Finite Group Databasebeta

← #7174905

Subgroups — #7174905

Order 512.

1812 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic, complemented
2#59815C211yesmaximal, characteristic
3#32014C214yesmaximal, complemented
4#31279C214yesmaximal, complemented
5#29259C211yesmaximal, characteristic, complemented
6#28202C211yesmaximal, characteristic, complemented
7#25886C212yesmaximal, complemented
8#25346C212yesmaximal, complemented
9#25010C212yesmaximal, complemented
10#24877C214yesmaximal, complemented
11#24046C214yesmaximal, complemented
12#23815C211yesmaximal, characteristic, complemented
13#22431C212yesmaximal, complemented
14#22264C211yesmaximal, characteristic, complemented
15#22221C211yesmaximal, characteristic, complemented
16C2×#550C2212yes—
17C2×#547C2212yes—
18C2×#487C2211yescharacteristic
19C2×#480C2214yes—
20C2×#476C2212yes—
21C2×#436C2212yes—
22C2×#417C2214yes—
23#2804—22no—
24#2682—22no—
25#2580—22no—
26#2580—22no—
27#2571—21no—
28#2563C2212yes—
29#2563C2212yes—
30#2541—21no—
31#2538C2212yes—
32#2537C2212yes—
33#2487C2214yes—
34#2445—22no—
35#2416—22no—
36#2407C2214yes—
37#2366—21no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2099C2212yes—
47#2095C2212yes—
48#2084C2212yes—
49#2077C2214yes—
50#2075C2214yes—
51#2071C2212yes—
52#2041C2212yes—
53#2041C2212yes—
54#2035C2214yes—
55#2035C2214yes—
56#2033C2214yes—
57#2033C2214yes—
58#2024C2212yes—
59#2023C2212yes—
60#1990C2211yescharacteristic
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1954C2212yes—
64#1947C2212yes—
65#1928C2214yes—
66#1926C2214yes—
67#1895C2214yes—
68#1894C2214yes—
69#1872C2214yes—
70#1870C2214yes—
71#1870C2214yes—
72#1863C2214yes—
73#1863C2214yes—
74#1852C2214yes—
75#1845C2211yescharacteristic
76#1831C2211yescharacteristic
77#1809C2214yes—
78#1802C2212yes—
79#1788C2214yes—
80#1788C2214yes—
81#1788C2214yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84#1744C2211yescharacteristic
85C22×#41C2314yes—
86C22×M16C2314yes—
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×#123—24no—
97C2×C22⋊Q8C2314yes—
98C2×#121—22no—
99C2×#121—22no—
100C2×#121—24no—
101C2×#121C2314yes—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C22×C4⋊C4—22no—
115C22×C4⋊C4C2314yes—
116#487—24no—
117#480—48no—
118#476—44no—
119#436—44no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8C2312yes—
124#417—48no—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2312yes—
131C2×#102C2312yes—
132C2×#102C2312yes—
133C2×#102C2314yes—
134C2×#102C2312yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140#400—24no—
141#399—24no—
142#396—24no—
143#394—24no—
144#394—24no—
145#393—24no—
146#393—24no—
147#392—24no—
148#391—21no—
149#391—21no—
150#391C2312yes—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—21no—
155#390—21no—
156#390D412yes—
157#390D412yes—
158#390D412yes—
159#390C2312yes—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389C2312yes—
168#389C2312yes—
169#386—24no—
170#386—24no—
171#385—24no—
172#385—24no—
173#385—24no—
174#385—24no—
175#384D411yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#382—22no—
181#382—22no—
182#380—22no—
183#380C2314yes—
184#380C2314yes—
185#380C2314yes—
186#380C2314yes—
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—24no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4C2414yes—
226C2×M16—44no—
227C2×M16—44no—
228C2×M16—44no—
229C22×C8C2412yes—
230C22×C8C2414yes—
231C22×C8C2412yes—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—42no—
236C4⋊Q8#4114yes—
237#127—24no—
238#127—24no—
239#127—48no—
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—82no—
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—
260C4×D4—84no—
261C4×D4—84no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×D414yes—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#107—44no—
314#107—44no—
315#106—44no—
316C4⋊C8—24no—
317#103—28no—
318#103—44no—
319#103—44no—
320#103—44no—
321#102—24no—
322#102—28no—
323#102—24no—
324#102—42no—
325#102—42no—
326#102—44no—
327C22⋊C8—44no—
328C22⋊C8—44no—
329C22⋊C8—44no—
330C4×C8—22no—
331C4×C8—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342C24—22no—
343C24C22×D414yes—
344#41—416no—
345#41—88no—
346#41—88no—
347#41—88no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—81no—
354C2×Q8—81no—
355C2×Q8—81no—
356C2×Q8—81no—
357C2×Q8C2×#4114yes—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—42no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4H(5,𝔽2)14yes—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D414yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C2511yescharacteristic
415M16—88no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—48no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—81no—
449C4⋊C4—84no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—82no—
453C4⋊C4—81no—
454C4⋊C4—82no—
455C22⋊C4—28no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—82no—
463C22⋊C4—84no—
464C22⋊C4—84no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—41no—
471C42—41no—
472C42—44no—
473C42—44no—
474C42—44no—
475C42#13714yes—
476C42C2×#4112yes—
477C42H(5,𝔽2)12yes—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C23×D411yescharacteristic
493Q8—28no—
494Q8—44no—
495Q8—84no—
496Q8—84no—
497D4—24no—
498D4—24no—
499D4—48no—
500D4—48no—
501D4—48no—
502D4—42no—
503D4—42no—
504D4—84no—
505D4—84no—
506D4—816no—
507D4—168no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—24no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
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—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—816no—
549C2×C4—88no—
550C2×C4—168no—
551C2×C4#54612yes—
552C2×C4#54612yes—
553C2×C4C2×#13712yes—
554C2×C4C2×#13712yes—
555C2×C4C2×#13712yes—
556C2×C4#57612yes—
557C2×C4C22×#4111yescharacteristic
558C2×C4C2×H(5,𝔽2)12yes—
559C2×C4C2×H(5,𝔽2)11yescharacteristic
560C2×C4#58512yes—
561C8—42no—
562C8—44no—
563C8—42no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22C2×#54611yescharacteristic
579C22#348411yescharacteristic
580C22#348511yescharacteristic
581C22C22×#13711yescharacteristic
582C22C2×#57611yescharacteristic
583C22#358511yescharacteristic
584C22#358611yescharacteristic
585C4—22no—
586C4—22no—
587C4—22no—
588C4—44no—
589C4—44no—
590C4—41no—
591C4—41no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4#333314yes—
599C4#351312yes—
600C4#355812yes—
601C2—84no—
602C2—84no—
603C2#4514011yescharacteristic
604C2C2×#333311yescharacteristic
605C2#6122011yescharacteristic
606C2#6158611yescharacteristic
607C2#6167911yescharacteristic
608C2#6171911yescharacteristic
609C2#6180711yescharacteristic
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. #59864C2semidirect
3. #32014C2semidirect
4. #31279C2semidirect
5. #29259C2semidirect
6. #28202C2semidirect
7. #25886C2semidirect
8. #25346C2semidirect
9. #25010C2semidirect
10. #24877C2semidirect
11. #24046C2semidirect
12. #23815C2semidirect
13. #22431C2semidirect
14. #22264C2semidirect
15. #22221C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#598152
#320142
#312792
#292592
#282022
#258862
#253462
#250102
#248772
#240462
#238152
#224312
#222642
#222212