Finite Group Databasebeta

← #6568687

Subgroups — #6568687

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36261C211yesmaximal, characteristic
2#36188C211yesmaximal, characteristic
3#30228C211yesmaximal, characteristic, complemented
4#30199C212yesmaximal, complemented
5#29335C212yesmaximal, complemented
6#29219C211yesmaximal, characteristic, complemented
7#29123C211yesmaximal, characteristic, complemented
8#27968C212yesmaximal, complemented
9#27823C212yesmaximal, complemented
10#27737C212yesmaximal, complemented
11#27722C214yesmaximal, complemented
12#27706C214yesmaximal, complemented
13#23816C211yesmaximal, characteristic, complemented
14#23747C212yesmaximal, complemented
15#22504C212yesmaximal
16#22504C212yesmaximal
17#20568C211yesmaximal, characteristic
18C2×#556C2212yes—
19C2×#554—21no—
20C2×#436C2212yes—
21#2868C2211yescharacteristic
22#2859C2211yescharacteristic
23#2857C2212yes—
24#2857C2212yes—
25#2584—21no—
26#2583—22no—
27#2568C2212yes—
28#2552C2211yescharacteristic
29#2551C2211yescharacteristic
30#2550—21no—
31#2549—22no—
32#2547—21no—
33#2547—21no—
34#2499C2214yes—
35#2496—22no—
36#2496—22no—
37#2440C2212yes—
38#2440C2212yes—
39#2365—21no—
40#2365—21no—
41#2349—21no—
42D4×C4⋊C4C2211yescharacteristic
43D4×C4⋊C4C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C2×#389C2212yes—
47#2285C2212yes—
48C42×D4C2211yescharacteristic
49#2101C2214yes—
50#2101C2214yes—
51#2100C2211yescharacteristic
52#2100C2211yescharacteristic
53#2100C2212yes—
54#2099C2214yes—
55#2099C2212yes—
56#2099C2212yes—
57#2099C2212yes—
58#2099C2212yes—
59#2072C2212yes—
60#2072C2214yes—
61#2043C2211yescharacteristic
62#2043C2211yescharacteristic
63#2043C2212yes—
64#2024C2212yes—
65#2024C2214yes—
66#1993C2214yes—
67#1990C2212yes—
68#1990C2212yes—
69#1990C2212yes—
70#1988C2212yes—
71#1988C2212yes—
72#1935C2214yes—
73#1935C2214yes—
74#1933C2214yes—
75#1894C2214yes—
76#1894C2214yes—
77#1872C2214yes—
78#1872C2214yes—
79#1836C2214yes—
80#1834C2212yes—
81#1825C2214yes—
82#1825C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1770C2212yes—
86C4×C4⋊C8C2211yescharacteristic
87C4×#103C2212yes—
88C4×#103C2212yes—
89#1753C2214yes—
90#1749C2212yes—
91#1744C2211yescharacteristic
92#1736C2214yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×M16C2314yes—
96#556—28no—
97#556—28no—
98#554—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#126—24no—
107C2×#125—21no—
108C2×#125—21no—
109C2×#124—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2314yes—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117C2314yes—
123C22×C4⋊C4—22no—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C42C2312yes—
127#436—28no—
128C2×C4⋊C8C2312yes—
129C2×#103C2312yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2312yes—
139#400—24no—
140#398—22no—
141#398—22no—
142#398—22no—
143#397—24no—
144#397—24no—
145#395—22no—
146#395—22no—
147#395—22no—
148#391—22no—
149#391—22no—
150#391—21no—
151#391—22no—
152#391C2312yes—
153#391C2314yes—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—21no—
167#389—22no—
168#389C2311yescharacteristic
169#389C2311yescharacteristic
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#387—24no—
175#387—24no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#384C2312yes—
189#382—21no—
190#382C2314yes—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380C2312yes—
197#380C2312yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4D412yes—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4C2312yes—
211C2×#95—22no—
212C43C2311yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2314yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2311yescharacteristic
232C22×Q8—22no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8C2412yes—
236C22×D4C2×D411yescharacteristic
237C23×C4—22no—
238C23×C4—22no—
