Finite Group Databasebeta

← #6345383

Subgroups — #6345383

Order 512.

1847 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2#60237C211yesmaximal, characteristic, complemented
3#60230C211yesmaximal, characteristic, complemented
4#18064C214yesmaximal, complemented
5#18054C214yesmaximal, complemented
6#18001C214yesmaximal, complemented
7#17994C214yesmaximal, complemented
8#17950C214yesmaximal, complemented
9#17922C214yesmaximal, complemented
10#17816C211yesmaximal, characteristic, complemented
11#17500C211yesmaximal, characteristic, complemented
12C2×#1490C211yesmaximal, characteristic, complemented
13#17327C211yesmaximal, characteristic, complemented
14C2×#532C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16#3100C412yes—
17#3100C414yes—
18#3100C412yes—
19#3100C2212yes—
20#3100C2212yes—
21#3100C2212yes—
22#3100C2212yes—
23C2×#480C412yes—
24C2×#474C412yes—
25C2×#459C412yes—
26C2×#447C412yes—
27#2981C2214yescomplemented
28#2980—22no—
29#2975C2212yescomplemented
30#2974—22no—
31#2974C2214yes—
32#2974C2214yescomplemented
33#2973—22no—
34#2973—22no—
35#2972C2212yescomplemented
36C2×C4×D8C2214yes—
37#2935C2214yescomplemented
38D4×M16—22no—
39D4×M16C2214yescomplemented
40#2932—22no—
41#2920C2211yescharacteristic
42#2920C2211yescharacteristic
43C2×#432C2211yescharacteristic
44C2×#431C2211yescharacteristic
45#2880—22no—
46C2×#407—22no—
47C2×C4×M16C2211yescharacteristic
48#1594C2218yescomplemented
49#1589C2218yescomplemented
50#1586C2218yescomplemented
51#1586C2218yescomplemented
52#1581C2218yescomplemented
53#1581C2218yescomplemented
54#1580C2218yescomplemented
55C8×D8C2218yescomplemented
56#1572C2214yescomplemented
57#1571C2212yescomplemented
58#1570C2212yescomplemented
59#1568C2214yescomplemented
60#1492C2214yescomplemented
61#1490C2212yes—
62#1490C2212yes—
63#1490C2212yescomplemented
64#1490C2212yescomplemented
65#1490C2212yescomplemented
66#1490C2212yescomplemented
67#1484C2214yescomplemented
68C2×D4⋊C8C2214yescomplemented
69#1453C2214yescomplemented
70#1451C2212yescomplemented
71#1451C2212yescomplemented
72C8×M16C2212yescomplemented
73C8×M16C2212yescomplemented
74C22×#41—21no—
75C23×D4C2×C411yescharacteristic
76C24×C4—21no—
77C22×D8C2×C414yes—
78C2×#131—24no—
79C22×M16—21no—
80C22×M16—24no—
81#532—21no—
82#532—21no—
83#532—21no—
84#532—21no—
85#532C2×C412yes—
86#532C2×C412yes—
87#532C2312yes—
88#532C2312yes—
89C2×C4⋊Q8C2×C411yescharacteristic
90C2×#127C2×C411yescharacteristic
91C2×#124—21no—
92C2×#121—21no—
93C2×#121C2×C412yes—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4C2×C411yescharacteristic
102C2×C4×D4C2×C412yes—
103C2×C4×D4C2×C411yescharacteristic
104C2×C4×D4C2×C412yes—
105C2×C4×D4C2312yes—
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2311yescharacteristic
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C22×C4⋊C4C2×C411yescharacteristic
111C22×C22⋊C4—21no—
112C22×C42C2311yescharacteristic
113#480C2×C414yes—
114#480C2×C414yes—
115#480C2×C414yes—
116#480C2×C414yes—
117#474C2×C414yes—
118#474C2×C414yes—
119#474C2×C412yes—
120#474C2×C412yes—
121#474C2×C412yes—
122#474C2×C412yes—
123#459C2×C414yes—
124#459C2×C412yes—
125#459C2×C412yes—
126#459C2×C414yes—
127#459C2×C412yes—
128#459C2×C412yes—
129#447C2×C414yes—
130#447C2×C414yes—
131#447C2×C414yes—
132#447C2×C414yes—
133C4×D8C2×C414yes—
