Finite Group Databasebeta

← #6345657

Subgroups — #6345657

Order 512.

1596 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60951C211yesmaximal, characteristic
2#60287C211yesmaximal, characteristic, complemented
3#60237C211yesmaximal, characteristic, complemented
4#18028C214yesmaximal, complemented
5#18026C214yesmaximal, complemented
6#18011C214yesmaximal, complemented
7#18007C214yesmaximal, complemented
8#17987C214yesmaximal, complemented
9#17968C214yesmaximal, complemented
10#17821C211yesmaximal, characteristic, complemented
11#17482C211yesmaximal, characteristic, complemented
12#17478C211yesmaximal, characteristic, complemented
13#17324C211yesmaximal, characteristic, complemented
14#3444C2211yescharacteristic
15C2×#532C2211yescharacteristic
16#3254C412yes—
17#3253C412yes—
18#3238C412yes—
19#3235C412yes—
20#3217C412yes—
21#3214C412yes—
22#3194C412yes—
23#3194C412yes—
24#3171C2214yes—
25#3106C2214yes—
26#3101C2212yes—
27#3100C2212yes—
28#2993—22no—
29#2993C2214yescomplemented
30#2986C2212yes—
31#2981C2214yes—
32#2980—21no—
33#2980—21no—
34#2979C2212yes—
35#2978—22no—
36#2974C2214yes—
37#2973—22no—
38#2971—21no—
39#2971C2212yescomplemented
40#2970—21no—
41#2970C2212yescomplemented
42#2935C2214yes—
43#2932—22no—
44#2923—22no—
45#2920C2211yescharacteristic
46#2920C2211yescharacteristic
47#2920C2211yescharacteristic
48C2×#432C2211yescharacteristic
49C2×#431C2211yescharacteristic
50#2880—22no—
51#1594C2214yescomplemented
52#1592C2214yescomplemented
53#1590C2214yescomplemented
54#1589C2214yescomplemented
55#1588C2214yescomplemented
56#1586C2214yescomplemented
57#1586C2214yescomplemented
58#1585C2214yescomplemented
59#1583C2214yescomplemented
60#1582C2214yescomplemented
61#1581C2214yescomplemented
62#1581C2214yescomplemented
63#1580C2214yescomplemented
64SD16⋊C8C2214yescomplemented
65C8×SD16C2214yescomplemented
66C8×D8C2214yescomplemented
67#1573C2212yescomplemented
68#1573C2212yescomplemented
69#1572C2214yes—
70#1569C2214yescomplemented
71#1497C2212yes—
72#1496C2212yes—
73#1496C2212yes—
74#1496C2212yes—
75#1494C2212yescomplemented
76#1494C2212yescomplemented
77#1492C2212yes—
78#1492C2212yes—
79#1478C2214yes—
80#1477C2214yescomplemented
81#1454C2214yes—
82#1453C2214yescomplemented
83#1451C2212yescomplemented
84C8×M16C2212yescomplemented
85C22×#41—21no—
86C2×#131—24no—
87#533C2×C411yescharacteristic
88#533C2×C411yescharacteristic
89#533C2311yescharacteristic
90#532—21no—
91#532—21no—
92#532—21no—
93#532—21no—
94#532C2×C411yescharacteristic
95#532C2×C412yes—
96#532C2×C412yes—
97#532C2×C411yescharacteristic
98#532C2311yescharacteristic
99#532C2312yes—
100#532C2312yes—
101C2×C4⋊Q8C2×C411yescharacteristic
102C2×#127C2×C411yescharacteristic
103#529—21no—
104#529—21no—
105#529—21no—
106#529—21no—
107#529C2×C412yes—
108#529C2312yes—
109C2×#124—21no—
110#525—21no—
111#525—21no—
112#525C2×C412yes—
113C2×#121—21no—
114C2×#121C2×C412yes—
115C4×#41—21no—
116C4×#41—21no—
117C4×#41C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4C2312yes—
120C22×C42C2311yescharacteristic
121#501C2×C414yes—
122#496C2×C414yes—
123#490C2×C414yes—
124#489C2×C414yes—
125#483C2×C412yes—
126#483C2×C412yes—
127#482C2×C414yes—
128#482C2×C414yes—
129#480C2×C412yes—
130#480C2×C412yes—
131#477C2×C414yes—
132#476C2×C414yes—
133#476C2×C414yes—
134#474C2×C414yes—
135#469C2×C414yes—
136#469C2×C414yes—
137#468C2×C414yes—
