Finite Group Databasebeta

← #7569880

Subgroups — #7569880

Order 512.

2039 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#36465C211yesmaximal, characteristic
3#32039C212yesmaximal, complemented
4#32026C212yesmaximal, complemented
5#28753C212yesmaximal, complemented
6#28610C214yesmaximal, complemented
7#26242C214yesmaximal, complemented
8#25896C214yesmaximal, complemented
9#24045C212yesmaximal, complemented
10#24041C212yesmaximal, complemented
11#23175C211yesmaximal, characteristic, complemented
12#22421C214yesmaximal, complemented
13#21616C212yesmaximal, complemented
14C2×#559—21no—
15C2×#550C2211yescharacteristic
16C2×D4×Q8C2211yescharacteristic
17C2×#548C2214yes—
18C2×#489C2214yes—
19C2×#481C2212yes—
20C2×#483C2212yes—
21C2×#477C2212yes—
22C2×#435C2214yes—
23C2×#417C2212yes—
24C22×#103C2212yes—
25#2858—21no—
26#2858C2211yescharacteristic
27#2724—21no—
28#2724—21no—
29#2582—21no—
30#2547C2214yes—
31#2542—21no—
32#2542C2211yescharacteristic
33#2541—21no—
34#2541C2211yescharacteristic
35#2540—24no—
36#2540C2214yes—
37#2486C2212yes—
38#2486C2212yes—
39#2447—24no—
40D4×C4⋊C4C2214yes—
41C4×C4⋊Q8C2212yes—
42C4×#127—21no—
43C4×#127C2211yescharacteristic
44C42×Q8C2211yescharacteristic
45#2043C2214yes—
46#2041C2214yes—
47#2036C2214yes—
48#2036C2214yes—
49#2035C2214yes—
50#2034C2214yes—
51#2033C2214yes—
52#2033C2214yes—
53#1987C2212yes—
54#1983C2214yes—
55#1983C2214yes—
56#1959C2214yes—
57#1955C2214yes—
58#1948C2212yes—
59#1918C2214yes—
60#1895C2212yes—
61#1894C2214yes—
62#1893C2214yes—
63#1892C2212yes—
64#1874C2214yes—
65#1872C2214yes—
66#1870C2212yes—
67#1869C2212yes—
68#1865C2214yes—
69#1803C2212yes—
70#1788C2212yes—
71#1788C2214yes—
72#1787C2212yes—
73#1787C2214yes—
74#1776C2214yes—
75#1767C2212yes—
76#1763C2214yes—
77C22×#41C2312yes—
78C23×Q8C2312yes—
79C22×M16C2312yes—
80C23×C8C2312yes—
81#559—48no—
82#550—42no—
83#550—42no—
84D4×Q8—42no—
85D4×Q8—42no—
86#548—216no—
87#548—48no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×#127—21no—
92C2×#127—21no—
93C2×#127D411yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#124—24no—
96C2×#124—24no—
97C2×#124C2314yes—
98C2×#123—22no—
99C2×#123—22no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2314yes—
104C2×#121—24no—
105C2×#121—22no—
106C2×#121C2312yes—
107C2×#121C2314yes—
108C2×C4×Q8—22no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×D4—21no—
112C2×C4×D4—24no—
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4C2314yes—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120#489—48no—
121#483—44no—
122#481—44no—
123#477—28no—
124#435—48no—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129#417—44no—
130C2×#103—44no—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×#97C2314yes—
146#399—24no—
147#399—22no—
148#398—22no—
149#398—22no—
150#397—22no—
151#397—22no—
152#397—24no—
153#397—24no—
154#392—24no—
155#392—22no—
156#391—24no—
157#391—22no—
158#390—21no—
159#390—21no—
160#390—24no—
161#390—24no—
162#390—21no—
163#390—22no—
164#390—21no—
165#390D411yescharacteristic
166#390D411yescharacteristic
167#390D414yes—
168#389—24no—
169#389—22no—
170#389C2314yes—
171#385—24no—
172#385—24no—
173#384—21no—
174#384D411yescharacteristic
175#384C2312yes—
176#381—22no—
177#381—22no—
178#380C2314yes—
179#380C2314yes—
180#380C2312yes—
181#380C2312yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2311yescharacteristic
187C4×C4⋊C4C2312yes—
188C43—21no—
189C43C2311yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195C2×#41—28no—
196C2×#41—24no—
197C2×#41—44no—
198C2×#41—42no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—28no—
202C22×Q8—24no—
203C22×Q8—44no—
204C22×Q8—42no—
205C22×Q8C2412yes—
206C22×Q8C2411yescharacteristic
207C22×D4—22no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C2×D414yes—
213C22×D4C2×D411yescharacteristic
214C22×D4C2×D411yescharacteristic
215C22×D4C2412yes—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C2×M16—42no—
223C2×M16—44no—
224C22×C8—42no—
225C22×C8—44no—
226C22×C8C2414yes—
227C22×C8C2414yes—
228C4⋊Q8—24no—
229C4⋊Q8—24no—
230C4⋊Q8—41no—
231C4⋊Q8—41no—
232C4⋊Q8—44no—
233#127—24no—
234#127—41no—
235#127—41no—
236#127—44no—
237#127—44no—
238#124—28no—
239#124—28no—
240#124—416no—
241#124—416no—
242#123—48no—
243#123—48no—
