Finite Group Databasebeta

← #6458515

Subgroups — #6458515

Order 512.

1965 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35306C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic, complemented
3#27985C214yesmaximal, complemented
4#27984C214yesmaximal, complemented
5#27980C212yesmaximal, complemented
6#27979C212yesmaximal, complemented
7#27976C212yesmaximal, complemented
8#27975C212yesmaximal, complemented
9#23634C212yesmaximal, complemented
10#23633C212yesmaximal, complemented
11#21205C212yesmaximal, complemented
12#20911C211yesmaximal, characteristic, complemented
13#20910C211yesmaximal, characteristic
14#20908C211yesmaximal, characteristic
15C2×#1873C211yesmaximal, characteristic, complemented
16#20763C212yesmaximal, complemented
17#20379C211yesmaximal, characteristic, complemented
18C2×#536C2211yescharacteristic
19C2×#525C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C2×#417C2212yes—
25C2×#419C2212yes—
26C2×#416C2212yes—
27C22×#103C2212yes—
28C2×#407C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C2×#405C2212yes—
31#2857C2212yes—
32#2850C2212yes—
33#2621C2212yes—
34#2620C2212yes—
35#2417—22no—
36#2415—22no—
37#2413—22no—
38#2412—22no—
39#2317C2212yes—
40#2317C2214yes—
41#2317C2212yes—
42#2317C2211yescharacteristic
43#2316—22no—
44#2315—22no—
45#2311C2211yescharacteristic
46#2310C2211yescharacteristic
47#2296—22no—
48#2296—22no—
49#2285C2212yes—
50#2285C2212yes—
51#2074C2214yescomplemented
52#2074C2214yescomplemented
53#2074C2214yescomplemented
54#2073C2214yescomplemented
55#2072C2214yescomplemented
56#2072C2214yescomplemented
57#2072C2214yescomplemented
58#2071C2214yescomplemented
59#1960C2214yescomplemented
60#1959C2214yescomplemented
61#1958C2214yescomplemented
62#1956C2214yescomplemented
63#1875C2211yescharacteristic
64#1875C2212yes—
65#1875C2211yescharacteristic
66#1875C2212yes—
67#1874C2212yes—
68#1873C2211yescharacteristic
69#1873C2212yes—
70#1873C2212yes—
71#1873C2212yes—
72#1873C2212yes—
73#1873C2212yes—
74#1873C2211yescharacteristic
75#1873C2214yes—
76#1873C2212yes—
77#1872C2214yes—
78#1872C2212yes—
79#1836C2214yescomplemented
80#1836C2214yescomplemented
81#1835C2214yescomplemented
82#1834C2214yescomplemented
83#1750C2214yes—
84#1749C2212yes—
85C4×C22⋊C8C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1735C2212yes—
89C2×#340C2212yes—
90#1729C2212yes—
91C22×#41C2311yescharacteristic
92C23×Q8C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C22×M16C2312yes—
95C23×C8C2312yes—
96#536—28no—
97C2×C4⋊Q8C2312yes—
98C2×#124C2312yes—
99#525—24no—
100#525—24no—
101C2×#123—21no—
102C2×#123—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2311yescharacteristic
112C2×#121C2311yescharacteristic
113C2×#121C2311yescharacteristic
114C2×#120—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×#117—28no—
118C2×#117—21no—
119C2×#117—21no—
120C2×#117—21no—
121C2×#117—21no—
122C2×#117—44no—
123C2×#117—44no—
124C2×#117D412yes—
125C2×#117D412yes—
126C2×#117C2311yescharacteristic
127C2×#117C2312yes—
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C2×#117C2312yes—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2312yes—
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4D412yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—21no—
142C22×C22⋊C4—21no—
143C22×C22⋊C4D412yes—
144C22×C42—21no—
145C22×C42—21no—
146C22×C42C2312yes—
147#419—44no—
148#417—44no—
149#416—44no—
150C2×#103—44no—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161#407—24no—
162C2×C22⋊C8—24no—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2311yescharacteristic
165C2×C22⋊C8C2311yescharacteristic
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C2312yes—
168#405—28no—
169C2×#97C2314yes—
170C2×C4×C8C2314yes—
171#398—24no—
172#397—24no—
173#396—24no—
174#394—24no—
175#391—22no—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#390C2312yes—
180#389—22no—
181#389—22no—
182#389—24no—
183#389—24no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—24no—
188#388—24no—
189#386—24no—
190#386—22no—
191#385—24no—
192#385—24no—
193#385—24no—
194#385—24no—
195#385—22no—
196#384C2314yes—
197#384C2314yes—
198#384C2312yes—
199#382C2314yes—
200#382C2314yes—
201#382C2312yes—
202#381—22no—
203#380—24no—
204#380—24no—
205#380—24no—
206#380—22no—
207C4×C4⋊C4—24no—
208C4×C4⋊C4—24no—
209#340C2314yes—
210#340C2314yes—
211#340C2312yes—
212#340C2312yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2312yes—
218#340C2312yes—
219#336C2314yes—
220#336C2314yes—
221C25C2×D411yescharacteristic
222C2×#41—22no—
223C2×#41—22no—
224C2×#41—24no—
225C2×#41—24no—
226C22×Q8—21no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—24no—
230C22×Q8—24no—
231C22×Q8C2412yes—
232C22×Q8C2411yescharacteristic
