Finite Group Databasebeta

← #6477455

Subgroups — #6477455

Order 512.

3500 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#553C211yesmaximal, characteristic, complemented
2C2×#3192C211yesmaximal, characteristic, complemented
3C22×#480C211yesmaximal, characteristic, complemented
4#35076C211yesmaximal, characteristic, complemented
5#33002C212yesmaximal, complemented
6#32980C212yesmaximal, complemented
7#29640C212yesmaximal, complemented
8#29592C212yesmaximal, complemented
9#25553C218yesmaximal, complemented
10#24092C214yesmaximal, complemented
11#23963C214yesmaximal, complemented
12#21761C211yesmaximal, characteristic, complemented
13#21740C211yesmaximal, characteristic, complemented
14#20700C211yesmaximal, characteristic, complemented
15C2×#553—28no—
16C2×#553—22no—
17C2×#553—22no—
18C2×#553—28no—
19C2×#553C2214yes—
20C2×#553C2214yescomplemented
21C22×#126—21no—
22C22×#125C2211yescharacteristic, complemented
23C22×#123—21no—
24C22×#121—21no—
25C22×#121C2211yescharacteristic
26C22×#121C2211yescharacteristic
27C22×#121C2211yescharacteristic, complemented
28C22×C4×D4—21no—
29C22×C4×D4C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic, complemented
31C22×#117C2211yescharacteristic, complemented
32C23×C4⋊C4C2211yescharacteristic
33#3192—28no—
34C2×#481C2212yes—
35C2×#481C2212yescomplemented
36C2×#480—28no—
37C2×#480C2212yes—
38C2×#480C2212yes—
39C2×#480C2214yescomplemented
40C2×#469C2214yescomplemented
41C2×#447C2212yes—
42C2×#447C2212yescomplemented
43C22×#107C2212yescomplemented
44C22×#102C2212yescomplemented
45C2×#407C2211yescharacteristic
46C22×C22⋊C8C2211yescharacteristic
47#2854C2212yes—
48#2847C2212yes—
49#2635—22no—
50#2591—21no—
51#2539—21no—
52#2487C2212yescomplemented
53#2414C2214yescomplemented
54#2408C2212yescomplemented
55#2405—21no—
56#2403—21no—
57#2321—22no—
58#2280C2211yescharacteristic
59#2059C2218yescomplemented
60#2033C2218yescomplemented
61#2025C2218yescomplemented
62#1999C2218yescomplemented
63#1893C2214yescomplemented
64#1879C2218yescomplemented
65#1875C2214yescomplemented
66#1875C2218yescomplemented
67#1870C2214yescomplemented
68#1869C2214yescomplemented
69#1862C2218yescomplemented
70#1854C2218yescomplemented
71#1824C2212yescomplemented
72#1824C2214yescomplemented
73#1823C2212yescomplemented
74#1823C2212yescomplemented
75#1817C2212yescomplemented
76#1786C2212yescomplemented
77#1779C2212yescomplemented
78#1736C2214yescomplemented
79C23×D4—21no—
80C23×D4—21no—
81C23×D4—21no—
82C23×D4D411yescharacteristic
83C23×D4C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C24×C4—21no—
86C24×C4—21no—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×D8C2314yes—
90C22×M16C2312yes—
91C23×C8C2312yes—
92#553—232no—
93#553—232no—
94#553—216no—
95#553—216no—
96#553—216no—
97#553—216no—
98C2×#126—216no—
99C2×#126—24no—
100C2×#126—24no—
101C2×#125—24no—
102C2×#125—24no—
103C2×#125—22no—
104C2×#125C2314yes—
105C2×#123—216no—
106C2×#123—22no—
107C2×#123—24no—
108C2×#123—24no—
109C2×#121—216no—
110C2×#121—24no—
111C2×#121—24no—
112C2×#121—24no—
113C2×#121—24no—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121—22no—
117C2×#121—21no—
118C2×#121—21no—
119C2×#121—24no—
120C2×#121—24no—
121C2×#121—24no—
122C2×#121D418yescomplemented
123C2×#121C2312yes—
124C2×#121C2312yes—
125C2×#121C2314yes—
126C2×#121C2312yes—
127C2×#121C2312yes—
128C2×#121C2314yes—
129C2×#120—22no—
130C2×#120—22no—
131C2×C4×D4—216no—
132C2×C4×D4—24no—
133C2×C4×D4—24no—
