Finite Group Databasebeta

← #6560767

Subgroups — #6560767

Order 512.

1431 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36560C211yesmaximal, characteristic, complemented
2#36312C211yesmaximal, characteristic
3#30383C212yesmaximal
4#30230C211yesmaximal, characteristic, complemented
5#29573C212yesmaximal, complemented
6#29513C212yesmaximal
7#29320C211yesmaximal, characteristic, complemented
8#29022C214yesmaximal
9#28520C211yesmaximal, characteristic
10#28008C214yesmaximal, complemented
11#25472C212yesmaximal, complemented
12#25469C212yesmaximal
13#23817C211yesmaximal, characteristic
14#23782C212yesmaximal, complemented
15#23066C212yesmaximal
16#22970C212yesmaximal, complemented
17#20572C211yesmaximal, characteristic, complemented
18C2×#557—21no—
19C2×#475C2212yes—
20C2×#474C2212yes—
21C2×C8×Q8C2212yes—
22#2868C2211yescharacteristic
23#2860C2212yes—
24#2859C2211yescharacteristic
25#2859C2212yes—
26#2582C2211yescharacteristic
27#2568—21no—
28#2565—22no—
29#2546—21no—
30#2546C2212yes—
31#2538C2211yescharacteristic
32#2529—22no—
33#2519—22no—
34#2508C2212yes—
35#2506—22no—
36#2506C2214yes—
37#2496—21no—
38#2448C2212yes—
39#2423—21no—
40#2365—21no—
41#2365C2212yes—
42#2362—21no—
43#2354C2211yescharacteristic
44Q8×C4⋊C4C2211yescharacteristic
45#2346—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#124C2211yescharacteristic
49#2295C2212yes—
50C42×Q8C2211yescharacteristic
51#2101C2212yes—
52#2100C2212yes—
53#2100C2214yes—
54#2100C2212yes—
55#2095C2212yes—
56#2094C2212yes—
57#2088C2214yes—
58#2086C2214yes—
59#2083C2212yes—
60#2072C2211yescharacteristic
61#2071C2211yescharacteristic
62#2067C2214yes—
63#2066C2214yes—
64#2058C2212yes—
65#2042C2212yes—
66#2024C2211yescharacteristic
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1990C2212yes—
70#1959C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1944C2214yes—
74#1944C2214yes—
75#1942C2214yes—
76#1941C2214yes—
77#1936C2214yes—
78#1934C2214yes—
79#1933C2214yes—
80#1926C2214yes—
81#1921C2214yes—
82#1919C2214yes—
83#1863C2214yes—
84#1847C2212yes—
85#1776C2214yes—
86#1770C2212yes—
87#1770C2214yes—
88#1767C2211yescharacteristic
89#1767C2212yes—
90#1763C2214yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1744C2211yescharacteristic
94#1744C2212yes—
95#557—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#126—22no—
102C2×#125—24no—
103C2×#125—22no—
104C2×#125—21no—
105C2×#124C2311yescharacteristic
106C2×C22⋊Q8—21no—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—21no—
117#475—28no—
118#474—28no—
119C8×Q8—28no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2312yes—
126C2×#103C2314yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#102C2314yes—
130C2×#102C2312yes—
131C2×#102C2312yes—
132C2×#97C2312yes—
133C2×#97C2314yes—
134C2×C4×C8C2314yes—
135C2×C4×C8C2312yes—
136#400—24no—
137#400—22no—
138#399—22no—
139#398—24no—
140#398—24no—
141#398—22no—
142#398—22no—
143#397—22no—
144#396—22no—
145#391—21no—
146#391—22no—
147#391—22no—
148#391—21no—
149#391—22no—
150#391C2312yes—
151#391C2311yescharacteristic
152#390C2312yes—
153#390C2311yescharacteristic
154#390C2312yes—
155#389—21no—
156#389—21no—
157#389—22no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389C2314yes—
164#389C2312yes—
165#389C2312yes—
166#389C2314yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#389C2312yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—21no—
177#388—21no—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—24no—
184#387—24no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#382—22no—
194#382—21no—
195#382—22no—
196#382C2311yescharacteristic
197#380—21no—
