Finite Group Databasebeta

← #6802860

Subgroups — #6802860

Order 512.

1361 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36599C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic
3#33109C211yesmaximal, characteristic
4#32403C214yesmaximal, complemented
5#31599C214yesmaximal, complemented
6#29756C212yesmaximal
7#28836C212yesmaximal, complemented
8#27919C212yesmaximal
9#27905C211yesmaximal, characteristic
10#27757C212yesmaximal, complemented
11#27653C212yesmaximal
12#27647C211yesmaximal, characteristic
13#26527C212yesmaximal, complemented
14#26089C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic
16#22084C212yesmaximal
17C4×#484C211yesmaximal, characteristic
18C2×#484C2211yescharacteristic
19C2×#484C2212yes—
20C2×#483C2212yes—
21C2×#483C2212yes—
22C2×#483C2212yes—
23#2857C2211yescharacteristic
24#2857C2212yes—
25#2585—22no—
26#2571—22no—
27#2571—21no—
28#2571—21no—
29#2571C2212yes—
30#2571C2214yes—
31#2571C2212yes—
32#2569—22no—
33#2552C2211yescharacteristic
34#2546C2211yescharacteristic
35#2511C2212yes—
36#2509—21no—
37#2446C2212yes—
38#2442—21no—
39C2×#400—22no—
40#2370—21no—
41#2365C2211yescharacteristic
42#2355C2211yescharacteristic
43#2345—21no—
44C4×C22⋊Q8C2212yes—
45C4×C22⋊Q8C2211yescharacteristic
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384C2211yescharacteristic
49C2×#382—21no—
50C2×#382C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2097C2214yes—
54#2097C2214yes—
55#2096C2214yes—
56#2085C2214yes—
57#2084C2214yes—
58#2084C2212yes—
59#2074C2214yes—
60#2072C2212yes—
61#2066C2214yes—
62#2066C2214yes—
63#2043C2214yes—
64#1981C2212yes—
65#1960C2214yes—
66#1956C2212yes—
67#1949C2214yes—
68#1948C2212yes—
69#1947C2212yes—
70#1944C2214yes—
71#1939C2212yes—
72#1935C2214yes—
73#1928C2212yes—
74#1928C2214yes—
75#1921C2212yes—
76#1920C2214yes—
77#1917C2214yes—
78#1872C2211yescharacteristic
79#1872C2212yes—
80#1863C2214yes—
81#1852C2211yescharacteristic
82#1847C2214yes—
83#1844C2212yes—
84#1824C2211yescharacteristic
85#1823C2212yes—
86#1805C2214yes—
87#1803C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91#1770C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×#125—21no—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2311yescharacteristic
100C2×C4×Q8C2311yescharacteristic
101C2×C4×Q8C2312yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C42—21no—
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113#484—24no—
114#484—44no—
115#483—28no—
116#483—44no—
117#483—44no—
118C2×#107C2312yes—
119C2×#107C2314yes—
120C2×#107C2312yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2312yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2314yes—
133C2×C22⋊C8C2311yescharacteristic
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2311yescharacteristic
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138#400—22no—
139#400—22no—
140#400—22no—
141#400—24no—
142#400—24no—
143#400—24no—
144#400—24no—
145#398—22no—
146#398—22no—
147#398—22no—
148#391C2312yes—
149#391C2312yes—
150#391C2311yescharacteristic
151#391C2312yes—
152#391C2312yes—
153#389—22no—
154#389—22no—
155#389—24no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389C2314yes—
167#389C2314yes—
168#389C2311yescharacteristic
169#389C2312yes—
170#387—22no—
171#387—22no—
172#387—21no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#385—21no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384C2312yes—
183#384C2312yes—
184#384C2312yes—
185#384C2311yescharacteristic
186#382—22no—
187#382—24no—
188#382—22no—
189#382—21no—
190#382—22no—
191#382—22no—
192#382C2314yes—
193#382C2314yes—
194#380—21no—
195#380—22no—
196#380—22no—
197#380—21no—
198#380C2311yescharacteristic
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95D412yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8C2411yescharacteristic
234C22×Q8C2412yes—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2411yescharacteristic
243C22×C8C2412yes—
244C22×C8C2414yes—
245C22×C8C2412yes—
246C4⋊Q8—24no—
247C4⋊Q8—44no—
248#125—44no—
249C22⋊Q8—24no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—24no—
253C4×Q8—24no—
254C4×Q8—44no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4—84no—
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2411yescharacteristic
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C42—24no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2412yes—
318C2×C42C2412yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321#107—24no—
322#107—42no—
323#107—44no—
324#106—28no—
325#106—44no—
326#106—44no—
327C4⋊C8—24no—
328C4⋊C8—44no—
329#103—28no—
330#103—24no—
331#103—44no—
332#103—42no—
333#103—44no—
334#103—44no—
335#103—44no—
336C22⋊C8—22no—
337C22⋊C8—24no—
338C22⋊C8#4114yes—
339#97—44no—
340C4×C8—24no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—21no—
350#95—21no—
351#95—22no—
352#95—22no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—22no—
358#95—22no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368C24C22×D411yescharacteristic
369C2×Q8—22no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4#13714yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4112yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C4C2511yescharacteristic
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8C2×#4114yes—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C22⋊C4—44no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—24no—
478C42—24no—
479C42—42no—
480C42—42no—
481C42—44no—
482C42—42no—
483C42—44no—
484C42—42no—
485C42—82no—
486C42—82no—
487C422−1+414yes—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#55012yes—
493C23C2×#13712yes—
494C23#57712yes—
495C23C23×D411yescharacteristic
496Q8—42no—
497Q8—44no—
498Q8—42no—
499Q8—44no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—88no—
557C2×C4#53612yes—
558C2×C4#53612yes—
559C2×C4#55012yes—
560C2×C4C2×#13712yes—
561C2×C4C2×#13712yes—
562C2×C4#57512yes—
563C2×C4C22×#4111yescharacteristic
564C2×C4C2×H(5,𝔽2)11yescharacteristic
565C2×C4C2×2−1+412yes—
566C2×C4#58512yes—
567C8—24no—
568C8—42no—
569C8—44no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22#340014yes—
577C22C2×#53611yescharacteristic
578C22C2×#55011yescharacteristic
579C22#348511yescharacteristic
580C22C22×#13711yescharacteristic
581C22C2×#57511yescharacteristic
582C22C2×#57711yescharacteristic
583C22#358611yescharacteristic
584C4—22no—
585C4—22no—
586C4—22no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—41no—
592C4—81no—
593C4—82no—
594C4—81no—
595C4—84no—
596C4—82no—
597C4—81no—
598C4—82no—
599C4—81no—
600C4—82no—
601C4#323614yes—
602C4#351512yes—
603C4#355912yes—
604C2—24no—
605C2#3913411yescharacteristic
606C2C2×#323611yescharacteristic
607C2#6087211yescharacteristic
608C2C2×#340011yescharacteristic
609C2#6140311yescharacteristic
610C2#6172311yescharacteristic
611C2#6173111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #32403C2semidirect
5. #31599C2semidirect
7. #28836C2semidirect
10. #27757C2semidirect
13. #26527C2semidirect
14. #26089C2semidirect

Maximal subgroups

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

subgroupindex
#365992
C2×#25712
#331092
#324032
#315992
#297562
#288362
#279192
#279052
#277572
#276532
#276472
#265272
#260892
#226292
#220842
C4×#4842