134C4×D8C2×C414yes—
135C4×D8C2×C414yes—
136C4×D8C2×C414yes—
137C4×D8C2314yes—
138C4×D8C2314yes—
139C4×D8C2314yes—
140C4×D8C2314yes—
141#436—24no—
142#436C2318yes—
143#436C2318yescomplemented
144#435—28no—
145#435—28no—
146#435—28no—
147#435—28no—
148#435—28no—
149C8×D4—24no—
150C8×D4C2318yes—
151C8×D4C2318yescomplemented
152#432—21no—
153#432—21no—
154#432—21no—
155#432D412yes—
156#432D412yes—
157#432C2312yes—
158#432C2312yes—
159#432C2312yes—
160#431—21no—
161#431C2×C412yes—
162#431C2×C412yes—
163#431C2312yes—
164#431C2311yescharacteristic
165#431C2311yescharacteristic
166#431C2312yes—
167#431C2312yes—
168#431C2312yes—
169C2×#107C2×C414yes—
170#423D412yes—
171#423C2312yes—
172C2×C4⋊C8C2×C412yes—
173C2×C4⋊C8C2312yes—
174C2×#102C2×C414yes—
175C2×#102C2×C414yes—
176#408—24no—
177#408—28no—
178#407—21no—
179#407—24no—
180#407—24no—
181#407—24no—
182#407—24no—
183#407D412yes—
184C2×C22⋊C8—21no—
185C2×C22⋊C8—24no—
186C2×C22⋊C8—24no—
187C2×C22⋊C8—24no—
188C2×C22⋊C8C2×C412yes—
189C4×M16—21no—
190C4×M16—21no—
191C4×M16—21no—
192C4×M16C2311yescharacteristic
193C4×M16C2311yescharacteristic
194C4×M16C2312yes—
195C4×M16C2312yes—
196C2×#97—21no—
197C2×#97—21no—
198C2×C4×C8C2312yes—
199C2×C4×C8C2312yes—
200#335C2318yescomplemented
201#334C2318yescomplemented
202D4⋊C8C2314yes—
203D4⋊C8C2314yes—
204D4⋊C8C2314yes—
205D4⋊C8C2314yes—
206D4⋊C8C2318yescomplemented
207D4⋊C8C2318yescomplemented
208#322C2314yescomplemented
209#322C2314yescomplemented
210#322C2314yescomplemented
211C82C2314yescomplemented
212C25—21no—
213C2×#41—22no—
214C2×#41—22no—
215C2×#41—22no—
216C2×#41—22no—
217C2×#41—84no—
218C22×Q8—21no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4—44no—
226C22×D4C22×C411yescharacteristic
227C22×D4C22×C412yes—
228C22×D4C22×C412yes—
229C22×D4C22×C411yescharacteristic
230C22×D4C22×C412yes—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—84no—
242C23×C4C22×C411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4#4111yescharacteristic
245C2×D8—48no—
246C2×D8C22×C414yes—
247C2×D8C22×C414yes—
248C2×D8C22×C414yes—
249C2×D8C22×C414yes—
250#131—88no—
251C2×M16—21no—
252C2×M16—22no—
253C2×M16—22no—
254C2×M16—21no—
255C2×M16—28no—
256C2×M16—28no—
257C2×M16—44no—
258C2×M16—88no—
259C2×M16C2×D414yes—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—28no—
263C22×C8—28no—
264C22×C8C22×C414yes—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8C22×C412yes—
268C4⋊Q8C22×C412yes—
269#127—21no—
270#127—21no—
271#127C22×C412yes—
272#127C22×C412yes—
273#124—22no—
274#124—22no—
275#124—22no—
276#124—22no—
277#121—22no—
278#121—22no—
279#121—22no—
280#121—22no—
281#121—22no—
282#121—22no—
283#121C22×C414yes—
284#121C22×C414yes—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—22no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—21no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—21no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—22no—
307C4×D4C22×C412yes—
308C4×D4C22×C412yes—
309C4×D4C22×C412yes—
310C4×D4C22×C412yes—
311C4×D4C22×C412yes—
312C4×D4C22×C412yes—
313C4×D4C22×C414yes—
314C4×D4C2412yes—
315C4×D4C2412yes—
316C4×D4C2414yes—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4C22×C412yes—
322C2×C4⋊C4C22×C412yes—
323C2×C4⋊C4C22×C411yescharacteristic
324C2×C4⋊C4C22×C411yescharacteristic
