Finite Group Databasebeta

← #6457222

Subgroups — #6457222

Order 512.

2218 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3096C212yesmaximal
2#34269C211yesmaximal, characteristic
3#34183C211yesmaximal, characteristic, complemented
4#27655C214yesmaximal, complemented
5#27654C214yesmaximal, complemented
6#27650C212yesmaximal, complemented
7#27650C212yesmaximal, complemented
8#27645C212yesmaximal, complemented
9#27644C212yesmaximal, complemented
10#23658C211yesmaximal, characteristic, complemented
11#23656C211yesmaximal, characteristic, complemented
12#23628C211yesmaximal, characteristic, complemented
13C2×#2071C211yesmaximal, characteristic, complemented
14#21175C212yesmaximal, complemented
15#20703C212yesmaximal, complemented
16#20650C212yesmaximal, complemented
17#20630C211yesmaximal, characteristic, complemented
18C2×#535C2211yescharacteristic
19C2×#520C2212yes—
20C2×#518C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22#3096—216no—
23C2×#481C2214yes—
24C2×#482C2214yes—
25C2×#482C2214yes—
26C2×#480C2214yes—
27C2×#428C2212yes—
28C2×#427C2212yes—
29C2×#424C2212yes—
30C2×#418C2212yes—
31C22×#102C2212yes—
32C2×#408C2212yes—
33#2367—21no—
34#2367—21no—
35#2366—21no—
36#2366—21no—
37#2358—22no—
38#2358—22no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4—21no—
41D4×C4⋊C4—22no—
42D4×C4⋊C4—22no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45#2324C2212yes—
46#2324C2212yes—
47#2317C2212yes—
48#2317C2214yescomplemented
49#2311C2211yescharacteristic
50C4×#121C2212yes—
51C4×#121C2212yes—
52#2296C2211yescharacteristic
53C2×#390C2211yescharacteristic
54#2285C2211yescharacteristic
55#2285C2212yescomplemented
56C2×#384C2211yescharacteristic
57C2×#384C2212yescomplemented
58#2274C2211yescharacteristic
59#2085C2212yescomplemented
60#2085C2212yescomplemented
61#2084C2211yescharacteristic, complemented
62#2084C2212yescomplemented
63#2084C2212yescomplemented
64#2084C2211yescharacteristic, complemented
65#2083C2211yescharacteristic, complemented
66#2083C2212yescomplemented
67#2083C2212yescomplemented
68#2083C2211yescharacteristic, complemented
69#2073C2214yes—
70#2071C2212yes—
71#2071C2212yes—
72#2071C2212yes—
73#2071C2214yes—
74#2071C2212yes—
75#1949C2214yescomplemented
76#1949C2214yescomplemented
77#1948C2214yescomplemented
78#1947C2214yescomplemented
79#1825C2214yescomplemented
80#1825C2214yescomplemented
81#1824C2214yescomplemented
82#1823C2214yescomplemented
83#1804C2214yes—
84#1804C2214yes—
85#1802C2214yes—
86#1802C2214yes—
87#1798C2214yescomplemented
88#1797C2212yescomplemented
89#1797C2212yescomplemented
90#1736C2212yes—
91#1736C2212yes—
92#1735C2212yes—
93C2×#340C2212yes—
94#1728C2211yescharacteristic
95C2×#336C2211yescharacteristic
96C22×#41C2311yescharacteristic
97C23×D4C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#535—24no—
101#535D418yes—
102C2×#127—21no—
103C2×#124—21no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#120—22no—
107#520—28no—
108#520—28no—
109#518—28no—
110C4×#41—28no—
111C2×C4×Q8—22no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×D4—24no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4D412yes—
121C2×C4×D4D412yes—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×#117C2311yescharacteristic
127C2×#117C2312yes—
128C2×#117C2314yes—
129C2×#117C2311yescharacteristic
130C2×#117C2312yes—
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2314yes—
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—22no—
142C22×C42C2312yes—
143C22×C42C2311yescharacteristic
144#482—216no—
145#482—216no—
146#481—216no—
147#480—216no—
148#428—28no—
149#427—28no—
150C2×#107C2312yes—
151C2×#107C2314yes—
152C2×#107C2312yes—
153C2×#106C2312yes—
