Finite Group Databasebeta

← #7533243

Subgroups — #7533243

Order 512.

1895 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#59815C211yesmaximal, characteristic
2#36594C211yesmaximal, characteristic, complemented
3#31400C214yesmaximal, complemented
4#29219C211yesmaximal, characteristic, complemented
5#28716C211yesmaximal, characteristic, complemented
6#28607C212yesmaximal, complemented
7#28150C212yesmaximal, complemented
8#26298C212yesmaximal, complemented
9#26229C214yesmaximal, complemented
10#25574C212yesmaximal, complemented
11#24966C212yesmaximal, complemented
12#24120C212yesmaximal, complemented
13#23176C211yesmaximal, characteristic, complemented
14#22504C212yesmaximal, complemented
15#22500C212yesmaximal, complemented
16C4×#475C211yesmaximal, characteristic, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#556C2212yes—
19C2×#550—21no—
20C2×#547C2212yes—
21C2×#486C2212yes—
22C2×#475C2211yescharacteristic
23C2×#460C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#436C2212yes—
26C2×#436C2212yes—
27#2868C2211yescharacteristic
28#2858—21no—
29#2682—22no—
30#2547C2212yes—
31#2545—22no—
32#2541C2212yes—
33#2538—21no—
34#2538C2212yes—
35#2538C2212yes—
36#2537C2212yes—
37#2512—22no—
38#2512—22no—
39#2445—22no—
40#2416C2214yes—
41#2407—22no—
42#2354C2212yes—
43#2354C2212yes—
44D4×C4⋊C4C2212yes—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#127C2211yescharacteristic
49C42×Q8C2211yescharacteristic
50#2099C2214yes—
51#2077C2214yes—
52#2077C2214yes—
53#2071C2214yes—
54#2034C2214yes—
55#2034C2214yes—
56#2000C2214yes—
57#2000C2214yes—
58#1990C2212yes—
59#1990C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1988C2211yescharacteristic
63#1982C2211yescharacteristic
64#1954C2212yes—
65#1954C2212yes—
66#1948C2211yescharacteristic
67#1948C2211yescharacteristic
68#1926C2212yes—
69#1919C2212yes—
70#1895C2214yes—
71#1895C2214yes—
72#1894C2214yes—
73#1880C2212yes—
74#1875C2214yes—
75#1875C2214yes—
76#1872C2214yes—
77#1870C2214yes—
78#1865C2212yes—
79#1853C2214yes—
80#1834C2212yes—
81#1825C2214yes—
82#1824C2214yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85#1802C2212yes—
86#1802C2212yes—
87C4×#106C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#102C2212yes—
92#1736C2214yes—
93C22×#41—22no—
94C22×SD16C2312yes—
95C22×M16C2314yes—
96#556—28no—
97#556—44no—
98#550—48no—
99#547—28no—
100#547—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127—22no—
106C2×#127D411yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#126—24no—
109C2×#124—22no—
110C2×#124—22no—
111C2×#123—24no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2314yes—
114C2×#121—22no—
115C2×#121—24no—
116C2×#121C2314yes—
117C2×#121C2314yes—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×#117C2314yes—
131C22×C4⋊C4C2314yes—
132#486—44no—
133#475—24no—
134#460—24no—
135C4×SD16—44no—
136#436—44no—
137#436—44no—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C4×C8C2312yes—
157C2×C4×C8C2312yes—
158#397—24no—
159#396—24no—
160#396—24no—
161#396—24no—
162#394—24no—
163#394—24no—
164#392—24no—
165#391—22no—
166#391—21no—
167#390—22no—
168#390—22no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390—21no—
173#390—21no—
174#390D412yes—
175#390D412yes—
176#390D412yes—
177#390C2312yes—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—21no—
182#389C2312yes—
183#388—24no—
184#388—24no—
185#386—24no—
186#385—24no—
187#384—21no—
188#384—21no—
189#384D411yescharacteristic
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#380—22no—
195#380C2314yes—
196#380C2314yes—
197#380C2314yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C43—21no—
209C43C2311yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2314yes—
216#340C2312yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#336C2312yes—
223#336C2312yes—
224C2×#41—48no—
225C2×#41—48no—
226C2×#41—84no—
227C22×Q8—21no—
228C22×Q8—22no—
229C22×Q8C2411yescharacteristic
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C2×D412yes—
235C22×D4C2×D412yes—
236C22×D4C2×D412yes—
237C22×D4C2412yes—
238C23×C4—24no—
239C23×C4—22no—
240C23×C4C2414yes—
241C2×SD16—24no—
242C2×SD16—42no—
243C2×SD16—42no—
244C2×M16—44no—
245C2×M16—44no—
246C2×M16—44no—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—44no—
255C4⋊Q8—82no—
256C4⋊Q8#4114yes—
257#127—24no—
258#127—22no—