244C22⋊Q8—28no—
245C22⋊Q8—24no—
246C22⋊Q8—416no—
247C22⋊Q8—44no—
248C22⋊Q8—42no—
249C22⋊Q8—48no—
250#121—28no—
251#121—24no—
252#121—44no—
253#121—42no—
254#121—416no—
255#121—48no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—24no—
259C4×Q8—48no—
260C4×Q8#4118yes—
261C4×D4—28no—
262C4×D4—28no—
263C4×D4—416no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—44no—
269C4×D4—44no—
270C4×D4—416no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2411yescharacteristic
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4C2×D414yes—
305C2×C22⋊C4C2×D414yes—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C42—22no—
309C2×C42—21no—
310C2×C42—24no—
311C2×C42—21no—
312C2×C42—22no—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2412yes—
318#107—28no—
319#106—28no—
320C4⋊C8—24no—
321C4⋊C8—24no—
322#103—28no—
323#103—28no—
324#103—44no—
325#103—44no—
326#103—44no—
327#103—44no—
328#102—44no—
329#102—44no—
330#102—44no—
331#102—44no—
332C22⋊C8—44no—
333C22⋊C8—44no—
334C22⋊C8—42no—
335C22⋊C8—42no—
336#97—28no—
337#95—21no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—22no—
344#95—22no—
345C24—22no—
346C24C22×D412yes—
347C24C22×D412yes—
348#41—48no—
349#41—88no—
350#41—84no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—48no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—48no—
363C2×Q8—88no—
364C2×Q8—84no—
365C2×Q8C2×#4114yes—
366C2×Q8H(5,𝔽2)12yes—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—44no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—44no—
385C2×D4—42no—
386C2×D4—48no—
387C2×D4—42no—
388C2×D4—82no—
389C2×D4—84no—
390C2×D4—84no—
391C2×D4AGL(1,ℤ/8)14yes—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C2511yescharacteristic
423M16—84no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—84no—
431C4⋊C4—24no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4C2×#4114yes—
453C4⋊C4H(5,𝔽2)12yes—
454C22⋊C4—28no—
455C22⋊C4—28no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—48no—
460C22⋊C4—44no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—44no—
464C22⋊C4—48no—
465C22⋊C4—44no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C42—24no—
469C42—28no—
470C42—22no—
471C42—21no—
472C42—21no—
473C42—42no—
474C42—41no—
475C42—48no—
476C42—42no—
477C42—82no—
478C42C2×#4114yes—
479C42H(5,𝔽2)14yes—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23C2×AGL(1,ℤ/8)14yes—
495C23C23×D411yescharacteristic
496Q8—24no—
497Q8—22no—
498Q8—42no—
499Q8—44no—
500Q8—42no—
501Q8—88no—
502Q8—88no—
503Q8#55914yescomplemented
504D4—44no—
505D4—416no—
506D4—42no—
507D4—44no—
508D4—42no—
509D4—82no—
510D4—88no—
511D4—84no—
512D4—816no—
513D4—164no—
514D4—162no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4—164no—
551C2×C4—164no—
552C2×C4#54614yes—
553C2×C4#54614yes—
554C2×C4C2×AGL(1,ℤ/8)12yes—
555C2×C4C22×#4112yes—
556C2×C4C2×H(5,𝔽2)11yescharacteristic
557C2×C4C2×H(5,𝔽2)11yescharacteristic
558C2×C4C2×H(5,𝔽2)12yes—
559C8—44no—
560C8—44no—
561C22—24no—
562C22—28no—
563C22—42no—
564C22—44no—
565C22—42no—
566C22—42no—
567C22—44no—
568C22—42no—
569C22—44no—
570C22—42no—
571C22—82no—
572C22—84no—
573C22—82no—
574C22—84no—
575C22—82no—
576C22C2×#54612yes—
577C22#348411yescharacteristic
578C22C22×AGL(1,ℤ/8)11yescharacteristic
579C22#358611yescharacteristic
580C22#358612yes—
581C4—24no—
582C4—24no—
583C4—22no—
584C4—24no—
585C4—42no—
586C4—42no—
587C4—44no—
588C4—41no—
589C4—42no—
590C4—82no—
591C4—84no—
592C4—84no—
593C4C2×#55911yescharacteristic
594C4#350812yes—
595C4#354311yescharacteristic
596C2—44no—
597C2—82no—
598C2—162no—
599C2#5914311yescharacteristic
600C2#6116112yes—
601C2#6122112yes—
602C2#6157211yescharacteristic
603C2#6167411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36465C2
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. #59827C2semidirect
3. #32039C2semidirect
4. #32026C2semidirect
5. #28753C2semidirect
6. #28610C2semidirect
7. #26242C2semidirect
8. #25896C2semidirect
9. #24045C2semidirect
10. #24041C2semidirect
11. #23175C2semidirect
12. #22421C2semidirect
13. #21616C2semidirect
503. Q8#559semidirect

Maximal subgroups

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

subgroupindex
#598272
#364652
#320392
#320262
#287532
#286102
#262422
#258962
#240452
#240412
#231752
#224212
#216162