Finite Group Databasebeta

← #6345748

Subgroups — #6345748

Order 512.

1549 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60701C211yesmaximal, characteristic
2#60266C211yesmaximal, characteristic
3#60251C211yesmaximal, characteristic, complemented
4#18072C212yesmaximal, complemented
5#18071C212yesmaximal, complemented
6#18068C212yesmaximal, complemented
7#18067C212yesmaximal, complemented
8#18053C214yesmaximal, complemented
9#18051C214yesmaximal, complemented
10#17988C214yesmaximal, complemented
11#17969C214yesmaximal, complemented
12C2×#1569C211yesmaximal, characteristic, complemented
13#17488C211yesmaximal, characteristic
14#17486C211yesmaximal, characteristic, complemented
15#17327C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3185C412yes—
19#3182C412yes—
20#3181C412yes—
21#3181C412yes—
22#3172C2212yes—
23#3172C2212yes—
24#3170C2212yes—
25#3167C2212yes—
26#3165C411yescharacteristic
27#3152C411yescharacteristic
28#3141C411yescharacteristic
29#3139C411yescharacteristic
30#3056C414yes—
31#2995—22no—
32#2993C2214yes—
33#2991C2212yes—
34#2990C2212yes—
35#2989—22no—
36#2987—22no—
37#2985C2214yes—
38#2984C2214yes—
39#2983—22no—
40#2962C2212yes—
41#2962C2212yes—
42#2962C2212yes—
43#2961—21no—
44#2961—21no—
45C4×#137C2214yes—
46#2935C2212yes—
47#2930—22no—
48#2925—21no—
49#2924—22no—
50#2920C2211yescharacteristic
51C2×#432C2211yescharacteristic
52C2×#431C2211yescharacteristic
53#2906—21no—
54C2×#423C2211yescharacteristic
55C2×C4×M16C2211yescharacteristic
56#1593C2214yescomplemented
57#1592C2214yescomplemented
58#1591C2214yescomplemented
59#1590C2214yescomplemented
60#1588C2214yescomplemented
61#1587C2214yescomplemented
62#1587C2214yescomplemented
63#1585C2214yescomplemented
64#1584C2214yescomplemented
65#1584C2214yescomplemented
66#1583C2214yescomplemented
67#1582C2214yescomplemented
68#1579C2214yescomplemented
69SD16⋊C8C2214yescomplemented
70C8×Q16C2214yescomplemented
71C8×SD16C2214yescomplemented
72#1569C2212yes—
73#1569C2212yes—
74#1569C2212yes—
75#1569C2212yes—
76C2×#335C2212yes—
77C2×#334C2212yes—
78#1494C2214yes—
79#1493C2214yes—
80#1488C2214yes—
81#1487C2212yes—
82#1487C2212yes—
83#1481C2212yes—
84#1481C2212yes—
85#1481C2211yescharacteristic
86#1481C2211yescharacteristic
87#1480C2211yescharacteristic
88#1480C2211yescharacteristic
89#1453C2214yes—
90#1451C2212yes—
91#1451C2212yes—
92C8×M16C2212yes—
93C8×M16C2212yes—
94C2×#137C2×C414yes—
95C2×#131—22no—
96C22×M16—21no—
97C22×M16C2×C412yes—
98#533—21no—
99#533—21no—
100#533—21no—
101#533—21no—
102#533C2×C412yes—
103#533C2×C412yes—
104#533C2×C411yescharacteristic
105#533C2×C411yescharacteristic
106#533C2312yes—
107#533C2311yescharacteristic
108#533C2312yes—
109#532C2×C411yescharacteristic
110#532C2×C411yescharacteristic
111#532C2311yescharacteristic
112C2×C4⋊Q8C2×C411yescharacteristic
