Finite Group Databasebeta

← #7174866

Subgroups — #7174866

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic
2#59817C211yesmaximal, characteristic
3#32000C214yesmaximal, complemented
4#31263C214yesmaximal, complemented
5#29259C211yesmaximal, characteristic, complemented
6#28206C211yesmaximal, characteristic, complemented
7#25888C212yesmaximal, complemented
8#25340C212yesmaximal, complemented
9#25016C212yesmaximal, complemented
10#24867C214yesmaximal, complemented
11#24042C214yesmaximal, complemented
12#23816C211yesmaximal, characteristic, complemented
13D4×#103C212yesmaximal, complemented
14#22260C211yesmaximal, characteristic
15#22221C211yesmaximal, characteristic, complemented
16C2×D4×Q8C2212yes—
17C2×D4×Q8C2212yes—
18C2×#487C2211yescharacteristic
19C2×#484C2214yes—
20C2×#477C2212yes—
21C2×C8×D4C2212yes—
22C22×#103C2214yes—
23#2804—22no—
24#2683—22no—
25#2580—22no—
26#2580—22no—
27#2571—21no—
28#2563C2212yes—
29#2563C2212yes—
30#2551—21no—
31#2538C2212yes—
32#2538C2212yes—
33#2487C2214yes—
34#2452—22no—
35#2452—22no—
36#2409C2214yes—
37#2366—21no—
38#2366—21no—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2100C2212yes—
47#2095C2212yes—
48#2083C2212yes—
49#2076C2214yes—
50#2076C2214yes—
51#2072C2212yes—
52#2043C2212yes—
53#2043C2212yes—
54#2036C2214yes—
55#2036C2214yes—
56#2036C2214yes—
57#2036C2214yes—
58#2024C2212yes—
59#2024C2212yes—
60#1990C2211yescharacteristic
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1954C2212yes—
64#1947C2212yes—
65#1928C2214yes—
66#1928C2214yes—
67#1892C2214yes—
68#1892C2214yes—
69#1872C2214yes—
70#1869C2214yes—
71#1869C2214yes—
72#1863C2214yes—
73#1863C2214yes—
74#1852C2214yes—
75#1845C2211yescharacteristic
76#1831C2211yescharacteristic
77#1808C2214yes—
78#1803C2212yes—
79#1787C2214yes—
80#1787C2214yes—
81#1787C2214yes—
82C4×#103C2212yes—
83C4×#103C2212yes—
84#1744C2211yescharacteristic
85C23×Q8C2314yes—
86C23×C8C2314yes—
87D4×Q8—28no—
88D4×Q8—44no—
89D4×Q8—48no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127D411yescharacteristic
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8C2314yes—
100C2×C22⋊Q8C2314yes—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C22×C4⋊C4C2314yes—
113#487—24no—
114#484—48no—
115#477—44no—
116C8×D4—44no—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×C4⋊C8C2312yes—
121C2×#103—48no—
122C2×#103C2314yes—
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2314yes—
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#400—24no—
138#399—24no—
139#396—24no—
140#396—24no—
141#395—24no—
142#393—24no—
143#393—24no—
144#393—24no—
145#391—21no—
146#391—21no—
147#391C2312yes—
148#391C2312yes—
149#390—22no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390D412yes—
154#390D412yes—
155#390D412yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2312yes—
168#386—24no—
169#386—24no—
170#385—24no—
171#385—24no—
172#385—24no—
173#385—24no—
174#384D411yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#382—22no—
180#382—22no—
181#380C2314yes—
182#380C2314yes—
183#380C2314yes—
184#380C2314yes—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
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#340C2312yes—
200#340C2314yes—
201#336C2312yes—
202#336C2312yes—
203#336C2312yes—
204#336C2312yes—
205C22×Q8—24no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8—28no—
209C22×Q8—28no—
210C22×Q8—44no—
211C22×Q8—44no—
212C22×Q8—44no—
213C22×Q8—44no—
214C22×Q8C2412yes—
215C22×Q8C2412yes—
216C22×D4C2×D412yes—
217C22×D4C2×D412yes—
218C22×D4C2×D412yes—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4C2414yes—
222C22×C8—44no—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8C2412yes—
226C22×C8C2414yes—
227C22×C8C2412yes—
228C4⋊Q8—24no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—42no—
233C4⋊Q8—48no—