259#127—22no—
260#127—48no—
261#126—416no—
262#124—48no—
263#124—84no—
264#123—416no—
265C22⋊Q8—28no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268#121—28no—
269#121—44no—
270#121—48no—
271#121—416no—
272#121—44no—
273#121—44no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—41no—
278C4×Q8—41no—
279C4×Q8—48no—
280C4×Q8—44no—
281C4×Q8#4114yescomplemented
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—42no—
289C4×D4—42no—
290C4×D4—48no—
291C4×D4—44no—
292C4×D4—82no—
293#117—28no—
294#117—44no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4C2×D414yes—
334C2×C22⋊C4C2×D414yes—
335C2×C22⋊C4C2×D414yes—
336C2×C42—21no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#106—24no—
352#106—42no—
353#106—42no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358#103—24no—
359#103—42no—
360#103—42no—
361#102—24no—
362#102—42no—
363#102—42no—
364#102—44no—
365#102—44no—
366C22⋊C8—44no—
367C22⋊C8—44no—
368C22⋊C8—44no—
369C4×C8—24no—
370C4×C8—24no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—24no—
379C24—22no—
380C24C22×D414yes—
381#41—832no—
382C2×Q8—24no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8C2×#4112yes—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—81no—
417C2×D4—81no—
418C2×D4—84no—
419C2×D4—84no—
420C2×D4—84no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—84no—
441C22×C4C22×D414yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449SD16—48no—
450M16—88no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—42no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4C2×#4112yes—
495C4⋊C4C2×#4114yes—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—44no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—82no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—21no—
513C42—21no—
514C42—41no—
515C42—48no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—44no—
520C42—42no—
521C42—42no—
522C42AGL(1,ℤ/8)14yes—
523C42C2×#4112yes—
524C42C2×#4112yes—
525C42H(5,𝔽2)12yes—
526C42H(5,𝔽2)12yes—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23C23×D411yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546Q8—48no—
547Q8—88no—
548D4—48no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—48no—
554D4—84no—
555D4—84no—
556D4—816no—
557D4—84no—
558D4—168no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—816no—
600C2×C4—88no—
601C2×C4#54612yes—
602C2×C4#54612yes—
603C2×C4#54612yes—
604C2×C4#54612yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4#57712yes—
609C2×C4#57712yes—
610C2×C4C22×#4111yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C2×H(5,𝔽2)11yescharacteristic
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4#58511yescharacteristic
615C2×C4#58511yescharacteristic
616C8—42no—
617C8—42no—
618C8—42no—
619C8—42no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C2×#54611yescharacteristic
635C22C2×#54611yescharacteristic
636C22#348511yescharacteristic
637C22C22×AGL(1,ℤ/8)11yescharacteristic
638C22C2×#57711yescharacteristic
639C22C2×#57711yescharacteristic
640C22#358611yescharacteristic
641C4—22no—
642C4—22no—
643C4—22no—
644C4—42no—
645C4—41no—
646C4—44no—
647C4—42no—
648C4—44no—
649C4—81no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4—84no—
654C4#332414yes—
655C4#333214yes—
656C4#348011yescharacteristic
657C4#350611yescharacteristic
658C4#351711yescharacteristic
659C4#354611yescharacteristic
660C2—84no—
661C2—162no—
662C2#5903311yescharacteristic
663C2C2×#332411yescharacteristic
664C2C2×#333211yescharacteristic
665C2#6120211yescharacteristic
666C2#6120911yescharacteristic
667C2#6172211yescharacteristic
668C2#6179211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59815C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36594C2semidirect
3. #31400C2semidirect
4. #29219C2semidirect
5. #28716C2semidirect
6. #28607C2semidirect
7. #28150C2semidirect
8. #26298C2semidirect
9. #26229C2semidirect
10. #25574C2semidirect
11. #24966C2semidirect
12. #24120C2semidirect
13. #23176C2semidirect
14. #22504C2semidirect
15. #22500C2semidirect
16. C4×#475C2semidirect
17. C4×#460C2semidirect
281. C4×Q8#41semidirect

Maximal subgroups

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

subgroupindex
#598152
#365942
#314002
#292192
#287162
#286072
#281502
#262982
#262292
#255742
#249662
#241202
#231762
#225042
#225002
C4×#4752
C4×#4602