Finite Group Databasebeta

← #6381917

Subgroups — #6381917

Order 512.

2578 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60444C211yesmaximal, characteristic
2C2×#2956C211yesmaximal, characteristic, complemented
3#60141C212yesmaximal, complemented
4C22×#430C211yesmaximal, characteristic, complemented
5#23021C214yesmaximal, complemented
6#23020C214yesmaximal, complemented
7#23019C214yesmaximal, complemented
8#23018C214yesmaximal, complemented
9#21241C211yesmaximal, characteristic, complemented
10#21165C211yesmaximal, characteristic, complemented
11#21071C212yesmaximal, complemented
12C2×#1871C212yesmaximal, complemented
13#20885C212yesmaximal, complemented
14#20445C211yesmaximal, characteristic, complemented
15C22⋊C4×M16C211yesmaximal, characteristic, complemented
16C2×#575C2211yescharacteristic
17C2×#568C2212yes—
18C23×M16C2212yes—
19C23×M16C2211yescharacteristic
20C2×#525C2212yes—
21#3056C414yescomplemented
22#3056C414yescomplemented
23#3055C414yescomplemented
24#3055C414yescomplemented
25#3054C2214yes—
26#3050C2214yes—
27#2956—28no—
28C2×#444C2214yescomplemented
29C2×#443C2214yescomplemented
30D4×M16—24no—
31D4×M16—24no—
32#2933C2218yescomplemented
33#2933C2218yescomplemented
34#2930—24no—
35#2929—24no—
36C2×#430—28no—
37C2×#430C2212yes—
38C2×#430C2211yescharacteristic
39C2×#430C2211yescharacteristic
40C2×#430C2214yescomplemented
41C22×#108C2214yescomplemented
42#2911C2211yescharacteristic
43#2905C2212yes—
44C22×#104C2212yes—
45C22×#104C2212yes—
46#2896C2212yes—
47#2879C2212yes—
48C2×#407C2211yescharacteristic
49#2873C2212yes—
50C2×C4×M16C2211yescharacteristic
51C2×C4×M16C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53#1975C2214yescomplemented
54#1974C2214yescomplemented
55#1946C2214yescomplemented
56#1945C2214yescomplemented
57#1929C2218yescomplemented
58#1929C2218yescomplemented
59#1871C2214yescomplemented
60#1871C2214yescomplemented
61#1871C2214yescomplemented
62#1871C2214yescomplemented
63#1866C2218yescomplemented
64#1866C2218yescomplemented
65#1773C2214yescomplemented
66#1772C2214yescomplemented
67#1753C2212yescomplemented
68#1753C2212yescomplemented
69#1752C2212yescomplemented
70C8×C22⋊C4C2212yescomplemented
71C2×#339C2212yes—
72C2×#339C2212yes—
73#1728C2211yescharacteristic
74#1728C2211yescharacteristic
75C22×#41C2312yes—
76C24×C4C2311yescharacteristic
77#575—24no—
78#575—24no—
79C2×#137C2×C414yes—
80C2×AGL(1,ℤ/8)C2×C414yes—
81C2×#135C2314yes—
82#568—28no—
83#568—44no—
84C2×#131—24no—
85C2×#131C2318yes—
86C22×M16—24no—
87C22×M16—24no—
88C22×M16—22no—
89C22×M16—22no—
90C22×M16—22no—
91C22×M16—24no—
92C22×M16—28no—
93C22×M16—44no—
94C22×M16C2×C414yes—
95C22×M16C2×C414yes—
96C22×M16C2312yes—
97C22×M16C2314yes—
98C22×M16C2314yes—
99C22×M16C2312yes—
100C22×M16C2311yescharacteristic
101C22×M16C2312yes—
102C22×M16C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8—22no—
106C23×C8C2312yes—
107#525—22no—
108#525—22no—
109#525—22no—
110#525—22no—
111#525C2×C414yes—
112#525C2×C414yes—
113#525C2314yes—
114#525C2314yes—
115C2×#123—22no—
116C2×C22⋊Q8C2×C412yes—
117C2×C22⋊Q8C2×C412yes—
118C2×#121C2×C412yes—
119C2×#121C2×C412yes—
120C2×#120—22no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×#117C2312yes—