113C2×C4⋊Q8C2×C411yescharacteristic
114#529—21no—
115#529—21no—
116#529—21no—
117#529—21no—
118#529C2×C412yes—
119#529C2312yes—
120C2×#125—21no—
121#525—21no—
122#525—21no—
123#525C2×C412yes—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8C2×C412yes—
126C4×#41—21no—
127C4×#41—21no—
128C4×#41C2312yes—
129C2×C4×Q8—21no—
130C2×C4×Q8C2312yes—
131C2×#117—21no—
132C22×C42C2311yescharacteristic
133#501C2×C412yes—
134#501C2×C412yes—
135#500C2×C412yes—
136#498C2×C412yes—
137#497C2×C412yes—
138#497C2×C412yes—
139#494C2×C412yes—
140#492C2×C412yes—
141#491C2×C412yes—
142#491C2×C412yes—
143#489C2×C412yes—
144#489C2×C412yes—
145#487C2×C412yes—
146#487C2×C412yes—
147#487C2×C412yes—
148#486C2×C412yes—
149#470C2×C414yes—
150#470C2×C414yes—
151#468C2×C414yes—
152#468C2×C414yes—
153#467C2×C414yes—
154#467C2×C414yes—
155#465C2×C414yes—
156#465C2×C414yes—
157#446—24no—
158#446—24no—
159#446—24no—
160#446—24no—
161#446—24no—
162#446—24no—
163#446—24no—
164#446C2314yes—
165#446C2314yes—
166#446C2314yes—
167C8×Q8C2314yes—
168C8×Q8C2314yes—
169C8×Q8C2314yes—
170#441C2×C414yes—
171#441C2×C414yes—
172#440C2×C414yes—
173#440C2×C414yes—
174C4×Q16C2314yes—
175C4×Q16C2314yes—
176C4×SD16C2314yes—
177C4×SD16C2314yes—
178#436—24no—
179#436—24no—
180#436C2314yes—
181#435—24no—
182#435—24no—
183#435—24no—
184C8×D4C2314yes—
185#433—24no—
186#433—24no—
187#433—24no—
188#432—21no—
189#432—21no—
190#432—22no—
191#432—24no—
192#432D412yes—
193#432D412yes—
194#432C2312yes—
195#432C2312yes—
196#431—21no—
197#431—21no—
198#431—22no—
199#431—24no—
200#431C2311yescharacteristic
201#431C2311yescharacteristic
202#431C2312yes—
203#431C2312yes—
204#431C2312yes—
205#428C2×C414yes—
206#424—22no—
207#424—22no—
208#424—22no—
209#424—24no—
210#423—21no—
211#423—21no—
212#423—22no—
213#423D411yescharacteristic
214#423D411yescharacteristic
215#423C2311yescharacteristic
216#423C2311yescharacteristic
217#423C2312yes—
218C2×C4⋊C8—21no—
219C2×C4⋊C8—21no—
220C2×C4⋊C8—22no—
221C2×C4⋊C8C2311yescharacteristic
222C2×C4⋊C8C2312yes—
223C2×C4⋊C8C2311yescharacteristic
224#419C2×C414yes—
225#417C2×C414yes—
226#408—22no—
227#407D412yes—
228C2×C22⋊C8—21no—
229C2×C22⋊C8—21no—
230#405—22no—
231#405—22no—
232#405—22no—
233C4×M16C2×C411yescharacteristic
234C4×M16C2×C411yescharacteristic
235C4×M16C2×C412yes—
236C4×M16C2311yescharacteristic
237C4×M16C2311yescharacteristic
238C4×M16C2312yes—
239C4×M16C2312yes—
240C2×#97—21no—
241C2×#97C2×C411yescharacteristic
242C2×#97C2×C411yescharacteristic
243C2×C4×C8C2312yes—
244C2×C4×C8C2312yes—
245#335C2312yes—
246#335C2312yes—
247#335C2312yes—
248#335C2312yes—
249#334C2312yes—
250#334C2312yes—
251#334C2312yes—
252#334C2312yes—
253Q8⋊C8C2314yes—
254Q8⋊C8C2314yes—
255Q8⋊C8C2314yes—
256Q8⋊C8C2314yes—
257Q8⋊C8C2314yes—
258Q8⋊C8C2314yes—
259D4⋊C8C2314yes—
260D4⋊C8C2314yes—
