Finite Group Databasebeta

← #6526709

Subgroups — #6526709

Order 512.

2082 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3234C211yesmaximal, characteristic, complemented
2C2×#3100C211yesmaximal, characteristic
3C2×#2566C211yesmaximal, characteristic, complemented
4#34236C211yesmaximal, characteristic, complemented
5#28117C212yesmaximal, complemented
6#28114C212yesmaximal, complemented
7#27651C212yesmaximal, complemented
8#27644C212yesmaximal, complemented
9#25462C214yesmaximal, complemented
10#25330C214yesmaximal, complemented
11#24820C214yesmaximal, complemented
12#24684C214yesmaximal, complemented
13#21953C211yesmaximal, characteristic, complemented
14#21836C211yesmaximal, characteristic, complemented
15#20739C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3234—28no—
20C2×#501C2212yescomplemented
21C2×#490C2212yescomplemented
22#3100—28no—
23C2×#481C2212yes—
24C2×#480C2212yes—
25C2×#480C2212yes—
26C2×#478C2214yescomplemented
27C2×#474C2212yes—
28C2×#459C2212yes—
29C2×#447C2212yes—
30C2×C4×D8C2214yes—
31C2×#432C2211yescharacteristic
32C2×#431C2211yescharacteristic
33#2566—21no—
34#2566—21no—
35#2566—21no—
36#2566—21no—
37#2566C2214yes—
38#2566C2214yes—
39C2×#396—22no—
40#2369C2212yescomplemented
41#2366C2212yescomplemented
42#2361—22no—
43#2357C2212yescomplemented
44#2349C2212yescomplemented
45#2339—22no—
46#2337—21no—
47#2337—21no—
48#2319—21no—
49C4×#121C2212yes—
50C4×#121C2212yes—
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#385—22no—
54C2×#384C2211yescharacteristic
55#2283C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2275—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59#2085C2214yescomplemented
60#2083C2214yescomplemented
61#2073C2214yescomplemented
62#2071C2214yescomplemented
63#2067C2214yes—
64#2065C2214yes—
65#2054C2214yes—
66#2054C2214yes—
67#1991C2212yescomplemented
68#1991C2212yescomplemented
69#1989C2214yescomplemented
70#1949C2214yescomplemented
71#1947C2214yescomplemented
72#1940C2214yescomplemented
73#1939C2214yescomplemented
74#1926C2214yes—
75#1925C2214yes—
76#1879C2214yes—
77#1879C2214yes—
78#1824C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1804C2212yes—
82#1804C2212yes—
83#1804C2214yescomplemented
84#1802C2214yescomplemented
85#1797C2212yes—
86#1797C2212yes—
87#1786C2212yes—
88#1762C2214yes—
89C22×#41—21no—
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C22×D8C2314yes—
93#532—24no—
94#532D418yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#125—21no—
98C2×#125—21no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125C2312yes—
102C2×#125C2312yes—
103C2×#124—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—24no—
113C2×C4×D4—24no—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4D412yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—22no—
128C22×C22⋊C4—22no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#501—28no—
133#490—28no—
134#481—28no—
135#480—28no—
136#480—28no—
137#478—216no—
138#474—28no—
139#459—28no—
140#447—28no—
141C4×D8—216no—
142#432—24no—
143#431—24no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×#97C2312yes—
160C2×C4×C8C2312yes—
161#396—24no—
162#396—24no—
163#396—24no—
164#396—24no—
165#391—22no—
166#391—22no—
167#390—22no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#390C2312yes—
174#390C2314yes—
175#390C2314yes—
176#389—22no—
177#389C2312yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—22no—
189#385—24no—
190#385—24no—
191#385—24no—
192#385—24no—
193#384—21no—
194#384—21no—
195#384C2314yes—
196#383—21no—
197#382—21no—
198#382—21no—
199#382C2314yes—
200#382C2311yescharacteristic
201#382C2311yescharacteristic
202#381—21no—
203#381—22no—
204#380—22no—
205#380C2312yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220#376—21no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232C25—21no—
233C2×#41—28no—
234C2×#41—84no—
235C22×Q8—21no—
236C22×D4—22no—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—42no—
244C22×D4C2412yes—
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—44no—
255C23×C4—84no—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×D8—28no—
262C2×D8—28no—
263C2×D8—44no—
264C22×C8C2414yes—
265C22×C8C2414yes—
266C4⋊Q8—22no—
267C4⋊Q8C2×D414yes—
268#127—22no—
269#127C2×D414yes—
270#125—24no—
271#125—24no—
272#125—24no—