124C22×C22⋊C4—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#470C2×C418yescomplemented
128#469C2×C418yescomplemented
129#468C2×C418yescomplemented
130#468C2×C418yescomplemented
131#467C2×C414yescomplemented
132#467C2×C414yescomplemented
133#466C2×C414yescomplemented
134#466C2×C414yescomplemented
135#465C2×C418yescomplemented
136#465C2×C418yescomplemented
137#444—216no—
138#443—216no—
139#436—28no—
140#436—28no—
141#435—28no—
142#435—28no—
143#435—28no—
144#435—28no—
145C8×D4—28no—
146C8×D4—28no—
147#430—28no—
148#430—24no—
149#430—216no—
150#430D418yescomplemented
151C2×#108—216no—
152C2×#108C2318yes—
153C2×#108C2314yes—
154C2×#108C2314yes—
155#428C2×C414yes—
156#428C2×C414yes—
157#427C2314yes—
158#424—24no—
159#424C2318yes—
160#423—24no—
161C2×#104—28no—
162C2×#104—44no—
163C2×#104C2314yes—
164C2×#104C2314yes—
165C2×#104C2314yes—
166C2×#104C2314yes—
167#419C2×C414yes—
168#418C2×C414yes—
169#417C2×C414yes—
170#417C2×C414yes—
171#416C2318yes—
172#408—24no—
173#408C2318yes—
174#407—24no—
175#407D414yes—
176#407D414yes—
177C2×C22⋊C8C2312yes—
178C2×C22⋊C8C2312yes—
179#405—24no—
180#405C2318yes—
181C4×M16—24no—
182C4×M16—24no—
183C4×M16—24no—
184C2×#97C2312yes—
185C2×#97C2312yes—
186C2×C4×C8C2312yes—
187C2×C4×C8C2312yes—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4C2311yescharacteristic
191C4×C22⋊C4C2311yescharacteristic
192C4×C22⋊C4C2311yescharacteristic
193C4×C22⋊C4C2311yescharacteristic
194C2×#95—21no—
195#339C2314yes—
196#339C2314yes—
197#339C2314yes—
198#339C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202#336C2312yes—
203C25#4111yescharacteristic
204C2×#41—22no—
205C2×#41—22no—
206C2×#41—24no—
207C2×#41—24no—
208C2×#41—24no—
209C2×#41—24no—
210C2×#41C22×C414yes—
211C2×#41C2414yes—
212C22×Q8C22×C412yes—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4C22×C412yes—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4C22×C412yes—
225C23×C4C22×C412yes—
226C23×C4C2×D412yes—
227C23×C4C2×D412yes—
228C23×C4#4111yescharacteristic
229C23×C4#4111yescharacteristic
230C23×C4#4111yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234#137—216no—
235AGL(1,ℤ/8)—216no—
236#135—28no—
237#135—28no—
238C2×Q16C22×C414yes—
239C2×SD16C22×C418yes—
240C2×D8C22×C414yes—
241#131—216no—
242#131—48no—
243#131—416no—
244C2×M16—22no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—22no—
248C2×M16—22no—
249C2×M16—24no—
250C2×M16—22no—
251C2×M16—22no—
252C2×M16—216no—
253C2×M16—216no—
254C2×M16—22no—
255C2×M16—24no—
256C2×M16—24no—
257C2×M16—24no—
258C2×M16—28no—
259C2×M16—24no—
260C2×M16—24no—
261C2×M16—24no—
262C2×M16—28no—
263C2×M16—28no—
264C2×M16—48no—
265C2×M16—44no—
266C2×M16—44no—
267C2×M16—416no—
268C2×M16—48no—
269C2×M16—48no—
270C2×M16C22×C414yes—
271C2×M16C22×C418yes—
272C2×M16C22×C414yes—
273C2×M16C2×D418yes—
274C2×M16C2×D414yes—
275C2×M16C2×D414yes—
276C2×M16C2×D414yes—
277C2×M16#4114yes—
278C2×M16C2414yes—
279C2×M16C2414yes—
280C22×C8—22no—
281C22×C8—24no—
282C22×C8—24no—
283C22×C8—22no—
284C22×C8—24no—
285C22×C8—24no—
286C22×C8—28no—
287C22×C8—48no—
288C22×C8C22×C414yes—
289C22×C8C22×C414yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294#123—24no—
295#123—24no—
296#123—24no—
297#123—24no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8C22×C414yes—
303C22⋊Q8C22×C414yes—