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2412yes—
242C2×M16—28no—
243C2×M16—44no—
244C2×M16—44no—
245C22×C8C2412yes—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C4⋊Q8—22no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—41no—
252C4⋊Q8—44no—
253C4⋊Q8—42no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257#126—416no—
258#125—44no—
259#125—44no—
260#124—48no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—48no—
264C22⋊Q8—48no—
265C22⋊Q8—48no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—48no—
269C4×Q8—44no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—44no—
273C4×D4—44no—
274C4×D4#4118yes—
275#117—28no—
276#117—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C42—24no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—44no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C4⋊C8—24no—
344#103—24no—
345#103—44no—
346#103—42no—
347#103—42no—
348#103—44no—
349#103—42no—
350C22⋊C8—28no—
351#97—44no—
352C4×C8—24no—
353C4×C8—42no—
354#95—21no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—21no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—21no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24C22×D412yes—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—42no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—81no—
387C2×Q8—82no—
388C2×Q8—82no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4#12414yes—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2×#4118yes—
426C22×C4C2511yescharacteristic
427M16—88no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4#12414yes—
481C4⋊C42−1+412yes—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4#12418yes—
489C42—21no—
490C42—21no—
491C42—24no—
492C42—22no—
493C42—42no—
494C42—42no—
495C42—41no—
496C42—41no—
497C42—48no—
498C42—44no—
499C42—42no—
500C42—42no—
501C42AGL(1,ℤ/8)14yes—
502C42#13714yes—
503C42C2×#4114yes—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23C2×#12414yes—
511C23C23×D411yescharacteristic
512Q8—42no—
513Q8—42no—
514Q8—42no—
515Q8—42no—
516Q8—84no—
517Q8—88no—
518Q8—164no—
519D4—28no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—88no—
570C2×C4—164no—
571C2×C4C2×#12414yes—
572C2×C4C2×#12412yes—
573C2×C4C2×#12412yes—
574C2×C4C2×AGL(1,ℤ/8)12yes—
575C2×C4C2×AGL(1,ℤ/8)12yes—
576C2×C4C2×AGL(1,ℤ/8)12yes—
577C2×C4C2×#13712yes—
578C2×C4C2×#13712yes—
579C2×C4C2×#13712yes—
580C2×C4C22×#4112yes—
581C2×C4C2×2−1+411yescharacteristic
582C2×C4#58512yes—
583C2×C4#58511yescharacteristic
584C8—42no—
585C8—44no—
586C8—42no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22C22×#12411yescharacteristic
595C22#348511yescharacteristic
596C22#348911yescharacteristic
597C22C22×AGL(1,ℤ/8)11yescharacteristic
598C22C22×#13711yescharacteristic
599C22#358611yescharacteristic
600C22#358611yescharacteristic
601C4—22no—
602C4—22no—
603C4—22no—
604C4—22no—
605C4—24no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—81no—
611C4—82no—
612C4—81no—
613C4—82no—
614C4—81no—
615C4—84no—
616C4—82no—
617C4—82no—
618C4#314114yes—
619C4#351512yes—
620C4#352112yes—
621C2—44no—
622C2#3604211yescharacteristic
623C2C2×#314111yescharacteristic
624C2#6062711yescharacteristic
625C2#6172211yescharacteristic
626C2#6172311yescharacteristic
627C2#6210911yescharacteristic
628C2#6211311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #30228C2semidirect
4. #30199C2semidirect
5. #29335C2semidirect
6. #29219C2semidirect
7. #29123C2semidirect
8. #27968C2semidirect
9. #27823C2semidirect
10. #27737C2semidirect
11. #27722C2semidirect
12. #27706C2semidirect
13. #23816C2semidirect
14. #23747C2semidirect

Maximal subgroups

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

subgroupindex
#362612
#361882
#302282
#301992
#293352
#292192
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