Finite Group Databasebeta

← #7174868

Subgroups — #7174868

Order 512.

1794 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59863C211yesmaximal, characteristic, complemented
2#59815C211yesmaximal, characteristic
3#32007C214yesmaximal, complemented
4#31269C214yesmaximal, complemented
5#29255C211yesmaximal, characteristic, complemented
6#28202C211yesmaximal, characteristic, complemented
7#25886C212yesmaximal, complemented
8#25339C212yesmaximal, complemented
9#25011C212yesmaximal, complemented
10#24870C214yesmaximal, complemented
11#24034C214yesmaximal, complemented
12#23815C211yesmaximal, characteristic, complemented
13#22417C212yesmaximal, complemented
14#22260C211yesmaximal, characteristic, complemented
15#22221C211yesmaximal, characteristic, complemented
16C2×#550C2212yes—
17C2×#547C2212yes—
18C2×#487C2211yescharacteristic
19C2×#482C2214yes—
20C2×#476C2212yes—
21C2×C8×D4C2212yes—
22C2×#416C2214yes—
23#2808—22no—
24#2682—22no—
25#2592—22no—
26#2563C2212yes—
27#2548—21no—
28#2547C2212yes—
29#2541—21no—
30#2538C2212yes—
31#2537C2212yes—
32#2513—22no—
33#2485C2214yes—
34#2445—22no—
35#2416—22no—
36#2407C2214yes—
37#2355—21no—
38#2354C2212yes—
39#2354C2212yes—
40D4×C4⋊C4—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2099C2212yes—
47#2096C2212yes—
48#2084C2212yes—
49#2077C2214yes—
50#2076C2214yes—
51#2071C2212yes—
52#2041C2212yes—
53#2041C2212yes—
54#2035C2214yes—
55#2035C2214yes—
56#2034C2214yes—
57#2033C2214yes—
58#2024C2212yes—
59#2023C2212yes—
60#1990C2211yescharacteristic
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1954C2212yes—
64#1948C2212yes—
65#1928C2214yes—
66#1926C2214yes—
67#1893C2214yes—
68#1892C2214yes—
69#1873C2214yes—
70#1869C2214yes—
71#1869C2214yes—
72#1865C2214yes—
73#1863C2214yes—
74#1854C2214yes—
75#1845C2211yescharacteristic
76#1831C2211yescharacteristic
77#1807C2214yes—
78#1802C2212yes—
79#1788C2214yes—
80#1788C2214yes—
81#1788C2214yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84#1744C2211yescharacteristic
85C22×#41C2314yes—
86C23×C8C2314yes—
87#550—48no—
88#547—28no—
89#547—44no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127—22no—
94C2×#127D411yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#124—22no—
97C2×#123—24no—
98C2×C22⋊Q8C2314yes—
99C2×#121—22no—
100C2×#121—22no—
101C2×#121—24no—
102C2×#121C2314yes—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117C2314yes—
114C22×C4⋊C4—22no—
115#487—24no—
116#482—48no—
117#476—44no—
118C8×D4—44no—
119C2×#107C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2312yes—
123#416—48no—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#102C2314yes—
129C2×#102C2312yes—
130C2×#102C2312yes—
131C2×#102C2312yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139#397—24no—
140#397—24no—
141#396—24no—
142#394—24no—
143#394—24no—
144#394—24no—
145#393—24no—
146#392—24no—
147#391—22no—
148#391—21no—
149#391—21no—
150#391C2312yes—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—21no—
156#390—21no—
157#390D412yes—
158#390D412yes—
159#390D412yes—
160#390C2312yes—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2312yes—
168#388—24no—
169#388—24no—
170#388—24no—
171#387—24no—
172#386—24no—
173#385—24no—
174#384D411yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#382—22no—
180#380—22no—
181#380C2314yes—
182#380C2314yes—
183#380C2314yes—
184#380C2314yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C43C2311yescharacteristic
194#340C2314yes—
195#340C2312yes—
196#340C2314yes—
197#340C2312yes—
198#340C2312yes—
199#340C2314yes—
200#340C2312yes—
201#336C2312yes—
202#336C2312yes—
203#336C2312yes—
204#336C2312yes—
205C2×#41—28no—
206C2×#41—28no—
207C2×#41—44no—
208C2×#41—44no—
209C2×#41—44no—
210C2×#41—44no—
211C22×Q8—21no—
212C22×Q8—21no—
213C22×Q8C2412yes—
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4C2×D412yes—
219C22×D4C2×D412yes—
220C22×D4C2×D412yes—