261#322C2314yes—
262#322C2314yes—
263#322C2314yes—
264C82C2314yes—
265C2×#41—21no—
266C2×#41—21no—
267C2×#41C22×C412yes—
268C22×Q8—21no—
269C22×Q8C22×C412yes—
270C23×C4C22×C411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4#4111yescharacteristic
273#137—48no—
274C2×Q16C22×C414yes—
275C2×Q16C22×C414yes—
276C2×SD16C22×C414yes—
277C2×SD16C22×C414yes—
278#131—84no—
279C2×M16—21no—
280C2×M16—22no—
281C2×M16—21no—
282C2×M16—24no—
283C2×M16—44no—
284C2×M16—82no—
285C2×M16C22×C412yes—
286C2×M16C22×C412yes—
287C2×M16C22×C412yes—
288C2×M16C22×C412yes—
289C2×M16C2×D412yes—
290C2×M16C2×D412yes—
291C22×C8—22no—
292C22×C8—21no—
293C22×C8—21no—
294C22×C8—22no—
295C22×C8—24no—
296C22×C8C22×C412yes—
297C22×C8C22×C412yes—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8C22×C412yes—
303C4⋊Q8C22×C412yes—
304C4⋊Q8C22×C411yescharacteristic
305C4⋊Q8C22×C412yes—
306C4⋊Q8C22×C412yes—
307C4⋊Q8C22×C411yescharacteristic
308C4⋊Q8C22×C411yescharacteristic
309#127C22×C411yescharacteristic
310#126—22no—
311#126—22no—
312#125—22no—
313#125—22no—
314#125—21no—
315#125—22no—
316#125—22no—
317#125—21no—
318#125C22×C412yes—
319#124—21no—
320#124—21no—
321#124C22×C412yes—
322#123—22no—
323#123—22no—
324#123—22no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—21no—
335C22⋊Q8C22×C414yes—
336C22⋊Q8C22×C414yes—
337C22⋊Q8C22×C412yes—
338C22⋊Q8C22×C412yes—
339#121—21no—
340#121—21no—
341#121—21no—
342#121—21no—
343#121C22×C412yes—
344#121C22×C412yes—
345#120—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—21no—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—21no—
355C4×Q8—22no—
356C4×Q8C22×C414yes—
357C4×Q8C22×C412yes—
358C4×Q8C2414yes—
359C4×Q8C2412yes—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4C22×C412yes—
371C4×D4C2412yes—
372#117—22no—
373#117—22no—
374#117—21no—
375#117—22no—
376#117—22no—
377#117—21no—
378#117—22no—
379#117—22no—
380#117—22no—
381#117C22×C412yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4C22×C412yes—
388C2×C22⋊C4—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42C22×C411yescharacteristic
392C2×C42C22×C411yescharacteristic
393C2×C42C22×C411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42#4111yescharacteristic
398C2×C42#4111yescharacteristic
399C2×C42#4111yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403#107C22×C414yes—
404#107C22×C414yes—
405#106C22×C414yes—
406#106C22×C414yes—
407C4⋊C8—24no—
408C4⋊C8—24no—
409C4⋊C8—24no—
410C4⋊C8—24no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413C4⋊C8—21no—
414C4⋊C8—21no—
415C4⋊C8—22no—
416C4⋊C8—22no—
417C4⋊C8—21no—
418C4⋊C8—21no—
419C4⋊C8—22no—
420C4⋊C8—21no—
421C4⋊C8—21no—
422C4⋊C8C2×D412yes—
423C4⋊C8C2×D412yes—
424C4⋊C8C2×D412yes—
425C4⋊C8C2×D412yes—
426C4⋊C8C2412yes—
427C4⋊C8C2412yes—
428C4⋊C8C2412yes—
429C4⋊C8C2412yes—
430#103C22×C414yes—
