Finite Group Databasebeta

← #6660593

Subgroups — #6660593

Order 512.

2048 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic, complemented
3#32790C212yesmaximal, complemented
4#32773C212yesmaximal, complemented
5#30364C214yesmaximal, complemented
6#29937C212yesmaximal, complemented
7#29891C214yesmaximal, complemented
8#29687C212yesmaximal, complemented
9#29334C212yesmaximal, complemented
10#29302C212yesmaximal, complemented
11#27678C212yesmaximal, complemented
12#24709C214yesmaximal, complemented
13#24298C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#525—21no—
18C22×C22⋊Q8—21no—
19C23×C42C2211yescharacteristic
20C2×#474C2214yescomplemented
21C2×#432C2212yes—
22#2857—21no—
23#2857—21no—
24#2857—21no—
25#2857—21no—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29#2857C2212yes—
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2856C2211yescharacteristic
33#2856C2211yescharacteristic
34#2854—21no—
35#2854C2212yes—
36#2850—21no—
37#2850—21no—
38#2847—21no—
39#2847C2212yes—
40C4×#121—21no—
41C4×#121—21no—
42C4×#121C2212yes—
43C4×#121C2212yes—
44#2297—21no—
45C2×#391—21no—
46C2×#389—21no—
47C2×#389C2212yes—
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#380—21no—
51C2×#380C2212yes—
52#2100C2214yescomplemented
53#2100C2214yescomplemented
54#2094C2214yescomplemented
55#2084C2214yescomplemented
56#2077C2218yes—
57#2076C2218yes—
58#2057C2214yescomplemented
59#2057C2214yescomplemented
60#2041C2214yescomplemented
61#2034C2218yes—
62#2033C2218yes—
63#2023C2214yescomplemented
64#1988C2214yescomplemented
65#1948C2214yescomplemented
66#1944C2218yescomplemented
67#1936C2218yescomplemented
68#1874C2214yes—
69#1874C2214yes—
70#1853C2214yes—
71#1852C2214yes—
72#1817C2212yes—
73#1802C2214yescomplemented
74#1797C2212yes—
75#1797C2212yes—
76#1779C2212yes—
77#1749C2212yes—
78#1749C2212yes—
79C22×#41—21no—
80C23×Q8—21no—
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84#532—44no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—44no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×#127C2311yescharacteristic
92C2×#124—21no—
93#525—48no—
94C2×#123—22no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—48no—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×#121C2312yes—
109C2×#121C2312yes—
110C2×#120—22no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4C2312yes—
115C2×#117—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4—22no—
123C22×C42—44no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#474—48no—
137#432—44no—
138C2×#107C2318yes—
139C2×#106C2318yes—
140C2×C4⋊C8C2314yes—
141C2×#102C2318yes—
142C2×#102C2318yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147#391—21no—
148#391—21no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2314yes—
169#389C2312yes—
170#389C2314yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#386—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#384—21no—
182#384—21no—
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#382—21no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380C2312yes—
198#380C2314yes—
199#380C2312yes—
200#380C2314yes—
201#380C2312yes—
202C4×C4⋊C4—21no—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C2×#95—22no—
207C2×#95—22no—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#336C2314yes—
213#336C2314yes—
214C25C2×D411yescharacteristic
215C2×#41—44no—
216C2×#41—44no—
217C2×#41—44no—
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×Q8—44no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2412yes—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C22×C8C2418yes—
261C4⋊Q8—28no—
262C4⋊Q8—44no—
263C4⋊Q8—42no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266#127—42no—
267#124—44no—
268#123—48no—
269C22⋊Q8—48no—