234C4⋊Q8—48no—
235C4⋊Q8#4114yes—
236#127Q1618yes—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239C22⋊Q8—416no—
240C22⋊Q8—44no—
241C22⋊Q8—44no—
242C22⋊Q8—416no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—44no—
246C4×Q8—82no—
247C4×Q8—82no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—44no—
251C4×D4—44no—
252C4×D4—48no—
253C4×D4—48no—
254C4×D4—84no—
255C4×D4—84no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
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—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4C2×D414yes—
295C2×C22⋊C4C2×D414yes—
296C2×C22⋊C4C2×D414yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308#107—44no—
309#107—44no—
310#107—44no—
311C4⋊C8—24no—
312#103—28no—
313#103—24no—
314#103—28no—
315#103—24no—
316#103—44no—
317#103—44no—
318#103—42no—
319#103—44no—
320#103—42no—
321#103—44no—
322C22⋊C8—44no—
323C22⋊C8—44no—
324C22⋊C8—44no—
325C4×C8—22no—
326C4×C8—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339C24C22×D414yes—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—22no—
346C2×Q8—24no—
347C2×Q8—22no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—416no—
353C2×Q8—81no—
354C2×Q8—81no—
355C2×Q8—81no—
356C2×Q8—81no—
357C2×Q8—84no—
358C2×Q8—88no—
359C2×Q8—88no—
360C2×Q8—88no—
361C2×Q8C2×#4114yes—
362C2×Q8H(5,𝔽2)14yes—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4C2×Q1618yes—
370C2×D4C2×Q1618yes—
371C2×D4C2×Q1618yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D414yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2511yescharacteristic
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—24no—
408C2×C8—22no—
409C2×C8—88no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—84no—
440C4⋊C4—81no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—81no—
444C22⋊C4—28no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—44no—
459C42—44no—
460C42—44no—
461C42—44no—
462C42C2×Q1614yes—
463C42C2×#4112yes—
464C42H(5,𝔽2)12yes—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23C22×Q1618yes—
472C23C23×D411yescharacteristic
473Q8—24no—
474Q8—28no—
475Q8—24no—
476Q8—42no—
477Q8—42no—
478Q8—44no—
479Q8—84no—
480Q8—816no—
481Q8—84no—
482D4—216no—
483D4—48no—
484D4—48no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—816no—
529C2×C4—168no—
530C2×C4#54612yes—
531C2×C4#54612yes—
532C2×C4C22×Q1612yes—
533C2×C4C22×Q1612yes—
534C2×C4C22×Q1612yes—
535C2×C4#57712yes—
536C2×C4C22×#4111yescharacteristic
537C2×C4C2×H(5,𝔽2)12yes—
538C2×C4C2×H(5,𝔽2)11yescharacteristic
539C2×C4#58512yes—
540C8—42no—
541C8—44no—
542C8—42no—
543C22—28no—
544C22—28no—
545C22—28no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22C2×#54611yescharacteristic
551C22#348411yescharacteristic
552C22#348511yescharacteristic
553C22C23×Q1611yescharacteristic
554C22C2×#57711yescharacteristic
555C22#358511yescharacteristic
556C22#358611yescharacteristic
557C4—22no—
558C4—22no—
559C4—22no—
560C4—44no—
561C4—44no—
562C4—44no—
563C4—41no—
564C4—41no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4#329914yes—
572C4#351312yes—
573C4#355812yes—
574C2—48no—
575C2#4514011yescharacteristic
576C2C2×#329911yescharacteristic
577C2#6122011yescharacteristic
578C2#6156311yescharacteristic
579C2#6168211yescharacteristic
580C2#6169811yescharacteristic
581C2#6180311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
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
3. #32000C2semidirect
4. #31263C2semidirect
5. #29259C2semidirect
6. #28206C2semidirect
7. #25888C2semidirect
8. #25340C2semidirect
9. #25016C2semidirect
10. #24867C2semidirect
11. #24042C2semidirect
12. #23816C2semidirect
13. D4×#103C2semidirect
15. #22221C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#598172
#320002
#312632
#292592
#282062
#258882
#253402
#250162
#248672
#240422
#238162
D4×#1032
#222602
#222212