154C2×#106C2314yes—
155C2×#106C2312yes—
156#424—28no—
157C2×C4⋊C8C2314yes—
158C2×C4⋊C8C2314yes—
159#418—28no—
160C2×#102—28no—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165#408—28no—
166C2×C22⋊C8C2314yes—
167C2×C22⋊C8C2314yes—
168#391—21no—
169#390—21no—
170#390—22no—
171#390—22no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#389—22no—
177#389—22no—
178#389—24no—
179#389—24no—
180#389C2311yescharacteristic
181#389C2311yescharacteristic
182#389C2311yescharacteristic
183#389C2311yescharacteristic
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—21no—
194#385—21no—
195#385—22no—
196#385—22no—
197#385—24no—
198#385—24no—
199#385—24no—
200#384—21no—
201#384—21no—
202#384—22no—
203#384—22no—
204#384C2314yes—
205#384C2312yes—
206#384C2312yes—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#382C2314yes—
210#382C2312yes—
211#380—21no—
212#380—21no—
213#380—22no—
214#380—22no—
215#380C2312yes—
216#380C2312yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#336C2311yescharacteristic
241#336C2312yes—
242#336C2311yescharacteristic
243#336C2312yes—
244#336C2312yes—
245C25—21no—
246C2×#41—24no—
247C2×#41—24no—
248C2×#41C2×D414yes—
249C2×#41#4114yes—
250C22×Q8C2×D411yescharacteristic
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4—24no—
255C22×D4—22no—
256C22×D4—24no—
257C22×D4C2×D414yes—
258C22×D4C2×D412yes—
259C22×D4C2×D411yescharacteristic
260C22×D4C2412yes—
261C23×C4—21no—
262C23×C4—24no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4C2×D412yes—
270C23×C4C2412yes—
271C23×C4C2411yescharacteristic
272C2×M16—24no—
273C2×M16—24no—
274C2×M16—24no—
275C22×C8—24no—
276C22×C8—24no—
277C22×C8—24no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2414yes—
281#127—28no—
282#124—28no—
283#121—24no—
284#121—24no—
285#121C2×D418yes—
286#121C2×D418yes—
287#120—216no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—48no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—416no—
298C4×D4—48no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—84no—
302C4×D4—84no—
303#117—22no—
304#117—28no—
305#117—24no—
306#117—28no—
307#117—24no—
308#117—28no—
309#117—24no—
310#117C2×D414yes—
311#117#4118yes—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—82no—
335C2×C4⋊C4—84no—
336C2×C4⋊C4—82no—
337C2×C4⋊C4—84no—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—48no—
365C2×C22⋊C4—82no—
366C2×C22⋊C4—82no—
367C2×C22⋊C4C2×D412yes—
368C2×C22⋊C4C2×D412yes—
369C2×C22⋊C4C2×D412yes—
370C2×C42—21no—
371C2×C42—24no—
372C2×C42—28no—
373C2×C42—21no—
374C2×C42—24no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42C2×D412yes—
379C2×C42C2×D412yes—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2412yes—
385#107—24no—
386#107—24no—
387#107—28no—
388#106—24no—
389#106—24no—
390#106—28no—
391C4⋊C8—28no—
392C4⋊C8—28no—
393#102—28no—
394#102—28no—
395#102—28no—
396#102—28no—
397C22⋊C8—28no—
398C22⋊C8—28no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—24no—
408#95—24no—
409#95—21no—
410#95—21no—
411C24—22no—
412C24—24no—
413C24—42no—
414C24—42no—
415C24—81no—
416C24—81no—
417C24—81no—
418C24—81no—
419C24C22×D412yes—
420C24C22×D411yescharacteristic
421#41—216no—
422#41—216no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8#12114yes—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—28no—
429C2×D4—28no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—216no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—48no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4#12118yes—
445C2×D4#12114yes—
446C2×D4C22×D414yes—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—42no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—41no—
