Finite Group Databasebeta

← #6498204

Subgroups — #6498204

Order 512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33212C212yesmaximal, complemented
4#33048C212yesmaximal
5#30305C211yesmaximal, characteristic, complemented
6#30276C211yesmaximal, characteristic, complemented
7#30228C211yesmaximal, characteristic, complemented
8#30203C211yesmaximal, characteristic, complemented
9#29578C214yesmaximal, complemented
10#29518C214yesmaximal, complemented
11#29163C214yesmaximal, complemented
12#27862C214yesmaximal, complemented
13#22248C212yesmaximal, complemented
14#22219C212yesmaximal, complemented
15#20364C211yesmaximal, characteristic, complemented
16C2×#533—21no—
17C2×#529—21no—
18C2×#500C2212yes—
19C2×#486C2212yes—
20C2×C4×M16C2212yes—
21#2868C2211yescharacteristic
22#2868C2211yescharacteristic
23#2862—21no—
24#2861—21no—
25#2860—21no—
26#2860C2212yes—
27#2859—21no—
28#2859C2211yescharacteristic
29#2859C2212yes—
30#2858C2211yescharacteristic
31#2857—21no—
32#2857—21no—
33#2857C2211yescharacteristic
34#2857C2212yes—
35#2857C2212yes—
36#2856C2211yescharacteristic
37#2854—21no—
38#2854C2212yes—
39#2850—21no—
40C4×C4⋊Q8—21no—
41C4×C4⋊Q8—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×#127C2211yescharacteristic
46C4×#124—21no—
47C4×#121—21no—
48C4×#121C2212yes—
49#2293—21no—
50#2293C2212yes—
51#2285—21no—
52#2285C2212yes—
53C2×C43C2211yescharacteristic
54#2099C2214yes—
55#2099C2214yes—
56#2093C2214yes—
57#2093C2214yes—
58#2084C2214yes—
59#2071C2214yes—
60#2058C2214yes—
61#2023C2214yes—
62#1990C2212yes—
63#1990C2212yes—
64#1990C2212yes—
65#1957C2212yes—
66#1944C2218yes—
67#1936C2218yes—
68#1926C2218yes—
69#1921C2218yes—
70#1921C2218yes—
71#1895C2218yes—
72#1864C2218yes—
73#1844C2212yes—
74#1834C2212yes—
75#1809C2218yes—
76#1745C2212yes—
77#1745C2212yes—
78#1744C2211yescharacteristic
79#1744C2211yescharacteristic
80#1744C2212yes—
81#1728C2212yes—
82#1728C2212yes—
83C22×M16C2314yes—
84#533—48no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×#127C2311yescharacteristic
93#529—48no—
94C2×#126—22no—
95C2×#125—22no—
96C2×#125—21no—
97C2×#124—21no—
98C2×#124—21no—
99C2×#123—22no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—22no—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C2×#117—22no—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#500—28no—
123#486—28no—
124C2×#107C2318yes—
125C2×#106C2318yes—
126C2×#102C2318yes—
127C2×#102C2318yes—
128C4×M16—28no—
129C2×#97C2314yes—
130C2×C4×C8C2312yes—
131C2×C4×C8C2312yes—
132#391—22no—
133#391—22no—
134#391—21no—
135#391—22no—
136#391—22no—
137#391—21no—
138#390—21no—
139#390—21no—
140#390—21no—
141#390—21no—
142#390—21no—
143#390—21no—
144#390C2312yes—
145#390C2312yes—
146#390C2312yes—
147#390C2312yes—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—21no—
153#389—22no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389C2312yes—
159#389C2314yes—
160#389C2314yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#386—22no—
173#386—22no—
174#384—21no—
175#384—21no—
176#384—21no—
177#384—21no—
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2312yes—
181#384C2312yes—
182#384C2312yes—
183#384C2312yes—
184#382—21no—
185#382—22no—
186#382—22no—
187#380—21no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380C2312yes—