273#125—24no—
274#125—44no—
275#125—44no—
276#124—28no—
277#121—28no—
278#121—24no—
279#121C2×D418yes—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—48no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—84no—
292C4×D4—84no—
293C4×D4C2×D418yes—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—48no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—84no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—42no—
348C2×C42—42no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D414yes—
356C2×C42#4114yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—28no—
362#107—28no—
363#106—28no—
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367C4⋊C8—44no—
368#102—28no—
369#102—28no—
370#102—28no—
371#102—28no—
372#102—28no—
373#102—44no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376#97—24no—
377C4×C8—24no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—24no—
395C24—22no—
396C24—22no—
397C24—44no—
398C24—42no—
399C24—42no—
400C24—44no—
401C24C22×D411yescharacteristic
402#41—816no—
403C2×Q8—24no—
404C2×Q8—82no—
405C2×D4—24no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—82no—
425C2×D4C22×D414yes—
426C2×D42−1+414yes—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—816no—
463C22×C4#12114yes—
464C22×C4#12114yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C2×#4112yes—
476C22×C4C2511yescharacteristic
477D8—416no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4C22×D414yes—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—24no—
526C42—24no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—44no—
532C42—82no—
533C42—82no—
534C42#12114yes—
535C42#12114yes—
536C42C22×D412yes—
537C42C22×D412yes—
538C42C2×#4112yes—
539C42C2×#4112yes—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—24no—
544C23—24no—
545C23—48no—
546C23—416no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—88no—
558C23—84no—
559C23—84no—
560C23C2×#12112yes—
561C23#57614yes—
562C23C23×D411yescharacteristic
563Q8—84no—
564D4—24no—
565D4—28no—
566D4—24no—
567D4—48no—
568D4—48no—
569D4—416no—
570D4—42no—
571D4—44no—
572D4—42no—
573D4—84no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—88no—
612C2×C4—88no—
613C2×C4—88no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—164no—
619C2×C4—164no—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4C2×#12112yes—
624C2×C4C2×#12112yes—
625C2×C4#57514yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×2−1+412yes—
629C2×C4#58512yes—
630C8—44no—
631C8—44no—
632C22—24no—
633C22—22no—
634C22—24no—
635C22—22no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—48no—
640C22—44no—
641C22—816no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—84no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22#304114yes—
650C22C22×#12111yescharacteristic
651C22#348511yescharacteristic
652C22#348611yescharacteristic
653C22C2×#57512yes—
654C22C2×#57612yes—
655C4—22no—
656C4—22no—
657C4—24no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—44no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#348912yes—
671C4#349512yes—
672C2—24no—
673C2—84no—
674C2—84no—
675C2#3563311yescharacteristic
676C2#6041511yescharacteristic
677C2C2×#304111yescharacteristic
678C2#6172712yes—
679C2#6183512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3100C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3234C2semidirect
3. C2×#2566C2semidirect
4. #34236C2semidirect
5. #28117C2semidirect
6. #28114C2semidirect
7. #27651C2semidirect
8. #27644C2semidirect
9. #25462C2semidirect
10. #25330C2semidirect
11. #24820C2semidirect
12. #24684C2semidirect
13. #21953C2semidirect
14. #21836C2semidirect
15. #20739C2semidirect
20. C2×#501C22semidirect
21. C2×#490C22semidirect
26. C2×#478C22semidirect
40. #2369C22semidirect
41. #2366C22semidirect
43. #2357C22semidirect
44. #2349C22semidirect
59. #2085C22semidirect
60. #2083C22semidirect
61. #2073C22semidirect
62. #2071C22semidirect
67. #1991C22semidirect
68. #1991C22semidirect
69. #1989C22semidirect
70. #1949C22semidirect
71. #1947C22semidirect
72. #1940C22semidirect
73. #1939C22semidirect
83. #1804C22semidirect
84. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32342
C2×#31002
C2×#25662
#342362
#281172
#281142
#276512
#276442
#254622
#253302
#248202
#246842
#219532
#218362
#207392