Finite Group Databasebeta

← #6556239

Subgroups — #6556239

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36419C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic, complemented
3#30203C211yesmaximal, characteristic, complemented
4#30200C212yesmaximal, complemented
5#29346C211yesmaximal, characteristic, complemented
6#29113C212yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#27978C212yesmaximal, complemented
9#27758C212yesmaximal, complemented
10#27737C212yesmaximal, complemented
11#27720C214yesmaximal, complemented
12#27274C214yesmaximal, complemented
13#23816C211yesmaximal, characteristic, complemented
14#23630C212yesmaximal, complemented
15#22504C212yesmaximal, complemented
16#22432C212yesmaximal
17#20568C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×#554—21no—
20C2×#550C2211yescharacteristic
21C2×#487C2212yes—
22C2×#436C2212yes—
23C2×#417C2214yes—
24#2868C2211yescharacteristic
25#2858C2211yescharacteristic
26#2857C2212yes—
27#2850C2212yes—
28#2568C2211yescharacteristic
29#2567—21no—
30#2559—22no—
31#2559—22no—
32#2552—21no—
33#2552C2212yes—
34#2548C2211yescharacteristic
35#2544—21no—
36#2544—21no—
37#2502—22no—
38#2500C2212yes—
39#2440—22no—
40#2440C2214yes—
41#2432C2212yes—
42#2367—21no—
43#2367—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48#2297C2212yes—
49C2×#384C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2214yes—
52#2100C2212yes—
53#2072C2211yescharacteristic
54#2072C2211yescharacteristic
55#2072C2212yes—
56#2072C2212yes—
57#2072C2214yes—
58#2071C2214yes—
59#2071C2212yes—
60#2071C2212yes—
61#2071C2212yes—
62#2043C2212yes—
63#2042C2214yes—
64#2024C2211yescharacteristic
65#2024C2211yescharacteristic
66#2023C2214yes—
67#2023C2212yes—
68#1990C2212yes—
69#1990C2212yes—
70#1988C2212yes—
71#1982C2212yes—
72#1959C2214yes—
73#1957C2212yes—
74#1936C2214yes—
75#1935C2214yes—
76#1934C2214yes—
77#1895C2214yes—
78#1894C2214yes—
79#1874C2214yes—
80#1872C2214yes—
81#1836C2214yes—
82#1825C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1788C2214yes—
86#1770C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88C4×#103C2212yes—
89C4×#103C2212yes—
90#1753C2214yes—
91#1749C2212yes—
92#1744C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C22×#41C2312yes—
96C22×M16C2314yes—
97#556—28no—
98#554—48no—
99#550—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127—21no—
107C2×#127C2311yescharacteristic
108C2×#126—22no—
109C2×#125—22no—
110C2×#124—21no—
111C2×#124—21no—
112C2×#124—21no—
113C2×#124C2312yes—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8C2312yes—
117C2×#121—22no—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4D411yescharacteristic
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117C2312yes—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2314yes—
132C22×C42C2312yes—
133#487—44no—
134#436—28no—
135C2×C4⋊C8C2312yes—
136#417—48no—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2314yes—
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2312yes—
151#400—22no—
152#398—24no—
153#398—24no—
154#395—24no—
155#394—22no—
156#394—22no—
157#394—22no—
158#394—22no—
159#394—22no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—22no—
164#391C2311yescharacteristic
165#391C2311yescharacteristic
166#391C2312yes—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389C2312yes—
181#389C2314yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—24no—
191#385—24no—
192#385—22no—
193#385—22no—
194#384—22no—
195#384—21no—
196#384—21no—
197#384—22no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#384C2314yes—
203#382C2312yes—
204#380—22no—
205#380C2314yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2314yes—
233#336C2312yes—
234C2×#41—24no—
235C2×#41—28no—
236C2×#41—42no—