431#103C22×C414yes—
432#103C22×C414yes—
433#103C22×C414yes—
434#103C22×C414yes—
435#103C22×C414yes—
436#102C22×C414yes—
437#102C22×C414yes—
438C22⋊C8—22no—
439C22⋊C8—22no—
440C22⋊C8—22no—
441C22⋊C8—21no—
442C22⋊C8—21no—
443C22⋊C8—24no—
444C22⋊C8—24no—
445C22⋊C8C2×D414yes—
446C22⋊C8C2×D414yes—
447#97—24no—
448#97—24no—
449#97—24no—
450#97—22no—
451#97—22no—
452#97C22×C412yes—
453#97C22×C412yes—
454#97C22×C411yescharacteristic
455#97C22×C411yescharacteristic
456#97C22×C411yescharacteristic
457#97C22×C411yescharacteristic
458#97C2×D414yes—
459C4×C8—24no—
460C4×C8—24no—
461C4×C8—24no—
462C4×C8—22no—
463C4×C8C22×C412yes—
464C4×C8C22×C412yes—
465C4×C8C2412yes—
466C4×C8C2412yes—
467C4×C8C2412yes—
468C4×C8C2412yes—
469C4×C8C2414yes—
470C24C4×D411yescharacteristic
471#41—44no—
472#41—42no—
473#41—42no—
474#41—82no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—44no—
481C2×Q8—82no—
482C2×Q8C23×C412yes—
483C2×Q8C23×C414yes—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4C23×C412yes—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—42no—
498C22×C4—81no—
499C22×C4C4×D411yescharacteristic
500C22×C4C4×D411yescharacteristic
501C22×C4C4×D411yescharacteristic
502C22×C4C4×D411yescharacteristic
503C22×C4C4×D411yescharacteristic
504C22×C4C4×D411yescharacteristic
505C22×C4C4×D411yescharacteristic
506C22×C4C4×D411yescharacteristic
507C22×C4C4×D411yescharacteristic
508C22×C4C23×C411yescharacteristic
509C22×C4C23×C411yescharacteristic
510C22×C4C23×C411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C2×#4111yescharacteristic
515C22×C4C2×#4111yescharacteristic
516C22×C4C2×#4111yescharacteristic
517Q16—28no—
518Q16—44no—
519SD16—28no—
520SD16—44no—
521M16—24no—
522M16—24no—
523M16—24no—
524M16—24no—
525M16—42no—
526M16—42no—
527M16—84no—
528M16—81no—
529M16—82no—
530M16—81no—
531C2×C8—24no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—42no—
539C2×C8—42no—
540C2×C8—44no—
541C2×C8—42no—
542C2×C8—84no—
543C2×C8—82no—
544C2×C8C23×C412yes—
545C2×C8C23×C412yes—
546C2×C8C23×C412yes—
547C2×C8C23×C412yes—
548C2×C8C22×D414yes—
549C2×C8C22×D414yes—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—24no—
557C4⋊C4—24no—
558C4⋊C4—24no—
559C4⋊C4—22no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—24no—
570C4⋊C4—22no—
571C4⋊C4C23×C414yes—
572C4⋊C4C23×C412yes—
573C4⋊C4C23×C412yes—
574C22⋊C4—24no—
575C22⋊C4—24no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—22no—
579C22⋊C4—22no—
580C22⋊C4—22no—
581C22⋊C4—22no—
582C22⋊C4—22no—
583C22⋊C4—22no—
584C22⋊C4—22no—
585C42—22no—
586C42—24no—
587C42—22no—
588C42—22no—
589C42—24no—
590C42C4×D411yescharacteristic
591C42C4×D411yescharacteristic
592C42C4×D411yescharacteristic
593C42C4×D411yescharacteristic
594C42C4×D411yescharacteristic
595C42C4×D411yescharacteristic
596C42C23×C411yescharacteristic
597C42C23×C411yescharacteristic