469C22×C4—48no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4—82no—
481C22×C4—84no—
482C22×C4—82no—
483C22×C4—84no—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4114yes—
497C22×C42−1+414yes—
498C22×C4C2511yescharacteristic
499M16—48no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—48no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—48no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—81no—
535C4⋊C4—81no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4#12314yes—
541C4⋊C4#12314yes—
542C4⋊C4C22×D412yes—
543C4⋊C4C22×D412yes—
544C22⋊C4—24no—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—42no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C22⋊C4#12318yes—
561C42—24no—
562C42—24no—
563C42—24no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—48no—
569C42—44no—
570C42—42no—
571C42—82no—
572C42—82no—
573C42C2×#4118yes—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—24no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—48no—
592C23—84no—
593C23—84no—
594C23—162no—
595C23—162no—
596C23—162no—
597C23—162no—
598C23C2×#12114yes—
599C23C2×#12112yes—
600C23C2×#12312yes—
601C23#53612yes—
602C23C23×D411yescharacteristic
603Q8—28no—
604D4—216no—
605D4—28no—
606D4—28no—
607D4—44no—
608D4—48no—
609D4—88no—
610D4—88no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—164no—
658C2×C4—164no—
659C2×C4—164no—
660C2×C4—168no—
661C2×C4—162no—
662C2×C4C2×#12114yes—
663C2×C4C2×#12112yes—
664C2×C4C2×#12314yes—
665C2×C4C2×#12314yes—
666C2×C4C2×#12312yes—
667C2×C4#53612yes—
668C2×C4#54718yes—
669C2×C4#55218yes—
670C2×C4#57612yes—
671C2×C4#57712yes—
672C2×C4C23×D411yescharacteristic
673C2×C4C22×#4112yes—
674C2×C4C22×#4111yescharacteristic
675C2×C4C2×2−1+412yes—
676C8—42no—
677C8—42no—
678C8—44no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—168no—
700C22#248614yes—
701C22C22×#12111yescharacteristic
702C22C22×#12311yescharacteristic
703C22C2×#53611yescharacteristic
704C22C2×#57611yescharacteristic
705C22C2×#57711yescharacteristic
706C22#358511yescharacteristic
707C22#358611yescharacteristic
708C4—22no—
709C4—24no—
710C4—22no—
711C4—24no—
712C4—24no—
713C4—42no—
714C4—42no—
715C4—41no—
716C4—41no—
717C4—84no—
718C4—82no—
719C4—82no—
720C4—81no—
721C4—81no—
722C4—82no—
723C4#248614yes—
724C4C2×#54712yes—
725C4C2×#55212yes—
726C2—24no—
727C2—44no—
728C2—84no—
729C2C2×#248611yescharacteristic
730C2#6042811yescharacteristic
731C2#6042911yescharacteristic
732C2#6052111yescharacteristic
733C2#6052511yescharacteristic
734C2#6087711yescharacteristic
735C2#6088511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #34183C2semidirect
4. #27655C2semidirect
5. #27654C2semidirect
6. #27650C2semidirect
7. #27650C2semidirect
8. #27645C2semidirect
9. #27644C2semidirect
10. #23658C2semidirect
11. #23656C2semidirect
12. #23628C2semidirect
13. C2×#2071C2semidirect
14. #21175C2semidirect
15. #20703C2semidirect
16. #20650C2semidirect
17. #20630C2semidirect
48. #2317C22semidirect
55. #2285C22semidirect
57. C2×#384C22semidirect
59. #2085C22semidirect
60. #2085C22semidirect
61. #2084C22semidirect
62. #2084C22semidirect
63. #2084C22semidirect
64. #2084C22semidirect
65. #2083C22semidirect
66. #2083C22semidirect
67. #2083C22semidirect
68. #2083C22semidirect
75. #1949C22semidirect
76. #1949C22semidirect
77. #1948C22semidirect
78. #1947C22semidirect
79. #1825C22semidirect
80. #1825C22semidirect
81. #1824C22semidirect
82. #1823C22semidirect
87. #1798C22semidirect
88. #1797C22semidirect
89. #1797C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30962
#342692
#341832
#276552
#276542
#276502
#276452
#276442
#236582
#236562
#236282
C2×#20712
#211752
#207032
#206502
#206302