237C2×#41—44no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8C2411yescharacteristic
242C22×Q8C2411yescharacteristic
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2412yes—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C2×M16—28no—
255C2×M16—44no—
256C2×M16—44no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—42no—
263C4⋊Q8—42no—
264C4⋊Q8—42no—
265C4⋊Q8—41no—
266C4⋊Q8—44no—
267C4⋊Q8—82no—
268#127—24no—
269#127—44no—
270#126—48no—
271#125—48no—
272#124—44no—
273#124—44no—
274#124—44no—
275#124—44no—
276C22⋊Q8—28no—
277C22⋊Q8—416no—
278C22⋊Q8—48no—
279#121—28no—
280#121—48no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—48no—
288C4×D4—44no—
289C4×D4#4118yes—
290#117—28no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C4⋊C8—24no—
356#103—24no—
357#103—28no—
358#103—42no—
359#103—42no—
360#103—42no—
361#103—44no—
362#102—44no—
363#102—44no—
364#102—44no—
365#102—44no—
366C22⋊C8—28no—
367#97—44no—
368C4×C8—24no—
369C4×C8—42no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—22no—
386C24C22×D412yes—
387#41—416no—
388#41—88no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×Q8H(5,𝔽2)12yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4#12414yes—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—82no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4118yes—
458C22×C4C2511yescharacteristic
459M16—88no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4#12414yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4#12418yes—
517C42—21no—
518C42—24no—
519C42—22no—
520C42—21no—
521C42—42no—
522C42—42no—
523C42—41no—
524C42—41no—
525C42—44no—
526C42—42no—
527C42—42no—
528C42—44no—
529C42AGL(1,ℤ/8)14yes—
530C42#13714yes—
531C42C2×#4114yes—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C2×#12414yes—
546C23C23×D411yescharacteristic
547Q8—24no—
548Q8—42no—
549Q8—42no—
550Q8—42no—
551Q8—84no—
552Q8—84no—
553Q8—164no—
554D4—28no—
555D4—44no—
556D4—44no—
557D4—88no—
558D4—84no—
559D4—162no—
560D4—164no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—88no—
608C2×C4—168no—
609C2×C4C2×#12414yes—
610C2×C4C2×#12412yes—
611C2×C4C2×#12412yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×AGL(1,ℤ/8)12yes—
614C2×C4C2×AGL(1,ℤ/8)12yes—
615C2×C4C2×#13712yes—
616C2×C4C2×#13712yes—
617C2×C4C2×#13712yes—
618C2×C4C22×#4112yes—
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4#58512yes—
621C2×C4#58511yescharacteristic
622C8—44no—
623C8—42no—
624C8—42no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22C22×#12411yescharacteristic
640C22#348511yescharacteristic
641C22#348811yescharacteristic
642C22C22×AGL(1,ℤ/8)11yescharacteristic
643C22C22×#13711yescharacteristic
644C22#358511yescharacteristic
645C22#358611yescharacteristic
646C4—24no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—81no—
656C4—82no—
657C4—84no—
658C4—82no—
659C4—82no—
660C4—81no—
661C4—81no—
662C4#314114yes—
663C4#351312yes—
664C4#352112yes—
665C2—44no—
666C2—162no—
667C2#3584711yescharacteristic
668C2C2×#314111yescharacteristic
669C2#6062511yescharacteristic
670C2#6171711yescharacteristic
671C2#6171911yescharacteristic
672C2#6196311yescharacteristic
673C2#6196411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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. #36180C2semidirect
3. #30203C2semidirect
4. #30200C2semidirect
5. #29346C2semidirect
6. #29113C2semidirect
7. #28717C2semidirect
8. #27978C2semidirect
9. #27758C2semidirect
10. #27737C2semidirect
11. #27720C2semidirect
12. #27274C2semidirect
13. #23816C2semidirect
14. #23630C2semidirect
15. #22504C2semidirect
17. #20568C2semidirect

Maximal subgroups

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

subgroupindex
#364192
#361802
#302032
#302002
#293462
#291132
#287172
#279782
#277582
#277372
#277202
#272742
#238162
#236302
#225042
#224322
#205682