—24no—
102#532C2×C418yescomplemented
103#532C2×C418yescomplemented
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#124—21no—
109C2×#124—21no—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×#121C2312yes—
114#520—48no—
115C2×C4×Q8—22no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—24no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2314yes—
128C2×#117—22no—
129C2×#117—22no—
130C2×#117D411yescharacteristic
131C2×#117D411yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4C2312yes—
138C22×C42—22no—
139C22×C42C2311yescharacteristic
140#476C2×C4116yescomplemented
141#476C2×C4116yescomplemented
142#469C2×C4132yescomplemented
143#459C2×C4116yescomplemented
144#459C2×C4116yescomplemented
145C4×D8—48no—
146#431—24no—
147#428—44no—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×#106C2314yes—
151#423C2×C418yescomplemented
152C2×C4⋊C8C2311yescharacteristic
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×C22⋊C8C2312yes—
165C2×#97C2314yes—
166C2×C4×C8C2312yes—
167C2×C4×C8C2314yes—
168#391—22no—
169#391—22no—
170#391—22no—
171#390—21no—
172#390—21no—
173#390—22no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390C2314yes—
178#390C2312yes—
179#390C2312yes—
180#389—21no—
181#389—21no—
182#389—22no—
183#389C2314yes—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#388—24no—
189#388—24no—
190#388—22no—
191#388—22no—
192#386—24no—
193#386—22no—
194#386—22no—
195#386—22no—
196#386—22no—
197#385—24no—
198#385—24no—
199#384—21no—
200#384—21no—
201#384C2312yes—
202#384C2312yes—
203#380—22no—
204#380—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229#376—21no—
230#376D412yes—
231C43—21no—
232C43C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#340C2312yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C2×#41—28no—
244C2×#41—28no—
245C2×#41—44no—
246C2×#41—44no—
247C2×#41—82no—
248C2×#41—82no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4C2412yes—
258C22×D4C2412yes—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—24no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×D8—44no—
269C2×D8—44no—
270C2×D8C22×C4116yes—
271C2×M16—42no—
272C2×M16—42no—
273C2×M16C22×C418yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2414yes—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8C22×C414yes—
280C4⋊Q8C22×C414yes—
281#127—22no—
282#127—22no—
283#127C22×C414yes—
284#127C22×C414yes—
285#124—28no—
286#124—28no—
287#121—28no—
288#121—28no—
289#121—24no—
290#121—24no—
291#121C22×C418yes—
292#121C22×C418yes—
293C4×Q8—48no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—28no—
297C4×D4—28no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4—48no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—416no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—84no—
309C4×D4C22×C418yes—
310C4×D4C22×C418yes—
311#117—24no—
312#117—22no—
313#117—22no—
314#117—42no—
315#117—41no—
316#117—41no—
317#117—48no—
318#117—84no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—48no—
333C2×C4⋊C4—82no—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C42—21no—
363C2×C42—24no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—84no—
372C2×C42C22×C414yes—
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2×D412yes—
376C2×C42C2×D412yes—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42#4114yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#107—28no—
385#106—44no—
386#106C22×C4116yes—
387C4⋊C8—22no—
388C4⋊C8—22no—
389C4⋊C8C22×C418yes—
390C4⋊C8C22×C418yes—
391#102—24no—
392#102—24no—
393#102—42no—
394#102—42no—
395#102—44no—
396#102—44no—
397#102C22×C4116yes—
398#102C22×C4116yes—
399C22⋊C8—24no—
400#97—44no—
401C4×C8—24no—
402C4×C8—44no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—24no—
412#95—24no—
413#95C2×D412yes—
414#95C2×D412yes—
415#95C2×D412yes—
416#95C2×D412yes—
417C24—22no—
418C24—22no—
419C24C22×D411yescharacteristic
420#41—432no—
421#41—816no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—81no—
427C2×Q8—81no—
428C2×D4—28no—
429C2×D4—28no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—81no—
453C2×D4—81no—
454C2×D4—82no—
455C2×D4—82no—
456C2×D4C23×C414yes—