304C22⋊Q8C22×C414yes—
305C22⋊Q8C22×C414yes—
306#121—22no—
307#121—22no—
308#121—22no—
309#121—22no—
310#121C22×C414yes—
311#121C22×C414yes—
312#121C22×C414yes—
313#121C22×C414yes—
314#120—24no—
315#120—24no—
316#120—24no—
317#120—24no—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4—24no—
326#117—22no—
327#117—22no—
328#117C22×C414yes—
329#117C2414yes—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4C22×C412yes—
333C2×C4⋊C4C22×C412yes—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—44no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#108—216no—
356#108—28no—
357#108C2×D4116yescomplemented
358#107C22×C414yes—
359#107C22×C414yes—
360#106C22×C418yes—
361C4⋊C8—28no—
362C4⋊C8—28no—
363#104—28no—
364#104—28no—
365#104—44no—
366#104—44no—
367#103C22×C418yes—
368#103C22×C418yes—
369#102C22×C418yes—
370#102C22×C418yes—
371C22⋊C8—28no—
372C22⋊C8—28no—
373C22⋊C8C2×D414yes—
374C22⋊C8C2×D418yes—
375C22⋊C8C2×D414yes—
376#97—24no—
377#97—28no—
378#97—24no—
379C4×C8—24no—
380C4×C8—28no—
381C4×C8—24no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—42no—
390C24—81no—
391C24C4×D412yes—
392C24C4×D412yes—
393C24C2×#4111yescharacteristic
394C24C2×#4111yescharacteristic
395C24C2×#4111yescharacteristic
396#41—28no—
397#41—28no—
398#41—24no—
399#41—24no—
400#41—24no—
401#41—24no—
402#41—44no—
403#41—44no—
404#41—44no—
405#41—44no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8C23×C414yes—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—84no—
422C2×D4—84no—
423C2×D4C23×C414yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4C4×D412yes—
464C22×C4C4×D412yes—
465C22×C4C4×D412yes—
466C22×C4C4×D412yes—
467C22×C4C4×D412yes—
468C22×C4C4×D412yes—
469C22×C4C23×C411yescharacteristic
470C22×C4C23×C411yescharacteristic
471C22×C4C23×C412yes—
472C22×C4C23×C412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4111yescharacteristic
479C22×C4C2×#4111yescharacteristic
480C22×C4C2×#4112yes—
481C22×C4C2×#4111yescharacteristic
482C22×C4H(5,𝔽2)12yes—
483C22×C42−1+412yes—
484C22×C4C2511yescharacteristic
485Q16—28no—
486SD16—216no—
487D8—28no—
488M16—28no—
489M16—28no—
490M16—28no—
491M16—28no—
492M16—24no—
493M16—216no—
494M16—24no—
495M16—24no—
496M16—416no—
497M16—48no—
498M16—48no—
499M16—44no—
500M16C4×D4116yes—
501M16C22×D418yes—
502C2×C8—28no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—28no—
511C2×C8—28no—
512C2×C8—24no—
513C2×C8—416no—
514C2×C8—48no—
515C2×C8—48no—
516C2×C8—44no—
517C2×C8C23×C414yes—
518C2×C8C23×C414yes—
519C2×C8C22×D414yes—
520C2×C8C22×D414yes—
521C2×C8C22×D414yes—
522C2×C8C2×#4114yes—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4C23×C414yes—
530C4⋊C4C23×C414yes—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C42—24no—
546C42—24no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—42no—
557C23—24no—
558C23—21no—
559C23—22no—
560C23—24no—
561C23—21no—
562C23—24no—
563C23—21no—
564C23—21no—
565C23—24no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—44no—
575C23—82no—
576C23—82no—
577C23—84no—
578C23—84no—
579C23C2×C4×D412yes—
580C23C2×C4×D411yescharacteristic
581C23C2×C4×D412yes—
582C23C2×C4×D411yescharacteristic
583C23#54812yes—
584C23#55312yes—
585C23C22×#4111yescharacteristic
586Q8—24no—
587Q8—24no—
588Q8—44no—
