Finite Group Databasebeta

← #6528572

Subgroups — #6528572

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35103C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#29335C211yesmaximal, characteristic, complemented
4#29334C211yesmaximal, characteristic
5#29330C212yesmaximal, complemented
6#27983C211yesmaximal, characteristic, complemented
7#27982C211yesmaximal, characteristic
8#27977C212yesmaximal, complemented
9#27761C214yesmaximal, complemented
10#27759C214yesmaximal, complemented
11#27745C214yesmaximal, complemented
12#27743C214yesmaximal, complemented
13#21867C212yesmaximal, complemented
14#21391C212yesmaximal, complemented
15#20534C211yesmaximal, characteristic, complemented
16C2×#529C2212yes—
17C2×#432C2212yes—
18#2857C2211yescharacteristic
19#2856C2211yescharacteristic
20#2850C2212yes—
21#2618—21no—
22#2618—21no—
23#2601—22no—
24#2601—22no—
25#2571—22no—
26#2571—22no—
27#2571C2214yes—
28#2571C2214yes—
29#2566C2212yes—
30#2565C2212yes—
31#2545C2212yes—
32#2544C2212yes—
33C2×#400—21no—
34C2×#400—21no—
35#2325—21no—
36#2325—21no—
37C2×#389—21no—
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#382—21no—
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#2101C2214yes—
47#2100C2212yes—
48#2099C2212yes—
49#2074C2214yes—
50#2074C2214yes—
51#2073C2214yes—
52#2073C2214yes—
53#2043C2212yes—
54#2042C2214yes—
55#2041C2212yes—
56#1992C2214yes—
57#1989C2214yes—
58#1983C2214yes—
59#1961C2212yes—
60#1960C2212yes—
61#1959C2214yes—
62#1958C2214yes—
63#1936C2214yes—
64#1935C2214yes—
65#1934C2214yes—
66#1933C2214yes—
67#1874C2212yes—
68#1874C2212yes—
69#1874C2212yes—
70#1874C2212yes—
71#1872C2212yes—
72#1872C2212yes—
73#1872C2212yes—
74#1872C2212yes—
75#1836C2212yes—
76#1836C2212yes—
77#1835C2214yes—
78#1803C2214yes—
79#1802C2214yes—
80#1771C2214yes—
81#1771C2214yes—
82#1763C2214yes—
83#1762C2214yes—
84#1749C2211yescharacteristic
85#1749C2211yescharacteristic
86C4×C22⋊C8C2212yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#127C2311yescharacteristic
89#529—28no—
90#529—28no—
91C2×#126—22no—
92C2×#125C2312yes—
93C2×#124C2312yes—
94C2×#124C2312yes—
95C2×#123—22no—
96C2×C22⋊Q8C2312yes—
97C2×#121—21no—
98C2×#121—21no—
99C2×#121C2312yes—
100C2×C4×Q8C2312yes—
101C2×C4×D4—22no—
102C2×C4×D4C2312yes—
103C2×#117—22no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#432—28no—
119C2×C4⋊C8C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×#97C2314yes—
131C2×#97C2312yes—
132C2×#97C2312yes—
133C2×C4×C8C2314yes—
134#400—24no—
135#400—24no—
136#400—24no—
137#400—24no—
138#398—22no—
139#398—22no—
140#396—22no—
141#396—22no—
142#396—22no—
143#396—22no—
144#394—22no—
145#394—22no—
146#391C2312yes—
147#391C2312yes—
148#390—21no—
149#390—21no—
150#390C2312yes—
151#390C2312yes—
152#389—24no—
153#389—24no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389C2312yes—
160#389C2312yes—
161#389C2314yes—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#387—22no—
167#387—22no—
168#386—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#385—22no—
173#385—22no—
174#384—22no—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2314yes—
178#382—24no—
179#382—24no—
180#382—22no—
181#382—21no—
182#382—21no—
183#382C2312yes—
184#382C2312yes—
185#382C2314yes—
186#382C2312yes—
187#380—21no—
188#380—21no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4D412yes—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95D411yescharacteristic
203C2×#95D411yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2312yes—
213#336C2314yes—
214#336C2312yes—
215C22×Q8C2412yes—
216C22×D4—22no—
217C22×D4—21no—
218C22×D4—21no—
219C22×D4C2412yes—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—22no—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2411yescharacteristic
232C22×C8C2414yes—
233C22×C8C2414yes—
234C4⋊Q8—42no—
235#127—42no—
236#126—48no—
237#125—24no—
238#125—24no—
239#124—24no—
240#124—24no—
241#124—44no—
242#123—48no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245#121—24no—
246#121—24no—
247#121—44no—
248#121—44no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—48no—
254#117—48no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
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×C42—22no—
306C2×C42—24no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—44no—
310C2×C42C2×D412yes—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D412yes—
314C2×C42#4118yes—
315C2×C42C2412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2412yes—
319C4⋊C8—28no—
320#103—44no—
321#103—44no—
322#103—44no—
323#103—44no—
324#102—44no—
325#102—44no—
326#102—44no—
327#102—44no—
328C22⋊C8—24no—
329C22⋊C8—24no—
330#97—28no—
331#97—42no—
332#97—42no—
333C4×C8—44no—
334#95—24no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95C2×D412yes—
351#95C2×D412yes—
352#95C2×D412yes—
353#95C2×D412yes—
354C24—22no—
355C24C22×D411yescharacteristic
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4—81no—
372C2×D4—81no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4#12414yes—
414C22×C4#12414yes—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4114yes—
425C22×C4C2511yescharacteristic
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—82no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C42—24no—
476C42—22no—
477C42—22no—
478C42—42no—
479C42—41no—
480C42—41no—
481C42—48no—
482C42—44no—
483C42#12418yes—
484C42C2×#4114yes—
485C42C2×#4114yes—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23C2×#12412yes—
498C23#57512yes—
499C23#57512yes—
500C23C23×D411yescharacteristic
501Q8—44no—
502Q8—44no—
503D4—44no—
504D4—44no—
505D4—88no—
506D4—162no—
507D4—162no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—28no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—88no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4C2×#12414yes—
566C2×C4C2×#12412yes—
567C2×C4C2×#12414yes—
568C2×C4C22×#4112yes—
569C2×C4#58512yes—
570C2×C4#58512yes—
571C8—44no—
572C8—44no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22#314214yes—
587C22C22×#12411yescharacteristic
588C22#348511yescharacteristic
589C22#348711yescharacteristic
590C22C2×#57511yescharacteristic
591C22C2×#57511yescharacteristic
592C22#358511yescharacteristic
593C22#358611yescharacteristic
594C4—22no—
595C4—24no—
596C4—24no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4#351214yes—
611C2—24no—
612C2—162no—
613C2#3564911yescharacteristic
614C2C2×#314211yescharacteristic
615C2#6062111yescharacteristic
616C2#6171011yescharacteristic
617C2#6171211yescharacteristic
618C2#6189411yescharacteristic
619C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
2. C2×#2571C2semidirect
3. #29335C2semidirect
5. #29330C2semidirect
6. #27983C2semidirect
8. #27977C2semidirect
9. #27761C2semidirect
10. #27759C2semidirect
11. #27745C2semidirect
12. #27743C2semidirect
13. #21867C2semidirect
14. #21391C2semidirect
15. #20534C2semidirect

Maximal subgroups

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

subgroupindex
#351032
C2×#25712
#293352
#293342
#293302
#279832
#279822
#279772
#277612
#277592
#277452
#277432
#218672
#213912
#205342