134C2×C4×D4—24no—
135C2×C4×D4—24no—
136C2×C4×D4—24no—
137C2×C4×D4—24no—
138C2×C4×D4—22no—
139C2×C4×D4D412yes—
140C2×C4×D4D412yes—
141C2×C4×D4C2314yes—
142C2×C4×D4C2312yes—
143C2×C4×D4C2312yes—
144C2×#117—22no—
145C2×#117—24no—
146C2×#117—24no—
147C2×#117C2314yes—
148C2×#117C2312yes—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—24no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—21no—
153C22×C4⋊C4—21no—
154C22×C4⋊C4—21no—
155C22×C4⋊C4D411yescharacteristic
156C22×C4⋊C4D418yescomplemented
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2311yescharacteristic
159C22×C4⋊C4C2311yescharacteristic
160C22×C4⋊C4C2312yes—
161C22×C4⋊C4C2311yescharacteristic
162C22×C4⋊C4C2312yes—
163C22×C22⋊C4—21no—
164C22×C22⋊C4—21no—
165C22×C22⋊C4—21no—
166C22×C22⋊C4—21no—
167C22×C22⋊C4—21no—
168C22×C22⋊C4D411yescharacteristic
169C22×C22⋊C4D411yescharacteristic
170C22×C42—21no—
171C22×C42C2311yescharacteristic
172#481—28no—
173#481—28no—
174#480—28no—
175#480—28no—
176#480—216no—
177#469—216no—
178#447—28no—
179#447—28no—
180C2×#107—28no—
181C2×#107C2314yes—
182C2×#107C2314yes—
183C2×#106C2314yes—
184C2×#102—28no—
185C2×#102C2314yes—
186C2×#102C2314yes—
187C2×#102C2314yes—
188C2×#102C2314yes—
189C2×#102C2318yes—
190#407—24no—
191C2×C22⋊C8—24no—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2312yes—
194C2×C22⋊C8C2312yes—
195C2×C22⋊C8C2312yes—
196C2×C22⋊C8C2318yes—
197#400—22no—
198#399—22no—
199#394—22no—
200#394—24no—
201#392—24no—
202#392—22no—
203#390—24no—
204#390C2314yes—
205#389C2314yes—
206#388—24no—
207#388—24no—
208#386—22no—
209#386—24no—
210#386—22no—
211#386—22no—
212#385—22no—
213#381—21no—
214#380C2312yes—
215#380C2314yes—
216#380C2312yes—
217#379—21no—
218#379—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221#340C2318yes—
222#340C2318yes—
223#340C2318yes—
224#336C2318yes—
225C25—21no—
226C25—21no—
227C25C2×D411yescharacteristic
228C25C2×D411yescharacteristic
229C22×D4—24no—
230C22×D4—28no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—28no—
234C22×D4—24no—
235C22×D4—22no—
236C22×D4—24no—
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—44no—
244C22×D4—44no—
245C22×D4—44no—
246C22×D4—48no—
247C22×D4—44no—
248C22×D4—84no—
249C22×D4—84no—
250C22×D4C2×D412yes—
251C22×D4C2×D412yes—
252C22×D4C2×D414yes—
253C22×D4C2412yes—
254C22×D4C2412yes—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—28no—
259C23×C4—28no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—24no—
265C23×C4—21no—
266C23×C4—44no—
267C23×C4—44no—
268C23×C4—84no—
269C23×C4—84no—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D412yes—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D414yes—
278C23×C4C2×D414yes—
279C23×C4C2412yes—
280C23×C4C2411yescharacteristic
281C23×C4C2411yescharacteristic
282C23×C4C2411yescharacteristic
283C2×D8—28no—
284C2×D8—48no—
285C2×M16—24no—
286C2×M16—44no—
287C22×C8—24no—
288C22×C8—44no—
289C22×C8C2414yes—
290C22×C8C2414yes—
291#126—232no—
292#126—232no—
293#125—216no—
294#125—28no—
295#125—28no—
296#123—232no—
297#123—232no—
298#123—84no—
299#121—232no—
300#121—232no—
301#121—216no—
302#121—28no—
303#121—28no—
304#121—28no—
305#121—24no—
306#121—24no—
307#121—28no—
308#121—24no—
309#121—24no—
310#121—24no—
311#121—24no—
312#121—44no—
313#121—44no—
314#121—48no—
315#121C2×D418yes—
316#121C2×D418yes—
317#120—48no—
318#120—84no—
319C4×D4—232no—
320C4×D4—232no—
321C4×D4—216no—
322C4×D4—28no—
323C4×D4—28no—
324C4×D4—28no—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—24no—