589D4—24no—
590D4—24no—
591D4—48no—
592D4—48no—
593D4—44no—
594D4—48no—
595D4—44no—
596D4—44no—
597D4—84no—
598D4—84no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4C2×C4×D411yescharacteristic
645C2×C4C2×C4×D411yescharacteristic
646C2×C4C2×C4×D412yes—
647C2×C4C2×C4×D412yes—
648C2×C4C2×C4×D414yes—
649C2×C4#52014yes—
650C2×C4D4214yes—
651C2×C4#54714yes—
652C2×C4#54812yes—
653C2×C4#55312yes—
654C2×C4C24×C411yescharacteristic
655C2×C4C23×D411yescharacteristic
656C2×C4C23×D411yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4C2×2−1+411yescharacteristic
660C8—24no—
661C8—28no—
662C8—24no—
663C8—24no—
664C8—44no—
665C8C2×C4×D418yes—
666C22—21no—
667C22—22no—
668C22—21no—
669C22—22no—
670C22—21no—
671C22—21no—
672C22—22no—
673C22—21no—
674C22—22no—
675C22—21no—
676C22—21no—
677C22—22no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—48no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—81no—
687C22—81no—
688C22—84no—
689C22—164no—
690C22#236114yes—
691C22#295612yescomplemented
692C22C22×C4×D411yescharacteristic
693C22C2×#54811yescharacteristic
694C22C2×#55311yescharacteristic
695C4—21no—
696C4—21no—
697C4—22no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—44no—
702C4—44no—
703C4—81no—
704C4—81no—
705C4#236114yes—
706C4C22×C4×D411yescharacteristic
707C4C2×#52011yescharacteristic
708C4C2×D4211yescharacteristic
709C4C2×#54711yescharacteristic
710C2—21no—
711C2—21no—
712C2—22no—
713C2—24no—
714C2—42no—
715C2—41no—
716C2—84no—
717C2C2×#236111yescharacteristic
718C2C2×#295611yescharacteristic
719C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radical#60444C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#407 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2956C2semidirect
3. #60141C2semidirect
4. C22×#430C2semidirect
5. #23021C2semidirect
6. #23020C2semidirect
7. #23019C2semidirect
8. #23018C2semidirect
9. #21241C2semidirect
10. #21165C2semidirect
11. #21071C2semidirect
12. C2×#1871C2semidirect
13. #20885C2semidirect
14. #20445C2semidirect
15. C22⋊C4×M16C2semidirect
21. #3056C4semidirect
22. #3056C4semidirect
23. #3055C4semidirect
24. #3055C4semidirect
28. C2×#444C22semidirect
29. C2×#443C22semidirect
32. #2933C22semidirect
33. #2933C22semidirect
40. C2×#430C22semidirect
41. C22×#108C22semidirect
53. #1975C22semidirect
54. #1974C22semidirect
55. #1946C22semidirect
56. #1945C22semidirect
57. #1929C22semidirect
58. #1929C22semidirect
59. #1871C22semidirect
60. #1871C22semidirect
61. #1871C22semidirect
62. #1871C22semidirect
63. #1866C22semidirect
64. #1866C22semidirect
65. #1773C22semidirect
66. #1772C22semidirect
67. #1753C22semidirect
68. #1753C22semidirect
69. #1752C22semidirect
70. C8×C22⋊C4C22semidirect
127. #470C2×C4semidirect
128. #469C2×C4semidirect
129. #468C2×C4semidirect
130. #468C2×C4semidirect
131. #467C2×C4semidirect
132. #467C2×C4semidirect
133. #466C2×C4semidirect
134. #466C2×C4semidirect
135. #465C2×C4semidirect
136. #465C2×C4semidirect
150. #430D4semidirect
357. #108C2×D4semidirect
691. C22#2956semidirect

Maximal subgroups

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

subgroupindex
#604442
C2×#29562
#601412
C22×#4302
#230212
#230202
#230192
#230182
#212412
#211652
#210712
C2×#18712
#208852
#204452
C22⋊C4×M162