Finite Group Databasebeta

← #6986870

Subgroups — #6986870

Order 512.

1681 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36492C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic, complemented
3#32789C212yesmaximal, complemented
4#32275C214yesmaximal, complemented
5#32020C214yesmaximal, complemented
6#30295C212yesmaximal, complemented
7#30166C212yesmaximal, complemented
8#29938C211yesmaximal, characteristic, complemented
9#29163C212yesmaximal, complemented
10#28713C212yesmaximal, complemented
11#26283C212yesmaximal, complemented
12#25835C212yesmaximal, complemented
13#25811C211yesmaximal, characteristic, complemented
14#25347C212yesmaximal, complemented
15#24298C211yesmaximal, characteristic, complemented
16#23754C211yesmaximal, characteristic, complemented
17#22473C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C2×#536—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#498C2212yes—
23C2×#474C2212yes—
24C2×#436C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2851C2212yes—
29#2621—22no—
30#2563C2211yescharacteristic
31#2562C2212yes—
32#2547—21no—
33#2547C2212yes—
34#2546—21no—
35#2546C2212yes—
36#2537C2211yescharacteristic
37#2535C2212yes—
38#2486—22no—
39#2486C2214yes—
40#2477—22no—
41#2463—22no—
42#2421—21no—
43#2407—21no—
44#2361—21no—
45#2355C2211yescharacteristic
46#2352—21no—
47#2310—21no—
48C4×#121C2211yescharacteristic
49C4×#121C2212yes—
50#2285C2211yescharacteristic
51#2101C2214yes—
52#2099C2211yescharacteristic
53#2099C2212yes—
54#2093C2211yescharacteristic
55#2085C2214yes—
56#2084C2212yes—
57#2083C2211yescharacteristic
58#2077C2214yes—
59#2077C2214yes—
60#2077C2214yes—
61#2076C2214yes—
62#2075C2212yes—
63#2071C2211yescharacteristic
64#2059C2214yes—
65#2059C2214yes—
66#2058C2212yes—
67#2058C2212yes—
68#2057C2212yes—
69#2034C2214yes—
70#2033C2212yes—
71#2023C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1987C2212yes—
75#1984C2212yes—
76#1948C2212yes—
77#1921C2214yes—
78#1920C2214yes—
79#1895C2214yes—
80#1874C2212yes—
81#1870C2214yes—
82#1864C2214yes—
83#1864C2214yes—
84#1853C2214yes—
85#1852C2212yes—
86#1817C2212yes—
87#1809C2214yes—
88#1809C2214yes—
89#1809C2214yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C22×M16C2314yes—
98#554—44no—
99#536—48no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#125C2312yes—
108C2×#124—22no—
109C2×#123—24no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—22no—
115C2×#121—21no—
116C2×#121C2311yescharacteristic
117C2×#121C2312yes—
118C4×#41—44no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D412yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2312yes—
129C2×#117—21no—
130C2×#117D411yescharacteristic
131C2×#117D412yes—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#498—44no—
138#474—44no—
139#436—44no—
140C2×#107C2312yes—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2314yes—
153C2×C4×C8C2312yes—
154#400—22no—
155#398—24no—
156#398—24no—
157#397—22no—
158#397—24no—
159#397—24no—
160#394—22no—
161#394—22no—
162#393—22no—
163#392—22no—
164#391—21no—
165#391—22no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391—21no—
170#390—21no—
171#390C2311yescharacteristic
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2311yescharacteristic
176#389C2312yes—
177#389C2314yes—
178#389C2314yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—21no—
183#387—22no—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—21no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385—22no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#382—21no—
196#382—21no—
197#382C2312yes—
198#381—24no—
199#380—21no—
200#380—22no—
201#380—22no—
202#380—21no—
203#380—21no—
204#380C2311yescharacteristic
205#380C2312yes—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209#380C2314yes—
210#380C2314yes—
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C2×#41—24no—
236C2×#41—42no—
237C2×#41—42no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8C2×D411yescharacteristic
245C22×D4—22no—
246C22×D4—21no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2×D412yes—
249C22×D4C2411yescharacteristic
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C23×C4C2412yes—
259C2×M16—28no—
260C2×M16—44no—
261C2×M16—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—48no—
272#125—44no—
273#124—48no—
274#123—416no—
275C22⋊Q8—48no—
276C22⋊Q8—48no—
277C22⋊Q8—48no—
278C22⋊Q8—44no—
279C22⋊Q8—82no—
280#121—22no—
281#121—24no—
282#121—42no—
283#121—41no—
284#121—44no—
285C4×Q8—24no—
286C4×Q8—42no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—42no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—84no—
295#117—44no—
296#117—42no—
297#117—44no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—44no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—24no—
359#107—42no—
360#107—44no—
361#106—44no—
362#106—44no—
363#106—44no—
364C4⋊C8—24no—
365#102—24no—
366#102—24no—
367#102—44no—
368C22⋊C8—24no—
369C22⋊C8—42no—
370C22⋊C8—44no—
371C4×C8—24no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—22no—
389C24—21no—
390C24C22×D412yes—
391C24C22×D411yescharacteristic
392#41—416no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×Q8—816no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—82no—
420C2×D4—81no—
421C2×D42−1+412yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4AGL(1,ℤ/8)14yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2511yescharacteristic
468M16—88no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C82−1+414yes—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—81no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—24no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—42no—
536C42—82no—
537C42H(5,𝔽2)14yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C2×AGL(1,ℤ/8)12yes—
559C23C23×D411yescharacteristic
560Q8—44no—
561Q8—88no—
562Q8—88no—
563Q8—162no—
564D4—24no—
565D4—48no—
566D4—44no—
567D4—42no—
568D4—84no—
569D4—88no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4—84no—
622C2×C4—164no—
623C2×C4#53412yes—
624C2×C4#53412yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4#57512yes—
628C2×C4#57712yes—
629C2×C4C2×H(5,𝔽2)12yes—
630C2×C4C2×2−1+411yescharacteristic
631C2×C4C2×2−1+412yes—
632C2×C4#58511yescharacteristic
633C8—24no—
634C8—42no—
635C8—44no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22C2×#53411yescharacteristic
658C22#348911yescharacteristic
659C22#349911yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C2×#57511yescharacteristic
662C22C2×#57711yescharacteristic
663C22#358611yescharacteristic
664C4—22no—
665C4—22no—
666C4—22no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—42no—
671C4—82no—
672C4—81no—
673C4—81no—
674C4—84no—
675C4—82no—
676C4—82no—
677C4—81no—
678C4—82no—
679C4#319414yes—
680C4#355212yes—
681C4#356112yes—
682C2—42no—
683C2—82no—
684C2—82no—
685C2#4291211yescharacteristic
686C2C2×#319411yescharacteristic
687C2#6075811yescharacteristic
688C2#6210711yescharacteristic
689C2#6214411yescharacteristic
690C2#6247711yescharacteristic
691C2#6251111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36492C2
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. #35316C2semidirect
3. #32789C2semidirect
4. #32275C2semidirect
5. #32020C2semidirect
6. #30295C2semidirect
7. #30166C2semidirect
8. #29938C2semidirect
9. #29163C2semidirect
10. #28713C2semidirect
11. #26283C2semidirect
12. #25835C2semidirect
13. #25811C2semidirect
14. #25347C2semidirect
15. #24298C2semidirect
16. #23754C2semidirect
17. #22473C2semidirect

Maximal subgroups

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

subgroupindex
#364922
#353162
#327892
#322752
#320202
#302952
#301662
#299382
#291632
#287132
#262832
#258352
#258112
#253472
#242982
#237542
#224732