Finite Group Databasebeta

← #6556201

Subgroups — #6556201

Order 512.

1763 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36303C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic, complemented
3#30205C211yesmaximal, characteristic, complemented
4#30188C212yesmaximal, complemented
5#29355C211yesmaximal, characteristic, complemented
6#29111C212yesmaximal, complemented
7#28726C211yesmaximal, characteristic, complemented
8#27978C212yesmaximal, complemented
9#27756C212yesmaximal, complemented
10#27737C212yesmaximal, complemented
11#27713C214yesmaximal, complemented
12#27266C214yesmaximal, complemented
13#23817C211yesmaximal, characteristic, complemented
14C2×#2072C212yesmaximal, complemented
15#22485C212yesmaximal
16D4×#103C212yesmaximal, complemented
17#20558C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×D4×Q8C2211yescharacteristic
20C2×#547C2211yescharacteristic
21C2×#489C2212yes—
22C2×C8×D4C2212yes—
23C22×#103C2214yes—
24#2868C2211yescharacteristic
25#2862C2211yescharacteristic
26#2857C2212yes—
27#2850C2212yes—
28#2568C2211yescharacteristic
29#2567—21no—
30#2552—21no—
31#2552C2212yes—
32#2547C2211yescharacteristic
33#2503C2212yes—
34#2502—22no—
35#2492—21no—
36#2492—21no—
37#2488—22no—
38#2488—22no—
39#2440—22no—
40#2440C2214yes—
41#2436C2212yes—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44#2339—21no—
45#2339—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#124C2211yescharacteristic
48C2×#391C2212yes—
49C2×#384C2212yes—
50C42×D4C2211yescharacteristic
51#2101C2212yes—
52#2101C2214yes—
53#2072C2211yescharacteristic
54#2072C2212yes—
55#2072C2214yes—
56#2072C2212yes—
57#2072C2214yes—
58#2072C2212yes—
59#2072C2212yes—
60#2071C2211yescharacteristic
61#2071C2212yes—
62#2042C2212yes—
63#2042C2214yes—
64#2024C2211yescharacteristic
65#2024C2214yes—
66#2024C2212yes—
67#2023C2211yescharacteristic
68#1993C2212yes—
69#1990C2212yes—
70#1988C2212yes—
71#1981C2212yes—
72#1959C2214yes—
73#1956C2212yes—
74#1935C2214yes—
75#1935C2214yes—
76#1935C2214yes—
77#1893C2214yes—
78#1892C2214yes—
79#1874C2214yes—
80#1872C2214yes—
81#1836C2214yes—
82#1823C2214yes—
83#1803C2212yes—
84#1802C2212yes—
85#1787C2214yes—
86#1770C2212yes—
87#1767C2211yescharacteristic
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1752C2214yes—
91#1749C2212yes—
92#1744C2211yescharacteristic
93C2×#340C2214yes—
94C2×#336C2212yes—
95C22×#41—21no—
96C23×Q8C2312yes—
97C23×C8C2314yes—
98#554—48no—
99D4×Q8—28no—
100#547—28no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#125—22no—
109C2×#124C2311yescharacteristic
110C2×#124C2312yes—
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2312yes—
117C2×#121C2312yes—
118C2×C4×Q8—21no—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2314yes—
130C22×C42C2312yes—
131#489—44no—
132C8×D4—28no—
133C2×C4⋊C8C2312yes—
134C2×#103—48no—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147C2×#97C2312yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#400—22no—
151#400—22no—
152#398—24no—
153#398—24no—
154#397—22no—
155#395—24no—
156#393—22no—
157#393—22no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—22no—
162#391—21no—
163#391—21no—
164#391—22no—
165#391—22no—
166#391C2312yes—
167#391C2312yes—
168#391C2311yescharacteristic
169#391C2312yes—
170#391C2312yes—
171#390C2311yescharacteristic
172#389—22no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389C2312yes—
177#389C2314yes—
178#388—22no—
179#388—22no—
180#388—21no—
181#388—21no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#385—24no—
187#385—24no—
188#384—22no—
189#384—21no—
190#384—21no—
191#384—22no—
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#384C2314yes—
196#382—21no—
197#382—21no—
198#382C2312yes—
199#382C2311yescharacteristic
200#381—22no—
201#381—22no—
202#380—22no—
203#380—21no—
204#380—21no—
205#380C2314yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C2×#95—22no—
217C2×#95—22no—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2312yes—
234C2×#41—44no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—24no—
242C22×Q8—28no—
243C22×Q8—42no—
244C22×Q8—44no—