457C2×D4C23×C414yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4C4×D414yes—
492C22×C4C4×D414yes—
493C22×C4C23×C412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C2×#4112yes—
504C22×C4C2511yescharacteristic
505D8—416no—
506M16—48no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8C23×C418yes—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—24no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—44no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4AGL(1,ℤ/8)14yes—
540C4⋊C4C23×C414yes—
541C4⋊C4C23×C414yes—
542C4⋊C4H(5,𝔽2)12yes—
543C4⋊C42−1+412yes—
544C22⋊C4—28no—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—28no—
548C22⋊C4—24no—
549C22⋊C4—42no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—48no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—84no—
560C22⋊C4—84no—
561C42—24no—
562C42—24no—
563C42—24no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—42no—
569C42—42no—
570C42—42no—
571C42—48no—
572C42—42no—
573C42—42no—
574C42—42no—
575C42—42no—
576C42—82no—
577C42—82no—
578C42C4×D414yes—
579C42C4×D414yes—
580C42C23×C412yes—
581C42C23×C412yes—
582C42C2×#4112yes—
583C42C2×#4112yes—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—24no—
591C23—24no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23C2×C4×D412yes—
605C23C23×D411yescharacteristic
606Q8—48no—
607Q8—84no—
608D4—24no—
609D4—24no—
610D4—24no—
611D4—24no—
612D4—48no—
613D4—416no—
614D4—48no—
615D4—48no—
616D4—42no—
617D4—42no—
618D4—42no—
619D4—42no—
620D4—84no—
621D4—88no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—28no—
633C2×C4—28no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—416no—
658C2×C4—416no—
659C2×C4—41no—
660C2×C4—44no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—44no—
664C2×C4—41no—
665C2×C4—44no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—88no—
669C2×C4—88no—
670C2×C4—168no—
671C2×C4—164no—
672C2×C4C2×C4×D412yes—
673C2×C4C2×C4×D412yes—
674C2×C4C2×C4×D412yes—
675C2×C4C2×C4×D412yes—
676C2×C4C2×C4×D412yes—
677C2×C4C2×AGL(1,ℤ/8)14yes—
678C2×C4C2×AGL(1,ℤ/8)12yes—
679C2×C4#57512yes—
680C2×C4#57612yes—
681C2×C4#57712yes—
682C2×C4C24×C411yescharacteristic
683C2×C4C22×#4111yescharacteristic
684C2×C4C2×H(5,𝔽2)11yescharacteristic
685C2×C4C2×2−1+411yescharacteristic
686C2×C4#58511yescharacteristic
687C2×C4#58511yescharacteristic
688C8—28no—
689C8—42no—
690C8—42no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22C22×C4×D411yescharacteristic
713C22#348811yescharacteristic
714C22#348911yescharacteristic
715C22C22×AGL(1,ℤ/8)11yescharacteristic
716C22C2×#57511yescharacteristic
717C22C2×#57611yescharacteristic
718C22C2×#57711yescharacteristic
719C4—24no—
720C4—22no—
721C4—22no—
722C4—22no—
723C4—42no—
724C4—42no—
725C4—42no—
726C4—42no—
727C4—42no—
728C4—44no—
729C4—44no—
730C4—84no—
731C4—81no—
732C4—81no—
733C4C4×AGL(1,ℤ/8)14yes—
734C4#294614yes—
735C4C4×H(5,𝔽2)11yescharacteristic
736C4C4×2−1+411yescharacteristic
737C4#351111yescharacteristic
738C4#351211yescharacteristic
739C2—42no—
740C2—84no—
741C2—84no—
742C2#3426711yescharacteristic
743C2C2×C4×AGL(1,ℤ/8)11yescharacteristic
744C2C2×#294611yescharacteristic
745C2#6197911yescharacteristic
746C2#6198711yescharacteristic
747C2#6214311yescharacteristic
748C2#6215011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3253C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34207C2semidirect
3. #34155C2semidirect
4. #23140C2semidirect
5. #23099C2semidirect
6. #23067C2semidirect
7. #23037C2semidirect
8. #22995C2semidirect
9. #22963C2semidirect
10. #22151C2semidirect
11. #22111C2semidirect
12. #21859C2semidirect
13. #21614C2semidirect
14. C4×#459C2semidirect
15. #21390C2semidirect
16. #21171C2semidirect
17. #20550C2semidirect
21. #3253C4semidirect
102. #532C2×C4semidirect
103. #532C2×C4semidirect
140. #476C2×C4semidirect
141. #476C2×C4semidirect
142. #469C2×C4semidirect
143. #459C2×C4semidirect
144. #459C2×C4semidirect
151. #423C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32532
#342072
#341552
#231402
#230992
#230672
#230372
#229952
#229632
#221512
#221112
#218592
#216142
C4×#4592
#213902
#211712
#205502