138#468C2×C414yes—
139#461C2×C414yes—
140#460C2×C414yes—
141#459C2×C414yes—
142#459C2×C414yes—
143#450C2×C412yes—
144#450C2×C412yes—
145#449C2×C414yes—
146#449C2×C414yes—
147#447C2×C412yes—
148#447C2×C412yes—
149#446—22no—
150#446C2314yes—
151C8×Q8—22no—
152C8×Q8C2314yes—
153C4×SD16C2314yes—
154C4×SD16C2314yes—
155C4×D8C2314yes—
156C4×D8C2314yes—
157#436—24no—
158#436—22no—
159#436—24no—
160#436C2314yes—
161#436C2314yes—
162#436C2314yes—
163#435—24no—
164#435—24no—
165#435—24no—
166#435—24no—
167#435—24no—
168#435—24no—
169#435—24no—
170#435—24no—
171C8×D4—22no—
172C8×D4C2314yes—
173C8×D4C2314yes—
174C8×D4C2314yes—
175#433—24no—
176#433—24no—
177#432—21no—
178#432—21no—
179#432—21no—
180#432—22no—
181#432—22no—
182#432C2×C412yes—
183#432C2×C412yes—
184#432D412yes—
185#432D411yescharacteristic
186#432D411yescharacteristic
187#432C2312yes—
188#432C2312yes—
189#432C2311yescharacteristic
190#432C2311yescharacteristic
191#431—21no—
192#431—21no—
193#431—21no—
194#431C2312yes—
195#431C2312yes—
196#431C2312yes—
197#431C2312yes—
198#431C2312yes—
199#431C2312yes—
200#431C2312yes—
201#424—24no—
202#423C2×C412yes—
203#423D412yes—
204#423C2312yes—
205C2×C4⋊C8—21no—
206C2×C4⋊C8C2312yes—
207#408—24no—
208#408—24no—
209#408—24no—
210#408—24no—
211#407—21no—
212#407—24no—
213#407C2×C411yescharacteristic
214#407C2×C411yescharacteristic
215#407D412yes—
216C2×C22⋊C8—21no—
217C2×C22⋊C8—21no—
218C2×C22⋊C8—24no—
219C4×M16—21no—
220C4×M16—22no—
221C4×M16C2312yes—
222C4×M16C2312yes—
223C2×#97—21no—
224C2×#97—22no—
225C2×C4×C8C2311yescharacteristic
226C2×C4×C8C2311yescharacteristic
227#335C2314yes—
228#335C2314yes—
229#334C2314yes—
230#334C2314yes—
231Q8⋊C8C2314yes—
232Q8⋊C8C2314yes—
233D4⋊C8C2314yes—
234D4⋊C8C2314yes—
235D4⋊C8C2314yes—
236D4⋊C8C2314yes—
237D4⋊C8C2314yes—
238D4⋊C8C2314yes—
239#322C2314yes—
240#322C2314yes—
241#322C2314yes—
242C82C2314yes—
243C2×#41—21no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—21no—
248C2×#41—22no—
249C2×#41—84no—
250C2×#41C22×C412yes—
251C22×Q8—21no—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4C22×C412yes—
256C23×C4—22no—
257C23×C4C22×C411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4#4111yescharacteristic
260C2×SD16C22×C414yes—
261C2×SD16C22×C414yes—
262C2×D8C22×C414yes—
263C2×D8C22×C414yes—
264#131—88no—
265C2×M16—22no—
266C2×M16—24no—
267C2×M16—24no—
268C2×M16—24no—
269C2×M16C22×C412yes—
270C2×M16C22×C412yes—
271C2×M16C2×D414yes—
272C22×C8—22no—
273C22×C8—21no—
274C22×C8—21no—
275C22×C8—24no—
276C22×C8—24no—
277C22×C8—24no—
278C4⋊Q8—21no—
279C4⋊Q8—21no—
280C4⋊Q8C22×C411yescharacteristic
281C4⋊Q8C22×C412yes—
282C4⋊Q8C22×C412yes—
283C4⋊Q8C22×C411yescharacteristic
284C4⋊Q8C22×C411yescharacteristic
285#127—21no—
286#127—21no—
287#127C22×C412yes—
288#127C22×C412yes—
289#127C22×C411yescharacteristic
290#126—22no—
291#126—22no—
292#125—21no—
293#125—21no—
294#125C22×C412yes—
295#124—22no—
296#124—22no—
297#124—21no—
298#124—22no—
299#124—22no—
300#124—21no—
301#124C22×C412yes—
302#123—22no—
303#123—22no—
304#123—22no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308C22⋊Q8—21no—
309C22⋊Q8C22×C412yes—
310C22⋊Q8C22×C412yes—
311#121—22no—
312#121—22no—
313#121—22no—
314#121—22no—
315#121—21no—
316#121—21no—
317#121—21no—
318#121—22no—
319#121—22no—
320#121—21no—
321#121C22×C414yes—
322#121C22×C414yes—
323#121C22×C412yes—