329C4×D4—28no—
330#117—216no—
331#117—28no—
332#117—28no—
333#117—44no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—28no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D414yes—
359C2×C4⋊C4C2×D414yes—
360C2×C4⋊C4C2×D418yes—
361C2×C4⋊C4C2×D418yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2414yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—28no—
369C2×C22⋊C4—28no—
370C2×C22⋊C4—28no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—28no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—44no—
389C2×C22⋊C4C2×D412yes—
390C2×C22⋊C4C2×D412yes—
391C2×C22⋊C4C2×D412yes—
392C2×C22⋊C4C2×D412yes—
393C2×C42—28no—
394C2×C42—24no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42C2×D412yes—
399C2×C42C2412yes—
400#107—28no—
401#107—28no—
402#106—28no—
403#102—28no—
404#102—28no—
405#102—28no—
406#102—28no—
407#102—48no—
408C22⋊C8—24no—
409C22⋊C8—24no—
410C22⋊C8—24no—
411C22⋊C8—24no—
412C22⋊C8—48no—
413#95—22no—
414#95—22no—
415#95—24no—
416#95—24no—
417C24—24no—
418C24—24no—
419C24—22no—
420C24—24no—
421C24—22no—
422C24—24no—
423C24—22no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—44no—
428C24—44no—
429C24—44no—
430C24—42no—
431C24—44no—
432C24—82no—
433C24—82no—
434C24#12014yes—
435C24C22×D412yes—
436C24C22×D411yescharacteristic
437C24C22×D411yescharacteristic
438C24C22×D411yescharacteristic
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—216no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—28no—
451C2×D4—28no—
452C2×D4—48no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—44no—
456C2×D4—48no—
457C2×D4—48no—
458C2×D4—416no—
459C2×D4—432no—
460C2×D4—48no—
461C2×D4—48no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—48no—
467C2×D4—42no—
468C2×D4—84no—
469C2×D4—82no—
470C2×D4—88no—
471C2×D4—82no—
472C2×D4—82no—
473C2×D4—816no—
474C2×D4—84no—
475C2×D4—84no—
476C2×D4—816no—
477C2×D4C22×D414yes—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—216no—
492C22×C4—28no—
493C22×C4—28no—
494C22×C4—24no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—28no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—44no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—42no—
514C22×C4—88no—
515C22×C4—816no—
516C22×C4—84no—
517C22×C4—84no—
518C22×C4—82no—
519C22×C4—816no—
520C22×C4#12014yes—
521C22×C4#12014yes—
522C22×C4#12014yes—
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C22×D412yes—
530C22×C4C22×D412yes—
531C22×C4C22×D412yes—
532C22×C4C22×D412yes—
533C22×C4C22×D412yes—
534C22×C4C22×D412yes—
535C22×C4C2511yescharacteristic
536D8—88no—
537M16—84no—
538C2×C8—24no—
539C2×C8—24no—
540C2×C8—28no—
541C2×C8—28no—
542C2×C8—48no—
543C4⋊C4—216no—
544C4⋊C4—216no—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—28no—
548C4⋊C4—28no—
549C4⋊C4—24no—
550C4⋊C4—24no—
551C4⋊C4—24no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—28no—
558C4⋊C4—42no—
559C4⋊C4—84no—
560C4⋊C4C22×D414yes—
561C4⋊C4C22×D414yes—
562C4⋊C4C22×D414yes—
563C22⋊C4—216no—
564C22⋊C4—216no—
565C22⋊C4—216no—
566C22⋊C4—24no—
567C22⋊C4—24no—
568C22⋊C4—28no—
569C22⋊C4—28no—
570C22⋊C4—24no—
571C22⋊C4—24no—
572C22⋊C4—48no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—84no—
578C22⋊C4—84no—
579C22⋊C4—82no—
580C22⋊C4—84no—
581C42—216no—
582C42—24no—
583C42—22no—
584C42—22no—
585C42—82no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—28no—
592C23—28no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—24no—