221C22×D4C2412yes—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4C2414yes—
225C22×C8—44no—
226C22×C8—44no—
227C22×C8—44no—
228C22×C8C2412yes—
229C22×C8C2414yes—
230C22×C8C2412yes—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—42no—
235C4⋊Q8#4114yes—
236#127—24no—
237#127—24no—
238#127—48no—
239#124—84no—
240#123—416no—
241C22⋊Q8—28no—
242C22⋊Q8—44no—
243#121—28no—
244#121—44no—
245#121—48no—
246#121—416no—
247#121—84no—
248C4×Q8—44no—
249C4×Q8—44no—
250C4×Q8—82no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—44no—
256C4×D4—44no—
257C4×D4—48no—
258C4×D4—44no—
259C4×D4—82no—
260#117—44no—
261#117—44no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×D414yes—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312#107—44no—
313#106—44no—
314#106—44no—
315C4⋊C8—24no—
316#103—28no—
317#103—44no—
318#103—44no—
319#103—44no—
320#102—24no—
321#102—28no—
322#102—24no—
323#102—42no—
324#102—42no—
325#102—44no—
326C22⋊C8—44no—
327C22⋊C8—44no—
328C22⋊C8—44no—
329C4×C8—22no—
330C4×C8—22no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—24no—
341C24—22no—
342C24C22×D414yes—
343#41—416no—
344#41—88no—
345#41—88no—
346#41—88no—
347C2×Q8—22no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—81no—
355C2×Q8—81no—
356C2×Q8C2×#4114yes—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—42no—
378C2×D4—81no—
379C2×D4—81no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4H(5,𝔽2)14yes—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—84no—
403C22×C4C22×D414yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2511yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—88no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—82no—
450C22⋊C4—28no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—82no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—41no—
467C42—41no—
468C42—44no—
469C42—44no—
470C42—44no—
471C42—84no—
472C42#13514yes—
473C42C2×#4112yes—
474C42H(5,𝔽2)12yes—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23C23×D411yescharacteristic
490Q8—28no—
491Q8—44no—
492Q8—84no—
493Q8—84no—
494D4—24no—
495D4—24no—
496D4—48no—
497D4—42no—
498D4—42no—
499D4—48no—
500D4—84no—
501D4—84no—
502D4—84no—
503D4—816no—
504D4—168no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
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—816no—
546C2×C4—88no—
547C2×C4#54612yes—
548C2×C4#54612yes—
549C2×C4C2×#13512yes—
550C2×C4C2×#13512yes—
551C2×C4C2×#13512yes—
552C2×C4#57712yes—
553C2×C4C22×#4111yescharacteristic
554C2×C4C2×H(5,𝔽2)12yes—
555C2×C4C2×H(5,𝔽2)11yescharacteristic
556C2×C4#58512yes—
557C8—42no—
558C8—44no—
559C8—42no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22C2×#54611yescharacteristic
575C22#348411yescharacteristic
576C22#348511yescharacteristic
577C22C22×#13511yescharacteristic
578C22C2×#57711yescharacteristic
579C22#358511yescharacteristic
580C22#358611yescharacteristic
581C4—22no—
582C4—22no—
583C4—22no—
584C4—44no—
585C4—44no—
586C4—41no—
587C4—41no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4#330414yes—
595C4#351312yes—
596C4#355812yes—
597C2—84no—
598C2—84no—
599C2#4514011yescharacteristic
600C2C2×#330411yescharacteristic
601C2#6122011yescharacteristic
602C2#6156511yescharacteristic
603C2#6168511yescharacteristic
604C2#6170011yescharacteristic
605C2#6180911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59815C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59863C2semidirect
3. #32007C2semidirect
4. #31269C2semidirect
5. #29255C2semidirect
6. #28202C2semidirect
7. #25886C2semidirect
8. #25339C2semidirect
9. #25011C2semidirect
10. #24870C2semidirect
11. #24034C2semidirect
12. #23815C2semidirect
13. #22417C2semidirect
14. #22260C2semidirect
15. #22221C2semidirect

Maximal subgroups

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

subgroupindex
#598632
#598152
#320072
#312692
#292552
#282022
#258862
#253392
#250112
#248702
#240342
#238152
#224172
#222602
#222212