Finite Group Databasebeta

← #6536244

Subgroups — #6536244

Order 512.

1877 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36127C211yesmaximal, characteristic
2#34332C211yesmaximal, characteristic, complemented
3#30176C212yesmaximal, complemented
4#29984C211yesmaximal, characteristic, complemented
5#29302C212yesmaximal, complemented
6#29208C211yesmaximal, characteristic, complemented
7#28202C211yesmaximal, characteristic, complemented
8#27175C214yesmaximal, complemented
9#26566C214yesmaximal, complemented
10#25010C212yesmaximal, complemented
11#24118C212yesmaximal, complemented
12#24060C212yesmaximal, complemented
13#23678C211yesmaximal, characteristic, complemented
14#23576C212yesmaximal, complemented
15#22969C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22671C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C2×#547—21no—
20C2×#532C2212yes—
21C2×#480C2214yes—
22C2×#476C2212yes—
23C2×#440C2214yes—
24C2×C4×SD16C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2547—21no—
28#2537—21no—
29#2537C2212yes—
30#2432—22no—
31#2422—22no—
32#2422C2214yes—
33#2370—21no—
34#2369C2211yescharacteristic
35#2360—22no—
36#2360C2212yes—
37Q8×C4⋊C4C2211yescharacteristic
38D4×C4⋊C4—21no—
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2212yes—
41D4×C4⋊C4C2211yescharacteristic
42#2345C2212yes—
43#2330—21no—
44#2329—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2295C2211yescharacteristic
49C2×#389C2212yes—
50#2275—21no—
51C42×D4C2211yescharacteristic
52#2100C2212yes—
53#2095C2212yes—
54#2084C2212yes—
55#2084C2211yescharacteristic
56#2075C2214yes—
57#2072C2211yescharacteristic
58#2065C2212yes—
59#2057C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2055C2212yes—
63#2041C2212yes—
64#2041C2211yescharacteristic
65#2035C2214yes—
66#2033C2214yes—
67#2024C2212yes—
68#2014C2211yescharacteristic
69#2000C2214yes—
70#1990C2212yes—
71#1954C2212yes—
72#1947C2214yes—
73#1947C2212yes—
74#1944C2214yes—
75#1944C2214yes—
76#1926C2214yes—
77#1924C2214yes—
78#1920C2214yes—
79#1874C2214yes—
80#1863C2214yes—
81#1862C2214yes—
82#1862C2214yes—
83#1853C2214yes—
84#1852C2214yes—
85#1845C2212yes—
86#1831C2211yescharacteristic
87#1817C2212yes—
88#1803C2212yes—
89#1802C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1762C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C22×#41—22no—
96C22×SD16C2312yes—
97#554—24no—
98#554—24no—
99#547—48no—
100#532—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#124—22no—
108C2×#123—24no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2312yes—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121C2314yes—
115C2×C4×Q8—21no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2314yes—
127C2×C4×D4C2311yescharacteristic
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C42C2312yes—
133#480—48no—
134#476—28no—
135#440—48no—
136C4×SD16—28no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155#397—24no—
156#394—24no—
157#394—24no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391C2311yescharacteristic
162#391C2311yescharacteristic
163#390—21no—
164#390—21no—
165#390C2312yes—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2311yescharacteristic
176#389C2311yescharacteristic
177#389C2312yes—
178#389C2314yes—
179#389C2312yes—
180#387—21no—
181#387—22no—
182#387—21no—
183#387—22no—
184#387—22no—
185#387—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—24no—
190#385—24no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#380C2314yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221#376—21no—
222C2×#95—22no—
223C43C2311yescharacteristic
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236C2×#41—48no—
237C2×#41—48no—
238C2×#41—84no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8C2411yescharacteristic
242C22×D4—24no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2×D411yescharacteristic
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—24no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—24no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—22no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264C4⋊Q8#4114yescomplemented
265#127—24no—
266#125—28no—
267#124—48no—
268#123—416no—
269C22⋊Q8—216no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8—48no—
273#121—28no—
274#121—44no—
275#121—48no—
276#121—48no—
277C4×Q8—22no—
278C4×Q8—44no—
279C4×Q8—82no—
280C4×Q8#4114yescomplemented
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—48no—
292C4×D4—44no—
293C4×D4—44no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4—84no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—44no—
358#107—44no—
359#107—44no—
360#106—24no—
361#106—28no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364#103—24no—
365#103—44no—
366#102—28no—
367#102—24no—
368#102—24no—
369#102—24no—
370#102—28no—
371C22⋊C8—28no—
372C22⋊C8—44no—
373#97—44no—
374C4×C8—24no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—24no—
389#95—24no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—22no—
394C24C22×D412yes—
395#41—832no—
396C2×Q8—24no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×Q8C2×#4112yes—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—84no—
422C2×D4AGL(1,ℤ/8)14yes—
423C2×D4H(5,𝔽2)14yes—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—84no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2511yescharacteristic
464SD16—216no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C82−1+414yes—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4C2×#4112yes—
508C4⋊C4C2×#4114yes—
509C22⋊C4—28no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—42no—
527C42—42no—
528C42—41no—
529C42—42no—
530C42—82no—
531C42—82no—
532C42—82no—
533C42#13714yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23C2×AGL(1,ℤ/8)14yes—
550C23C23×D411yescharacteristic
551Q8—24no—
552Q8—42no—
553Q8—84no—
554Q8—88no—
555D4—28no—
556D4—24no—
557D4—24no—
558D4—48no—
559D4—48no—
560D4—42no—
561D4—42no—
562D4—816no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—48no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—84no—
611C2×C4—88no—
612C2×C4—816no—
613C2×C4—88no—
614C2×C4—164no—
615C2×C4#52512yes—
616C2×C4#52512yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4#57612yes—
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4C2×2−1+412yes—
626C8—24no—
627C8—42no—
628C8—44no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22C2×#52511yescharacteristic
644C22#348411yescharacteristic
645C22#349911yescharacteristic
646C22C22×AGL(1,ℤ/8)11yescharacteristic
647C22C22×#13711yescharacteristic
648C22C2×#57611yescharacteristic
649C22#358611yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—41no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—81no—
662C4—82no—
663C4—84no—
664C4—82no—
665C4—81no—
666C4#311214yes—
667C4#349212yes—
668C4#352212yes—
669C2—44no—
670C2—84no—
671C2#3569311yescharacteristic
672C2C2×#311211yescharacteristic
673C2#6059611yescharacteristic
674C2#6158611yescharacteristic
675C2#6159111yescharacteristic
676C2#6248011yescharacteristic
677C2#6250311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34332C2semidirect
3. #30176C2semidirect
4. #29984C2semidirect
5. #29302C2semidirect
6. #29208C2semidirect
7. #28202C2semidirect
8. #27175C2semidirect
9. #26566C2semidirect
10. #25010C2semidirect
11. #24118C2semidirect
12. #24060C2semidirect
13. #23678C2semidirect
14. #23576C2semidirect
15. #22969C2semidirect
16. #22839C2semidirect
17. #22671C2semidirect
264. C4⋊Q8#41semidirect
280. C4×Q8#41semidirect

Maximal subgroups

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

subgroupindex
#361272
#343322
#301762
#299842
#293022
#292082
#282022
#271752
#265662
#250102
#241182
#240602
#236782
#235762
#229692
#228392
#226712