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4C2411yescharacteristic
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D414yes—
248C23×C4#4114yes—
249C23×C4#4114yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×M16—24no—
254C2×M16—24no—
255C2×M16—24no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8—24no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C4⋊Q8—28no—
263#124—28no—
264#123—44no—
265#123—48no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—48no—
277#121—22no—
278#121—22no—
279#121—22no—
280#121—22no—
281#120—44no—
282C4×D4—44no—
283C4×D4—44no—
284#117—24no—
285#117—28no—
286#117—28no—
287#117—28no—
288#117—22no—
289#117—22no—
290#117—24no—
291#117—24no—
292#117—44no—
293#117—44no—
294#117—42no—
295#117—42no—
296#117—44no—
297#117—44no—
298#117—44no—
299#117—44no—
300#117#4118yes—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C22⋊C4—28no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D414yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—28no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—44no—
360C2×C42—44no—
361C2×C42C2×D414yes—
362C2×C42C2412yes—
363C2×C42C2412yes—
364#103—44no—
365#103—44no—
366#103—42no—
367#103—42no—
368#103—44no—
369#103—44no—
370#103—44no—
371#102—42no—
372#102—42no—
373#102—44no—
374C22⋊C8—24no—
375C22⋊C8—22no—
376C22⋊C8—22no—
377C22⋊C8—24no—
378C22⋊C8—24no—
379#97—28no—
380C4×C8—28no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389C24—22no—
390C24—22no—
391C24—21no—
392C24—42no—
393C24—42no—
394C24#12414yes—
395C24C22×D411yescharacteristic
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398#41—48no—
399#41—48no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—82no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—28no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4#12414yes—
462C22×C4#12414yes—
463C22×C4#12414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4C2511yescharacteristic
478M16—48no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—48no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4#12318yes—
517C22⋊C4—24no—
518C22⋊C4—28no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—84no—
533C22⋊C4#12318yes—
534C42—28no—
535C42—24no—
536C42—44no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42—42no—
541C42—44no—
542C42C2×#4118yes—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—28no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—84no—
562C23—84no—
563C23C2×#12312yes—
564C23C2×#12412yes—
565C23C2×#12412yes—
566C23C2×#12412yes—
567C23#53512yes—
568C23C23×D411yescharacteristic
569Q8—44no—
570Q8—44no—
571Q8—42no—
572Q8—42no—
573Q8—84no—
574D4—42no—
575D4—42no—
576D4—84no—
577D4—84no—
578D4—84no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—88no—
617C2×C4—88no—
618C2×C4—84no—
619C2×C4—88no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4C2×#12318yes—
623C2×C4C2×#12312yes—
624C2×C4C2×#12412yes—
625C2×C4C2×#12412yes—
626C2×C4C2×#12412yes—
627C2×C4#53512yes—
628C2×C4#55318yes—
629C2×C4#55318yes—
630C2×C4C22×#4112yes—
631C2×C4C22×#4111yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4C2×H(5,𝔽2)12yes—
634C8—42no—
635C8—42no—
636C8—44no—
637C22—24no—
638C22—24no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—88no—
652C22—82no—
653C22—82no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22#250414yes—
658C22C22×#12311yescharacteristic
659C22C22×#12411yescharacteristic
660C22C2×#53511yescharacteristic
661C22#358412yes—
662C22#358612yes—
663C4—28no—
664C4—22no—
665C4—22no—
666C4—44no—
667C4—84no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4—82no—
675C4#250414yes—
676C4C2×#55312yes—
677C4C2×#55312yes—
678C2—24no—
679C2—42no—
680C2—42no—
681C2—161no—
682C2C2×#250411yescharacteristic
683C2#6052412yes—
684C2#6061211yescharacteristic
685C2#6062011yescharacteristic
686C2#6081612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34052C2semidirect
3. #27985C2semidirect
4. #27984C2semidirect
5. #27980C2semidirect
6. #27979C2semidirect
7. #27976C2semidirect
8. #27975C2semidirect
9. #23634C2semidirect
10. #23633C2semidirect
11. #21205C2semidirect
12. #20911C2semidirect
15. C2×#1873C2semidirect
16. #20763C2semidirect
17. #20379C2semidirect
51. #2074C22semidirect
52. #2074C22semidirect
53. #2074C22semidirect
54. #2073C22semidirect
55. #2072C22semidirect
56. #2072C22semidirect
57. #2072C22semidirect
58. #2071C22semidirect
59. #1960C22semidirect
60. #1959C22semidirect
61. #1958C22semidirect
62. #1956C22semidirect
79. #1836C22semidirect
80. #1836C22semidirect
81. #1835C22semidirect
82. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#353062
#340522
#279852
#279842
#279802
#279792
#279762
#279752
#236342
#236332
#212052
#209112
#209102
#209082
C2×#18732
#207632
#203792