Finite Group Databasebeta

← #6570703

Subgroups — #6570703

Order 512.

1742 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36411C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29938C211yesmaximal, characteristic, complemented
4#29906C212yesmaximal, complemented
5#27960C211yesmaximal, characteristic, complemented
6#27862C212yesmaximal, complemented
7#27677C211yesmaximal, characteristic, complemented
8#27615C212yesmaximal, complemented
9#26282C212yesmaximal, complemented
10#26214C214yesmaximal, complemented
11#24992C212yesmaximal, complemented
12#24951C214yesmaximal, complemented
13#23754C211yesmaximal, characteristic, complemented
14#23745C212yesmaximal, complemented
15#22713C212yesmaximal, complemented
16#22503C212yesmaximal, complemented
17#22473C211yesmaximal, characteristic, complemented
18C2×#555C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#436C2212yes—
22#2857C2211yescharacteristic
23#2854C2212yes—
24#2567C2211yescharacteristic
25#2538C2211yescharacteristic
26#2485C2212yes—
27#2474—22no—
28#2461—22no—
29#2422—21no—
30#2414C2212yes—
31#2407—21no—
32#2362—21no—
33#2354—21no—
34#2354C2211yescharacteristic
35Q8×C4⋊C4—21no—
36Q8×C4⋊C4C2212yes—
37D4×C4⋊C4—21no—
38D4×C4⋊C4—22no—
39D4×C4⋊C4C2214yes—
40D4×C4⋊C4C2212yes—
41#2317—21no—
42#2317—22no—
43C4×#121C2211yescharacteristic
44#2296C2212yes—
45#2285C2211yescharacteristic
46#2285C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49#2100C2214yes—
50#2100C2214yes—
51#2099C2212yes—
52#2099C2212yes—
53#2099C2212yes—
54#2099C2211yescharacteristic
55#2099C2212yes—
56#2093C2212yes—
57#2093C2211yescharacteristic
58#2084C2212yes—
59#2083C2214yes—
60#2083C2211yescharacteristic
61#2077C2214yes—
62#2075C2212yes—
63#2071C2214yes—
64#2071C2211yescharacteristic
65#2033C2212yes—
66#1990C2212yes—
67#1987C2212yes—
68#1984C2212yes—
69#1948C2212yes—
70#1947C2212yes—
71#1926C2214yes—
72#1921C2214yes—
73#1921C2214yes—
74#1919C2214yes—
75#1895C2214yes—
76#1874C2212yes—
77#1874C2214yes—
78#1870C2214yes—
79#1854C2214yes—
80#1852C2212yes—
81#1852C2214yes—
82#1834C2212yes—
83#1825C2214yes—
84#1823C2214yes—
85#1817C2212yes—
86#1809C2214yes—
87#1802C2212yes—
88#1797C2211yescharacteristic
89C4×#107C2212yes—
90C4×#102C2212yes—
91C4×C22⋊C8C2211yescharacteristic
92#1736C2214yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C22×M16C2314yes—
97#555—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#124—22no—
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×#121C2311yescharacteristic
106#520—48no—
107C4×#41—28no—
108C2×C4×Q8—22no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—24no—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D412yes—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×#117—22no—
120C2×#117—21no—
121C2×#117—21no—
122C2×#117D411yescharacteristic
123C2×#117D412yes—
124C2×#117C2312yes—
125C2×#117C2312yes—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#436—28no—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×#107C2312yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8C2312yes—
147#397—22no—
148#396—22no—
149#395—22no—
150#394—22no—
151#394—22no—
152#392—22no—
153#391—21no—
154#391—21no—
155#390—21no—
156#390—22no—
157#390C2312yes—
158#390C2311yescharacteristic
159#389—22no—
160#389—24no—
161#389—24no—
162#389—22no—
163#389—21no—
164#389C2312yes—
165#389C2312yes—
166#389C2311yescharacteristic
167#389C2312yes—
168#389C2312yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—21no—
175#387—22no—
176#386—22no—
177#386—21no—
178#385—22no—
179#385—22no—
180#385—24no—
181#385—24no—
182#385—24no—
183#384—21no—
184#384—22no—
185#384—21no—
186#384—21no—
187#384—22no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2314yes—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2311yescharacteristic
194#382—22no—
195#382—21no—
196#380—21no—
197#380—21no—
198#380—22no—
199#380—21no—
200#380C2312yes—
201#380C2314yes—
202#380C2311yescharacteristic
203#380C2312yes—
204#380C2312yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C22⋊C4—21no—
214#340C2314yes—
215#340C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#336C2312yes—
228#336C2314yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232C2×#41—24no—
233C2×#41—24no—
234C2×#41—24no—
235C22×Q8—21no—
236C22×Q8C2×D411yescharacteristic