245C22×Q8C2412yes—
246C22×Q8C2411yescharacteristic
247C22×D4—22no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2411yescharacteristic
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C22×C8—28no—
258C22×C8—44no—
259C22×C8—44no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—82no—
271#125—48no—
272#124—22no—
273#124—41no—
274#124—44no—
275#123—48no—
276C22⋊Q8—28no—
277C22⋊Q8—416no—
278C22⋊Q8—48no—
279C22⋊Q8—44no—
280C22⋊Q8—48no—
281#121—28no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—44no—
288C4×D4—48no—
289C4×D4#4118yes—
290C2×C4⋊C4—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—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
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—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—84no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—44no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C4⋊C8—24no—
350#103—24no—
351#103—28no—
352#103—44no—
353#103—44no—
354#103—42no—
355#103—44no—
356#103—44no—
357#102—42no—
358#102—42no—
359#102—44no—
360C22⋊C8—28no—
361#97—44no—
362#97—42no—
363C4×C8—24no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24C22×D412yes—
387#41—816no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—416no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—88no—
409C2×Q8H(5,𝔽2)12yes—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4#12414yes—
418C2×D4AGL(1,ℤ/8)14yes—
419C2×D4#13714yes—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—82no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4118yes—
459C22×C4C2511yescharacteristic
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—
469C2×C8—88no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4#12414yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4#12418yes—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—21no—
519C42—24no—
520C42—22no—
521C42—21no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—44no—
526C42C2×#4114yes—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23C2×#12414yes—
539C23C2×AGL(1,ℤ/8)14yes—
540C23C2×#13714yes—
541C23C23×D411yescharacteristic
542Q8—24no—
543Q8—42no—
544Q8—44no—
545Q8—44no—
546Q8—88no—
547Q8—84no—
548Q8—84no—
549Q8—164no—
550D4—42no—
551D4—42no—
552D4—88no—
553D4#489116yescomplemented
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—88no—
603C2×C4—88no—
604C2×C4—84no—
605C2×C4—168no—
606C2×C4C2×#12412yes—
607C2×C4C2×#12414yes—
608C2×C4C2×#12412yes—
609C2×C4C2×AGL(1,ℤ/8)12yes—
610C2×C4C2×#13712yes—
611C2×C4#57712yes—
612C2×C4#57812yes—
613C2×C4C22×#4112yes—
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4#58512yes—
616C2×C4#58511yescharacteristic
617C8—44no—
618C8—42no—
619C8—42no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C2×#48918yes—
634C22C22×#12411yescharacteristic
635C22#348511yescharacteristic
636C22#348811yescharacteristic
637C22C22×AGL(1,ℤ/8)11yescharacteristic
638C22C22×#13711yescharacteristic
639C22C2×#57711yescharacteristic
640C22C2×#57811yescharacteristic
641C4—24no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—82no—
651C4—82no—
652C4—84no—
653C4—82no—
654C4—81no—
655C4—82no—
656C4—81no—
657C4C2×#48914yes—
658C4#351312yes—
659C4#352112yes—
660C2—28no—
661C2—161no—
662C2#3584711yescharacteristic
663C2C22×#48911yescharacteristic
664C2#6062311yescharacteristic
665C2#6173511yescharacteristic
666C2#6173711yescharacteristic
667C2#6198311yescharacteristic
668C2#6198411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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. #30205C2semidirect
4. #30188C2semidirect
5. #29355C2semidirect
6. #29111C2semidirect
7. #28726C2semidirect
8. #27978C2semidirect
9. #27756C2semidirect
10. #27737C2semidirect
11. #27713C2semidirect
12. #27266C2semidirect
13. #23817C2semidirect
14. C2×#2072C2semidirect
16. D4×#103C2semidirect
17. #20558C2semidirect
553. D4#489semidirect

Maximal subgroups

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

subgroupindex
#363032
#361802
#302052
#301882
#293552
#291112
#287262
#279782
#277562
#277372
#277132
#272662
#238172
C2×#20722
#224852
D4×#1032
#205582