325C2×C4⋊C4C22×C412yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C42C22×C411yescharacteristic
337C2×C42C22×C411yescharacteristic
338C2×C42C22×C411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42#4111yescharacteristic
343C2×C42#4111yescharacteristic
344C2×C42#4111yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107C22×C414yes—
349#107C22×C414yes—
350#107C22×C414yes—
351#107C22×C414yes—
352C4⋊C8—22no—
353C4⋊C8—22no—
354C4⋊C8C22×C412yes—
355C4⋊C8C22×C412yes—
356C4⋊C8C22×C414yes—
357C4⋊C8C2×D414yes—
358C4⋊C8C2×D414yes—
359C4⋊C8C2414yes—
360C4⋊C8C2414yes—
361#102C22×C414yes—
362#102C22×C414yes—
363#102C22×C414yes—
364#102C22×C414yes—
365#102C22×C414yes—
366#102C22×C414yes—
367#102C22×C414yes—
368#102C22×C414yes—
369C22⋊C8—22no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—28no—
374C22⋊C8—28no—
375C22⋊C8—28no—
376C22⋊C8—28no—
377C22⋊C8C22×C412yes—
378C22⋊C8C22×C412yes—
379C22⋊C8C22×C414yes—
380C22⋊C8C2×D414yes—
381C22⋊C8C2×D414yes—
382#97—22no—
383#97—22no—
384#97—22no—
385#97—21no—
386#97—21no—
387#97—22no—
388#97C2×D414yes—
389C4×C8—22no—
390C4×C8—22no—
391C4×C8C22×C414yes—
392C4×C8C2412yes—
393C4×C8C2412yes—
394C4×C8C2412yes—
395C4×C8C2412yes—
396C4×C8C2414yes—
397C24—21no—
398C24—22no—
399C24—22no—
400C24—21no—
401C24—22no—
402C24—44no—
403C24—82no—
404C24—82no—
405C24C4×D411yescharacteristic
406#41—48no—
407#41—84no—
408#41—84no—
409#41—84no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—82no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4—84no—
437C2×D4C23×C412yes—
438C2×D4C23×C414yes—
439C2×D4C23×C412yes—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—48no—
459C22×C4—84no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—84no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—82no—
468C22×C4C4×D411yescharacteristic
469C22×C4C4×D411yescharacteristic
470C22×C4C4×D411yescharacteristic
471C22×C4C4×D411yescharacteristic
472C22×C4C4×D411yescharacteristic
473C22×C4C4×D411yescharacteristic
474C22×C4C4×D411yescharacteristic
475C22×C4C4×D411yescharacteristic
476C22×C4C4×D411yescharacteristic
477C22×C4C23×C411yescharacteristic
478C22×C4C23×C411yescharacteristic
479C22×C4C23×C411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4111yescharacteristic
484C22×C4C2×#4111yescharacteristic
485C22×C4C2×#4111yescharacteristic
486D8—28no—
487D8—28no—
488D8—44no—
489D8—44no—
490M16—24no—
491M16—28no—
492M16—44no—
493M16—42no—
494M16—44no—
495M16—88no—
496M16—88no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—44no—
502C2×C8—44no—
503C2×C8—44no—
504C2×C8—88no—
505C2×C8—88no—
506C2×C8C23×C414yes—
507C2×C8C23×C414yes—
508C2×C8C22×D418yes—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4C23×C412yes—
517C4⋊C4C23×C412yes—
518C4⋊C4C23×C414yes—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—24no—
531C42C4×D411yescharacteristic
532C42C4×D411yescharacteristic
533C42C4×D411yescharacteristic
534C42C4×D411yescharacteristic
535C42C4×D411yescharacteristic
536C42C4×D411yescharacteristic
537C42C23×C411yescharacteristic
538C42C23×C411yescharacteristic
539C42C23×C411yescharacteristic
540C42C22×D411yescharacteristic
541C42C22×D411yescharacteristic
542C42C22×D411yescharacteristic
543C42C2×#4111yescharacteristic
544C42C2×#4111yescharacteristic
545C42C2×#4111yescharacteristic
546C42C2511yescharacteristic