237C22×D4—21no—
238C22×D4—22no—
239C22×D4C2×D412yes—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2412yes—
251C2×M16—28no—
252C2×M16—44no—
253C2×M16—44no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258#124—48no—
259#123—84no—
260C22⋊Q8—44no—
261C22⋊Q8—82no—
262#121—28no—
263#121—22no—
264#121—41no—
265#121—44no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—48no—
269C4×D4—24no—
270C4×D4—28no—
271C4×D4—28no—
272C4×D4—24no—
273C4×D4—416no—
274C4×D4—48no—
275C4×D4—44no—
276C4×D4#4118yes—
277#117—24no—
278#117—28no—
279#117—28no—
280#117—24no—
281#117—42no—
282#117—48no—
283#117—44no—
284#117—44no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4—84no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—28no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—24no—
356#107—42no—
357#107—44no—
358#106—44no—
359C4⋊C8—24no—
360#102—42no—
361#102—42no—
362#102—44no—
363C22⋊C8—28no—
364C22⋊C8—42no—
365C22⋊C8—42no—
366C4×C8—24no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380C24—21no—
381C24C22×D412yes—
382C24C22×D411yescharacteristic
383#41—232no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—42no—
387C2×Q8—81no—
388C2×Q8—81no—
389C2×Q8#13714yes—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4—81no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4#12314yes—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—82no—
438C22×C4—84no—
439C22×C4—82no—
440C22×C4—84no—
441C22×C4AGL(1,ℤ/8)14yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2×#4118yes—
451C22×C4C2511yescharacteristic
452M16—88no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—84no—
495C4⋊C4—81no—
496C4⋊C4—82no—
497C4⋊C4#12314yes—
498C4⋊C42−1+412yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4#12318yes—
516C42—24no—
517C42—22no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—44no—
523C42—48no—
524C42—44no—
525C42—44no—
526C42—42no—
527C42C2×#4114yes—
528C23—22no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C2×#12314yes—
548C23C2×AGL(1,ℤ/8)12yes—
549C23C23×D411yescharacteristic
550Q8—28no—
551Q8—162no—
552D4—216no—
553D4—28no—
554D4—42no—
555D4—42no—
556D4—84no—
557D4—164no—
558C2×C4—216no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—84no—
607C2×C4—164no—
608C2×C4—168no—
609C2×C4C2×#12314yes—
610C2×C4C2×#12312yes—
611C2×C4C2×#12312yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×AGL(1,ℤ/8)12yes—
614C2×C4C2×#13714yes—
615C2×C4C2×#13712yes—
616C2×C4#57612yes—
617C2×C4C22×#4112yes—
618C2×C4C2×2−1+411yescharacteristic
619C2×C4#58512yes—
620C2×C4#58511yescharacteristic
621C8—42no—
622C8—44no—
623C8—42no—
624C22—22no—
625C22—24no—
626C22—22no—
627C22—24no—
628C22—22no—
629C22—24no—
630C22—42no—
631C22—41no—
632C22—41no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—41no—
637C22—41no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22C22×#12311yescharacteristic
646C22#348911yescharacteristic
647C22#350011yescharacteristic
648C22C22×AGL(1,ℤ/8)11yescharacteristic
649C22C22×#13711yescharacteristic
650C22C2×#57611yescharacteristic
651C22#358611yescharacteristic
652C4—22no—
653C4—24no—
654C4—22no—
655C4—22no—
656C4—24no—
657C4—42no—
658C4—42no—
659C4—41no—
660C4—81no—
661C4—84no—
662C4—82no—
663C4—81no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4#309214yes—
668C4#351512yes—
669C4#352112yes—
670C2—42no—
671C2—44no—
672C2—161no—
673C2#3606711yescharacteristic
674C2C2×#309211yescharacteristic
675C2#6052111yescharacteristic
676C2#6210711yescharacteristic
677C2#6214911yescharacteristic
678C2#6253511yescharacteristic
679C2#6258911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36411C2
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
2. #34183C2semidirect
3. #29938C2semidirect
4. #29906C2semidirect
5. #27960C2semidirect
6. #27862C2semidirect
7. #27677C2semidirect
8. #27615C2semidirect
9. #26282C2semidirect
10. #26214C2semidirect
11. #24992C2semidirect
12. #24951C2semidirect
13. #23754C2semidirect
14. #23745C2semidirect
15. #22713C2semidirect
16. #22503C2semidirect
17. #22473C2semidirect

Maximal subgroups

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

subgroupindex
#364112
#341832
#299382
#299062
#279602
#278622
#276772
#276152
#262822
#262142
#249922
#249512
#237542
#237452
#227132
#225032
#224732