192#380C2314yes—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#336C2312yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217#336C2314yes—
218#336C2314yes—
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×D4—22no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2412yes—
226C23×C4—22no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2411yescharacteristic
234C2×M16—28no—
235C2×M16—28no—
236C2×M16—44no—
237C22×C8C2414yes—
238C22×C8C2414yes—
239C4⋊Q8—24no—
240C4⋊Q8—24no—
241C4⋊Q8—24no—
242C4⋊Q8—44no—
243C4⋊Q8—44no—
244C4⋊Q8—44no—
245C4⋊Q8—44no—
246C4⋊Q8—44no—
247#127—24no—
248#126—48no—
249#125—48no—
250#125—44no—
251#124—44no—
252#124—44no—
253#123—48no—
254C22⋊Q8—44no—
255C22⋊Q8—48no—
256C22⋊Q8—48no—
257#121—28no—
258#121—44no—
259#121—44no—
260C4×Q8—44no—
261C4×Q8—48no—
262C4×Q8—48no—
263C4×D4—44no—
264C4×D4—44no—
265C4×D4—48no—
266C4×D4—44no—
267#117—48no—
268#117—48no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—216no—
345#106—48no—
346#102—216no—
347#102—48no—
348#97—28no—
349C4×C8—24no—
350C4×C8#4118yes—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—24no—
360#95—22no—
361C24—22no—
362C24C22×D411yescharacteristic
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—42no—
371C2×Q8—44no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C42−1+414yes—
426C22×C4C2511yescharacteristic
427M16—416no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8C2×#4118yes—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—44no—
498C42—48no—
499C42AGL(1,ℤ/8)14yes—
500C42AGL(1,ℤ/8)14yes—
501C42#13714yes—
502C42C2×#4114yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23#53612yes—
517C23C23×D411yescharacteristic
518Q8—88no—
519Q8—84no—
520Q8—88no—
521D4—48no—
522D4—88no—
523D4—84no—
524D4—84no—
525C2×C4—28no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—88no—
574C2×C4#53212yes—
575C2×C4#53212yes—
576C2×C4#53612yes—
577C2×C4C2×AGL(1,ℤ/8)12yes—
578C2×C4C2×AGL(1,ℤ/8)12yes—
579C2×C4C2×AGL(1,ℤ/8)12yes—
580C2×C4C2×AGL(1,ℤ/8)12yes—
581C2×C4C2×AGL(1,ℤ/8)12yes—
582C2×C4C2×AGL(1,ℤ/8)12yes—
583C2×C4C2×#13712yes—
584C2×C4C2×#13712yes—
585C2×C4C2×#13712yes—
586C2×C4#57512yes—
587C2×C4C22×#4112yes—
588C2×C4C2×2−1+412yes—
589C2×C4#58512yes—
590C8—28no—
591C8—44no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22C2×#53211yescharacteristic
607C22C2×#53611yescharacteristic
608C22#344911yescharacteristic
609C22C22×AGL(1,ℤ/8)11yescharacteristic
610C22C22×AGL(1,ℤ/8)11yescharacteristic
611C22C22×#13711yescharacteristic
612C22C2×#57511yescharacteristic
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—82no—
625C4—84no—
626C4—84no—
627C4—82no—
628C4—82no—
629C4#317214yes—
630C4#318414yes—
631C4#349514yes—
632C2—42no—
633C2—162no—
634C2#3532611yescharacteristic
635C2C2×#317211yescharacteristic
636C2C2×#318411yescharacteristic
637C2#6086011yescharacteristic
638C2#6086411yescharacteristic
639C2#6096511yescharacteristic
640C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #33212C2semidirect
5. #30305C2semidirect
6. #30276C2semidirect
7. #30228C2semidirect
8. #30203C2semidirect
9. #29578C2semidirect
10. #29518C2semidirect
11. #29163C2semidirect
12. #27862C2semidirect
13. #22248C2semidirect
14. #22219C2semidirect
15. #20364C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#366312
#332122
#330482
#303052
#302762
#302282
#302032
#295782
#295182
#291632
#278622
#222482
#222192
#203642