598C42C23×C411yescharacteristic
599C42C22×D411yescharacteristic
600C42C22×D411yescharacteristic
601C42C22×D411yescharacteristic
602C42C2×#4111yescharacteristic
603C42C2×#4111yescharacteristic
604C42C2×#4111yescharacteristic
605C42C2511yescharacteristic
606C23—22no—
607C23—22no—
608C23—41no—
609C23—41no—
610C23C2×C4×D411yescharacteristic
611C23C2×C4×D411yescharacteristic
612C23C2×C4×D411yescharacteristic
613Q8—24no—
614Q8—24no—
615Q8—24no—
616Q8—42no—
617Q8—82no—
618Q8—82no—
619Q8—82no—
620Q8—81no—
621D4—24no—
622D4—44no—
623D4—42no—
624D4—81no—
625D4—82no—
626D4—82no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—82no—
656C2×C4—82no—
657C2×C4C2×C4×D411yescharacteristic
658C2×C4C2×C4×D411yescharacteristic
659C2×C4C2×C4×D411yescharacteristic
660C2×C4C2×C4×D411yescharacteristic
661C2×C4C2×C4×D411yescharacteristic
662C2×C4C2×C4×D411yescharacteristic
663C2×C4C2×C4×D411yescharacteristic
664C2×C4C2×C4×D411yescharacteristic
665C2×C4C2×C4×D411yescharacteristic
666C2×C4#57712yes—
667C2×C4#57812yes—
668C2×C4C24×C411yescharacteristic
669C2×C4C23×D411yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C8—22no—
672C8—22no—
673C8—22no—
674C8—24no—
675C8—24no—
676C8—42no—
677C8—42no—
678C8—42no—
679C8—42no—
680C8—42no—
681C8—42no—
682C8#56814yes—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—82no—
695C22#294712yes—
696C22C22×C4×D411yescharacteristic
697C4—21no—
698C4—21no—
699C4—21no—
700C4—21no—
701C4—21no—
702C4—21no—
703C4—21no—
704C4—21no—
705C4—21no—
706C4—42no—
707C4—42no—
708C4—44no—
709C4—42no—
710C4—44no—
711C4#294712yes—
712C4C2×#56812yes—
713C4C2×#57711yescharacteristic
714C4C2×#57811yescharacteristic
715C2—22no—
716C2—41no—
717C2—41no—
718C2—82no—
719C2#6014411yescharacteristic
720C2C2×#294711yescharacteristic
721C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#3444C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #2920 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #60251C2semidirect
4. #18072C2semidirect
5. #18071C2semidirect
6. #18068C2semidirect
7. #18067C2semidirect
8. #18053C2semidirect
9. #18051C2semidirect
10. #17988C2semidirect
11. #17969C2semidirect
12. C2×#1569C2semidirect
14. #17486C2semidirect
15. #17327C2semidirect
56. #1593C22semidirect
57. #1592C22semidirect
58. #1591C22semidirect
59. #1590C22semidirect
60. #1588C22semidirect
61. #1587C22semidirect
62. #1587C22semidirect
63. #1585C22semidirect
64. #1584C22semidirect
65. #1584C22semidirect
66. #1583C22semidirect
67. #1582C22semidirect
68. #1579C22semidirect
69. SD16⋊C8C22semidirect
70. C8×Q16C22semidirect
71. C8×SD16C22semidirect

Maximal subgroups

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

subgroupindex
#607012
#602662
#602512
#180722
#180712
#180682
#180672
#180532
#180512
#179882
#179692
C2×#15692
#174882
#174862
#173272