597C23—24no—
598C23—22no—
599C23—22no—
600C23—24no—
601C23—44no—
602C23—416no—
603C23—44no—
604C23—48no—
605C23—44no—
606C23—48no—
607C23—44no—
608C23—48no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—42no—
613C23—44no—
614C23—44no—
615C23—42no—
616C23—42no—
617C23—88no—
618C23—816no—
619C23—84no—
620C23—84no—
621C23—84no—
622C23—168no—
623C23C2×#12012yes—
624C23C2×#12012yes—
625C23C2×#12012yes—
626C23#57514yes—
627C23#57614yes—
628C23C23×D411yescharacteristic
629D4—28no—
630D4—48no—
631D4—48no—
632D4—416no—
633D4—48no—
634D4—416no—
635D4—44no—
636D4—44no—
637D4—44no—
638D4—84no—
639D4—816no—
640D4—88no—
641D4—88no—
642D4—168no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—24no—
655C2×C4—22no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—24no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—48no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—48no—
669C2×C4—44no—
670C2×C4—48no—
671C2×C4—48no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—88no—
684C2×C4—88no—
685C2×C4—816no—
686C2×C4—84no—
687C2×C4C2×#12012yes—
688C2×C4C2×#12012yes—
689C2×C4C2×#12012yes—
690C2×C4C23×D411yescharacteristic
691C2×C4C23×D411yescharacteristic
692C2×C4#58514yes—
693C8—44no—
694C8—44no—
695C22—22no—
696C22—22no—
697C22—28no—
698C22—28no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—416no—
705C22—44no—
706C22—42no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—42no—
711C22—44no—
712C22—44no—
713C22—42no—
714C22—88no—
715C22—816no—
716C22—88no—
717C22—82no—
718C22—84no—
719C22—84no—
720C22—82no—
721C22—82no—
722C22—168no—
723C22#300514yes—
724C22#300814yes—
725C22C22×#12011yescharacteristic
726C22#348512yes—
727C22C2×#57512yes—
728C22C2×#57612yes—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—42no—
733C4—42no—
734C4—44no—
735C4—42no—
736C4—44no—
737C4—44no—
738C4—84no—
739C4—82no—
740C4#344912yes—
741C4#348512yes—
742C2—24no—
743C2—24no—
744C2—42no—
745C2—82no—
746C2—84no—
747C2—162no—
748C2#3508811yescharacteristic
749C2C2×#300511yescharacteristic
750C2C2×#300811yescharacteristic
751C2#6172714yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#553C2semidirect
2. C2×#3192C2semidirect
3. C22×#480C2semidirect
4. #35076C2semidirect
5. #33002C2semidirect
6. #32980C2semidirect
7. #29640C2semidirect
8. #29592C2semidirect
9. #25553C2semidirect
10. #24092C2semidirect
11. #23963C2semidirect
12. #21761C2semidirect
13. #21740C2semidirect
14. #20700C2semidirect
20. C2×#553C22semidirect
22. C22×#125C22semidirect
27. C22×#121C22semidirect
30. C22×C4×D4C22semidirect
31. C22×#117C22semidirect
35. C2×#481C22semidirect
39. C2×#480C22semidirect
40. C2×#469C22semidirect
42. C2×#447C22semidirect
43. C22×#107C22semidirect
44. C22×#102C22semidirect
52. #2487C22semidirect
53. #2414C22semidirect
54. #2408C22semidirect
59. #2059C22semidirect
60. #2033C22semidirect
61. #2025C22semidirect
62. #1999C22semidirect
63. #1893C22semidirect
64. #1879C22semidirect
65. #1875C22semidirect
66. #1875C22semidirect
67. #1870C22semidirect
68. #1869C22semidirect
69. #1862C22semidirect
70. #1854C22semidirect
71. #1824C22semidirect
72. #1824C22semidirect
73. #1823C22semidirect
74. #1823C22semidirect
75. #1817C22semidirect
76. #1786C22semidirect
77. #1779C22semidirect
78. #1736C22semidirect
122. C2×#121D4semidirect
156. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5532
C2×#31922
C22×#4802
#350762
#330022
#329802
#296402
#295922
#255532
#240922
#239632
#217612
#217402
#207002