324#121C22×C412yes—
325#120—22no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8C22×C412yes—
331C4×Q8C2412yes—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—21no—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—21no—
347C4×D4—22no—
348C4×D4C22×C414yes—
349C4×D4C22×C412yes—
350C4×D4C2414yes—
351C4×D4C2412yes—
352#117—21no—
353#117—21no—
354#117—22no—
355#117—22no—
356#117—22no—
357#117C22×C412yes—
358C2×C4⋊C4—21no—
359C2×C4⋊C4C22×C412yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C42—22no—
363C2×C42C22×C411yescharacteristic
364C2×C42C22×C411yescharacteristic
365C2×C42C22×C411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42#4111yescharacteristic
370C2×C42#4111yescharacteristic
371C2×C42#4111yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107C22×C414yes—
376#107C22×C414yes—
377#106C22×C414yes—
378#106C22×C414yes—
379C4⋊C8—24no—
380C4⋊C8—24no—
381C4⋊C8—22no—
382C4⋊C8—22no—
383C4⋊C8—22no—
384C4⋊C8—22no—
385C4⋊C8C22×C414yes—
386C4⋊C8C22×C414yes—
387C4⋊C8C2×D414yes—
388C4⋊C8C2×D412yes—
389C4⋊C8C2×D412yes—
390C4⋊C8C2412yes—
391C4⋊C8C2412yes—
392C4⋊C8C2414yes—
393#103C22×C414yes—
394#103C22×C414yes—
395#102C22×C414yes—
396#102C22×C414yes—
397#102C22×C414yes—
398#102C22×C414yes—
399#102C22×C414yes—
400#102C22×C414yes—
401C22⋊C8—22no—
402C22⋊C8—22no—
403C22⋊C8—22no—
404C22⋊C8—21no—
405C22⋊C8—21no—
406C22⋊C8—22no—
407C22⋊C8—22no—
408C22⋊C8—24no—
409C22⋊C8—24no—
410C22⋊C8—24no—
411C22⋊C8—24no—
412C22⋊C8—24no—
413C22⋊C8—24no—
414C22⋊C8C22×C412yes—
415C22⋊C8C22×C412yes—
416C22⋊C8C22×C412yes—
417C22⋊C8C22×C412yes—
418C22⋊C8C2×D414yes—
419C22⋊C8C2×D412yes—
420C22⋊C8C2×D412yes—
421#97—24no—
422#97—21no—
423#97—22no—
424#97—22no—
425#97—21no—
426#97C22×C414yes—
427#97C2×D412yes—
428#97C2×D412yes—
429C4×C8—24no—
430C4×C8—22no—
431C4×C8—21no—
432C4×C8—21no—
433C4×C8C2412yes—
434C4×C8C2412yes—
435C4×C8C2414yes—
436C4×C8C2414yes—
437C24—22no—
438C24C4×D411yescharacteristic
439#41—48no—
440#41—44no—
441#41—82no—
442#41—82no—
443#41—82no—
444#41—84no—
445#41—82no—
446#41—84no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—82no—
452C2×Q8C23×C412yes—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—44no—
466C2×D4—82no—
467C2×D4—82no—
468C2×D4—84no—
469C2×D4C23×C412yes—
470C2×D4C23×C414yes—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—84no—
480C22×C4—81no—
481C22×C4C4×D411yescharacteristic
482C22×C4C4×D411yescharacteristic
483C22×C4C4×D411yescharacteristic
484C22×C4C4×D411yescharacteristic
485C22×C4C4×D411yescharacteristic
486C22×C4C4×D411yescharacteristic
487C22×C4C4×D411yescharacteristic
488C22×C4C4×D411yescharacteristic
489C22×C4C4×D411yescharacteristic
490C22×C4C23×C411yescharacteristic
491C22×C4C23×C411yescharacteristic
492C22×C4C23×C411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C2×#4111yescharacteristic
497C22×C4C2×#4111yescharacteristic
498C22×C4C2×#4111yescharacteristic
499SD16—44no—
500SD16—44no—
501D8—44no—
502D8—44no—
503M16—42no—
504M16—42no—
505M16—44no—
506M16—84no—
507M16—84no—
508M16—84no—
509M16—82no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—24no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—44no—
516C2×C8—84no—
517C2×C8—84no—
518C2×C8—84no—
519C2×C8—81no—
520C2×C8—81no—
521C2×C8C23×C414yes—
522C2×C8C23×C414yes—
523C2×C8C22×D414yes—
524C2×C8C22×D414yes—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4C23×C412yes—