547C23—24no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—41no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—82no—
561C23—81no—
562C23—81no—
563C23—82no—
564C23—81no—
565C23—81no—
566C23—82no—
567C23—82no—
568C23—816no—
569C23C2×C4×D411yescharacteristic
570C23C2×C4×D411yescharacteristic
571C23C2×C4×D411yescharacteristic
572Q8—44no—
573Q8—82no—
574D4—24no—
575D4—22no—
576D4—24no—
577D4—24no—
578D4—22no—
579D4—22no—
580D4—44no—
581D4—44no—
582D4—42no—
583D4—48no—
584D4—44no—
585D4—82no—
586D4—82no—
587D4—82no—
588D4—82no—
589D4—84no—
590D4#56814yes—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—84no—
622C2×C4—82no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—82no—
626C2×C4C2×C4×D411yescharacteristic
627C2×C4C2×C4×D411yescharacteristic
628C2×C4C2×C4×D411yescharacteristic
629C2×C4C2×C4×D411yescharacteristic
630C2×C4C2×C4×D411yescharacteristic
631C2×C4C2×C4×D411yescharacteristic
632C2×C4C2×C4×D411yescharacteristic
633C2×C4C2×C4×D411yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4#57614yes—
636C2×C4C24×C411yescharacteristic
637C2×C4C23×D411yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C8—24no—
640C8—24no—
641C8—24no—
642C8—28no—
643C8—42no—
644C8—44no—
645C8—44no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—24no—
654C22—44no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—44no—
659C22—44no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—88no—
664C22—88no—
665C22—84no—
666C22—84no—
667C22—164no—
668C22#294812yes—
669C22#295612yes—
670C22C22×C4×D411yescharacteristic
671C4—21no—
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4#294812yes—
684C4C2×#56812yes—
685C4C2×#57612yes—
686C2—22no—
687C2—22no—
688C2—41no—
689C2—44no—
690C2—82no—
691C2—84no—
692C2#6014111yescharacteristic
693C2C2×#294811yescharacteristic
694C2C2×#295611yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radicalC2×#3100C2
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. #60237C2semidirect
3. #60230C2semidirect
4. #18064C2semidirect
5. #18054C2semidirect
6. #18001C2semidirect
7. #17994C2semidirect
8. #17950C2semidirect
9. #17922C2semidirect
10. #17816C2semidirect
11. #17500C2semidirect
12. C2×#1490C2semidirect
13. #17327C2semidirect
27. #2981C22semidirect
29. #2975C22semidirect
32. #2974C22semidirect
35. #2972C22semidirect
37. #2935C22semidirect
39. D4×M16C22semidirect
48. #1594C22semidirect
49. #1589C22semidirect
50. #1586C22semidirect
51. #1586C22semidirect
52. #1581C22semidirect
53. #1581C22semidirect
54. #1580C22semidirect
55. C8×D8C22semidirect
56. #1572C22semidirect
57. #1571C22semidirect
58. #1570C22semidirect
59. #1568C22semidirect
60. #1492C22semidirect
63. #1490C22semidirect
64. #1490C22semidirect
65. #1490C22semidirect
66. #1490C22semidirect
67. #1484C22semidirect
68. C2×D4⋊C8C22semidirect
69. #1453C22semidirect
70. #1451C22semidirect
71. #1451C22semidirect
72. C8×M16C22semidirect
73. C8×M16C22semidirect
143. #436C23semidirect
151. C8×D4C23semidirect
200. #335C23semidirect
201. #334C23semidirect
206. D4⋊C8C23semidirect
207. D4⋊C8C23semidirect
208. #322C23semidirect
209. #322C23semidirect
210. #322C23semidirect
211. C82C23semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
#602372
#602302
#180642
#180542
#180012
#179942
#179502
#179222
#178162
#175002
C2×#14902
#173272