198#380—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—21no—
214#376—22no—
215C43C2312yes—
216C43C2311yescharacteristic
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8C2×D411yescharacteristic
235C22×Q8C2411yescharacteristic
236C22×D4C2411yescharacteristic
237C23×C4—21no—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C22×C8C2414yes—
241C4⋊Q8—24no—
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244C4⋊Q8—44no—
245C4⋊Q8—44no—
246C4⋊Q8—48no—
247#126—84no—
248#125—48no—
249#125—416no—
250#125—82no—
251#124—22no—
252#124—41no—
253C22⋊Q8—44no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—42no—
259C4×Q8—42no—
260C4×Q8—48no—
261C4×Q8—44no—
262C4×Q8—48no—
263C4×D4—24no—
264#117—44no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—21no—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2412yes—
324C2×C42C2412yes—
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328#107—28no—
329#107—44no—
330#106—28no—
331#106—28no—
332C4⋊C8—28no—
333C4⋊C8#4118yes—
334#103—24no—
335#103—28no—
336#103—24no—
337#102—28no—
338#102—24no—
339#102—42no—
340#97—44no—
341#97—42no—
342C4×C8—28no—
343C4×C8—24no—
344#95—21no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365C24—21no—
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—42no—
377C2×Q8AGL(1,ℤ/8)14yes—
378C2×Q82−1+412yes—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2511yescharacteristic
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8C2×#4118yes—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C42−1+414yes—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—44no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—24no—
468C42—22no—
469C42—24no—
470C42—24no—
471C42—22no—
472C42—21no—
473C42—22no—
474C42—22no—
475C42—21no—
476C42—42no—
477C42—44no—
478C42—44no—
479C42—48no—
480C42—42no—
481C42—44no—
482C42C2×#4114yes—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C23×D411yescharacteristic
491Q8—28no—
492Q8—24no—
493Q8—48no—
494Q8—42no—
495Q8—88no—
496D4—42no—
497D4—42no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4#53212yes—
547C2×C4#53212yes—
548C2×C4C2×AGL(1,ℤ/8)14yes—
549C2×C4C2×AGL(1,ℤ/8)12yes—
550C2×C4#57512yes—
551C2×C4#57712yes—
552C2×C4#57812yes—
553C2×C4C22×#4112yes—
554C2×C4C2×2−1+412yes—
555C2×C4C2×2−1+411yescharacteristic
556C2×C4#58511yescharacteristic
557C8—28no—
558C8—42no—
559C8—42no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22C2×#53211yescharacteristic
568C22#348611yescharacteristic
569C22#348911yescharacteristic
570C22C22×AGL(1,ℤ/8)11yescharacteristic
571C22C2×#57511yescharacteristic
572C22C2×#57711yescharacteristic
573C22C2×#57811yescharacteristic
574C4—24no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—42no—
579C4—42no—
580C4—44no—
581C4—42no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—81no—
586C4—84no—
587C4—81no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4#318014yes—
592C4#351512yes—
593C4#352312yes—
594C2—161no—
595C2#3594911yescharacteristic
596C2C2×#318011yescharacteristic
597C2#6072311yescharacteristic
598C2#6183611yescharacteristic
599C2#6183711yescharacteristic
600C2#6214611yescharacteristic
601C2#6215011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36560C2semidirect
4. #30230C2semidirect
5. #29573C2semidirect
7. #29320C2semidirect
10. #28008C2semidirect
11. #25472C2semidirect
14. #23782C2semidirect
16. #22970C2semidirect
17. #20572C2semidirect

Maximal subgroups

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

subgroupindex
#365602
#363122
#303832
#302302
#295732
#295132
#293202
#290222
#285202
#280082
#254722
#254692
#238172
#237822
#230662
#229702
#205722