270C22⋊Q8—48no—
271C22⋊Q8—48no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8—48no—
275#121—28no—
276#121—44no—
277#121—44no—
278#121—44no—
279#121—44no—
280#121—44no—
281#121—44no—
282#121—44no—
283#120—48no—
284C4×D4—28no—
285C4×D4—44no—
286C4×D4—48no—
287C4×D4—48no—
288#117—44no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—48no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—48no—
361#106—48no—
362C4⋊C8—28no—
363#102—48no—
364#102—48no—
365C22⋊C8—28no—
366C22⋊C8—44no—
367C22⋊C8—44no—
368#97—44no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—41no—
386C24—41no—
387C24C22×D411yescharacteristic
388C24C22×D411yescharacteristic
389C24C22×D411yescharacteristic
390#41—816no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—816no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—42no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—416no—
475C22×C4—416no—
476C22×C4—416no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4AGL(1,ℤ/8)14yes—
487C22×C4AGL(1,ℤ/8)14yes—
488C22×C4#13714yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C42−1+414yes—
501C22×C4C2511yescharacteristic
502C2×C8—28no—
503C2×C8—28no—
504C2×C8—28no—
505C4⋊C4—28no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—48no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—44no—
557C42H(5,𝔽2)14yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—28no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23#53612yes—
589C23C2×AGL(1,ℤ/8)12yes—
590C23C2×AGL(1,ℤ/8)12yes—
591C23C2×#13712yes—
592C23#57512yes—
593C23C23×D411yescharacteristic
594Q8—88no—
595Q8—84no—
596Q8—88no—
597D4—48no—
598D4—84no—
599D4—84no—
600D4—88no—
601D4—88no—
602D4—88no—
603C2×C4—28no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—88no—
661C2×C4—88no—
662C2×C4—88no—
663C2×C4#53412yes—
664C2×C4#53412yes—
665C2×C4#53612yes—
666C2×C4C2×AGL(1,ℤ/8)12yes—
667C2×C4C2×AGL(1,ℤ/8)12yes—
668C2×C4C2×AGL(1,ℤ/8)12yes—
669C2×C4C2×AGL(1,ℤ/8)12yes—
670C2×C4C2×#13712yes—
671C2×C4C2×#13712yes—
672C2×C4C2×H(5,𝔽2)12yes—
673C2×C4C2×2−1+412yes—
674C2×C4#58512yes—
675C8—48no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—416no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22#323514yes—
705C22C2×#53411yescharacteristic
706C22C2×#53611yescharacteristic
707C22#344911yescharacteristic
708C22C22×AGL(1,ℤ/8)11yescharacteristic
709C22C22×AGL(1,ℤ/8)11yescharacteristic
710C22C22×#13711yescharacteristic
711C22C2×#57511yescharacteristic
712C4—22no—
713C4—22no—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—41no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—84no—
723C4—82no—
724C4—82no—
725C4—84no—
726C4—82no—
727C4#319214yes—
728C4#354114yes—
729C2—24no—
730C2—42no—
731C2—42no—
732C2—162no—
733C2#3743811yescharacteristic
734C2C2×#319211yescharacteristic
735C2#6075111yescharacteristic
736C2C2×#323511yescharacteristic
737C2#6086011yescharacteristic
738C2#6096211yescharacteristic
739C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2857C2semidirect
3. #32790C2semidirect
4. #32773C2semidirect
5. #30364C2semidirect
6. #29937C2semidirect
7. #29891C2semidirect
8. #29687C2semidirect
9. #29334C2semidirect
10. #29302C2semidirect
11. #27678C2semidirect
12. #24709C2semidirect
13. #24298C2semidirect
14. #20715C2semidirect
20. C2×#474C22semidirect
52. #2100C22semidirect
53. #2100C22semidirect
54. #2094C22semidirect
55. #2084C22semidirect
58. #2057C22semidirect
59. #2057C22semidirect
60. #2041C22semidirect
63. #2023C22semidirect
64. #1988C22semidirect
65. #1948C22semidirect
66. #1944C22semidirect
67. #1936C22semidirect
73. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#28572
#327902
#327732
#303642
#299372
#298912
#296872
#293342
#293022
#276782
#247092
#242982
#207152