539C4⋊C4C23×C412yes—
540C4⋊C4C23×C414yes—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—24no—
553C22⋊C4—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42C4×D411yescharacteristic
558C42C4×D411yescharacteristic
559C42C4×D411yescharacteristic
560C42C4×D411yescharacteristic
561C42C4×D411yescharacteristic
562C42C4×D411yescharacteristic
563C42C23×C411yescharacteristic
564C42C23×C411yescharacteristic
565C42C23×C411yescharacteristic
566C42C22×D411yescharacteristic
567C42C22×D411yescharacteristic
568C42C22×D411yescharacteristic
569C42C2×#4111yescharacteristic
570C42C2×#4111yescharacteristic
571C42C2×#4111yescharacteristic
572C42C2511yescharacteristic
573C23—22no—
574C23—24no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—44no—
579C23—82no—
580C23—82no—
581C23C2×C4×D411yescharacteristic
582C23C2×C4×D411yescharacteristic
583C23C2×C4×D411yescharacteristic
584Q8—24no—
585Q8—44no—
586Q8—42no—
587Q8—82no—
588Q8—81no—
589D4—24no—
590D4—24no—
591D4—24no—
592D4—44no—
593D4—42no—
594D4—48no—
595D4—82no—
596D4—82no—
597D4—82no—
598D4—82no—
599D4—81no—
600D4—84no—
601D4—82no—
602D4—82no—
603D4—82no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—84no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4C2×C4×D411yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C2×C4×D411yescharacteristic
636C2×C4C2×C4×D411yescharacteristic
637C2×C4C2×C4×D411yescharacteristic
638C2×C4C2×C4×D411yescharacteristic
639C2×C4C2×C4×D411yescharacteristic
640C2×C4C2×C4×D411yescharacteristic
641C2×C4C2×C4×D411yescharacteristic
642C2×C4C24×C411yescharacteristic
643C2×C4C23×D411yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C8—24no—
646C8—24no—
647C8—24no—
648C8—24no—
649C8—42no—
650C8—42no—
651C8—44no—
652C8—42no—
653C8—42no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—42no—
668C22—44no—
669C22—82no—
670C22—164no—
671C22C22×C4×D411yescharacteristic
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—21no—
681C4—21no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4#357312yes—
687C4#358512yes—
688C2—41no—
689C2—41no—
690C2—41no—
691C2—82no—
692C2—84no—
693C2#6014311yescharacteristic
694C2#6016811yescharacteristic
695C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#60951C2
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
2. #60287C2semidirect
3. #60237C2semidirect
4. #18028C2semidirect
5. #18026C2semidirect
6. #18011C2semidirect
7. #18007C2semidirect
8. #17987C2semidirect
9. #17968C2semidirect
10. #17821C2semidirect
11. #17482C2semidirect
12. #17478C2semidirect
13. #17324C2semidirect
29. #2993C22semidirect
39. #2971C22semidirect
41. #2970C22semidirect
51. #1594C22semidirect
52. #1592C22semidirect
53. #1590C22semidirect
54. #1589C22semidirect
55. #1588C22semidirect
56. #1586C22semidirect
57. #1586C22semidirect
58. #1585C22semidirect
59. #1583C22semidirect
60. #1582C22semidirect
61. #1581C22semidirect
62. #1581C22semidirect
63. #1580C22semidirect
64. SD16⋊C8C22semidirect
65. C8×SD16C22semidirect
66. C8×D8C22semidirect
67. #1573C22semidirect
68. #1573C22semidirect
70. #1569C22semidirect
75. #1494C22semidirect
76. #1494C22semidirect
80. #1477C22semidirect
82. #1453C22semidirect
83. #1451C22semidirect
84. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
#609512
#602872
#602372
#180282
#180262
#180112
#180072
#179872
#179682
#178212
#174822
#174782
#173242