Finite Group Databasebeta

← #7156340

Subgroups — #7156340

Order 512.

1638 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59863C211yesmaximal, characteristic, complemented
2#59827C211yesmaximal, characteristic, complemented
3#32609C214yesmaximal, complemented
4#30739C214yesmaximal, complemented
5#28460C212yesmaximal, complemented
6#28161C211yesmaximal, characteristic, complemented
7#28156C211yesmaximal, characteristic
8#27195C214yesmaximal, complemented
9#25635C214yesmaximal, complemented
10#24731C212yesmaximal, complemented
11#24629C212yesmaximal, complemented
12#23799C211yesmaximal, characteristic, complemented
13#22770C212yesmaximal, complemented
14#22307C211yesmaximal, characteristic, complemented
15#22268C211yesmaximal, characteristic, complemented
16C2×#559C2212yes—
17C2×#495C2214yes—
18C2×#492C2211yescharacteristic
19C2×#462C2212yes—
20C2×#463C2214yes—
21C2×#459C2212yes—
22C2×C4×SD16C2212yes—
23C2×C4×SD16C2212yes—
24#2808—22no—
25#2724—22no—
26#2592C2214yes—
27#2582C2212yes—
28#2563—21no—
29#2548C2212yes—
30#2547—21no—
31#2547C2212yes—
32#2547C2212yes—
33#2513—22no—
34#2486—22no—
35#2485—22no—
36#2447—22no—
37#2447C2214yes—
38#2355C2212yes—
39#2354—21no—
40#2354C2212yes—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#127C2211yescharacteristic
48#2096C2212yes—
49#2094C2212yes—
50#2072C2212yes—
51#2071C2212yes—
52#2065C2214yes—
53#2056C2214yes—
54#2042C2214yes—
55#2016C2212yes—
56#2015C2214yes—
57#2015C2214yes—
58#2014C2212yes—
59#2013C2212yes—
60#2012C2214yes—
61#2011C2212yes—
62#1984C2214yes—
63#1983C2214yes—
64#1968C2214yes—
65#1957C2211yescharacteristic
66#1956C2211yescharacteristic
67#1939C2212yes—
68#1933C2214yes—
69#1927C2214yes—
70#1919C2214yes—
71#1916C2214yes—
72#1880C2214yes—
73#1879C2214yes—
74#1879C2214yes—
75#1878C2214yes—
76#1877C2214yes—
77#1877C2214yes—
78#1834C2211yescharacteristic
79#1831C2211yescharacteristic
80#1803C2212yes—
81#1802C2212yes—
82C8×C4⋊C4C2212yes—
83C4×#102C2214yes—
84#1744C2211yescharacteristic
85C22×Q16C2314yes—
86C22×SD16C2312yes—
87C22×SD16C2314yes—
88C22×SD16C2312yes—
89C22×D8C2314yes—
90#559—48no—
91C2×C4⋊Q8D411yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#124—22no—
97C2×#124C2314yes—
98C2×#124C2314yes—
99C2×#123—24no—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2312yes—
103C2×C4×D4—22no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2314yes—
110C2×C4×D4C2312yes—
111C2×#117—22no—
112C22×C4⋊C4—22no—
113#495—48no—
114#492—24no—
115#463—48no—
116#462—44no—
117#459—44no—
118C4×SD16—28no—
119C4×SD16—44no—
120C2×#106C2312yes—
121C2×#106C2312yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2314yes—
124C2×#103C2312yes—
125C2×#103C2312yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2312yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2314yes—
136C2×C4×C8C2311yescharacteristic
137#398—24no—
138#397—24no—
139#397—24no—
140#397—24no—
141#397—24no—
142#397—24no—
143#394—24no—
144#393—24no—
145#391—22no—
146#391D412yes—
147#391C2312yes—
148#391C2312yes—
149#391C2314yes—
150#391C2314yes—
151#390—22no—
152#390—22no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#390C2314yes—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389D412yes—
166#389D412yes—
167#389C2314yes—
168#388—24no—
169#388—24no—
170#388—24no—
171#387—24no—
172#385—24no—
173#385—24no—
174#384D411yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#382—22no—
180#381—24no—
181#380—22no—
182#380—22no—
183#380—22no—
184#380—22no—
185#380—22no—
186#380—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2314yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C43C2311yescharacteristic
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202#336C2312yes—
203#336C2314yes—
204#336C2314yes—
205C22×Q8C2×D412yes—
206C22×Q8C2412yes—
207C22×Q8C2412yes—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C2412yes—
213C22×D4C2412yes—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C2×Q16—44no—
218C2×Q16—44no—
219C2×Q16—44no—
220C2×SD16—24no—
221C2×SD16—24no—
222C2×SD16—24no—
223C2×SD16—28no—
224C2×SD16—42no—
225C2×SD16—42no—
226C2×SD16—42no—
227C2×SD16—44no—
228C2×SD16—44no—
229C2×D8—28no—
230C2×D8—44no—
231C2×D8—44no—
232C22×C8C2414yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C4⋊Q8—24no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—42no—
239#127—24no—
240#127—22no—
241#127—22no—
242#124—28no—
243#124—44no—
244#124—44no—
245#124—44no—
246#124—48no—
247#123—416no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—44no—
251C4×Q8—44no—
252C4×D4—28no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—44no—
256C4×D4—44no—
257C4×D4—48no—
258C4×D4—44no—
259C4×D4—44no—
260C4×D4—82no—
261C4×D4—82no—
262#117—84no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
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—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—84no—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2414yes—
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#106—24no—
314#106—42no—
315C4⋊C8—24no—
316C4⋊C8—44no—
317#103—24no—
318#103—44no—
319#103—42no—
320#103—44no—
321#103—44no—
322#102—24no—
323#102—44no—
324#102—42no—
325#102—44no—
326#102—44no—
327C4×C8—22no—
328C4×C8—44no—
329C4×C8#4114yes—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95C2×D414yes—
341#95C2×D414yes—
342C24—22no—
343C24—22no—
344C2×Q8—24no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—81no—
375C2×D4—81no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4H(5,𝔽2)14yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4C22×D414yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D414yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2511yescharacteristic
403Q16—88no—
404SD16—48no—
405SD16—48no—
406SD16—88no—
407D8—88no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8C2×#4114yes—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—81no—
440C4⋊C4—81no—
441C4⋊C4—84no—
442C4⋊C4—81no—
443C4⋊C4—81no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C42—22no—
458C42—24no—
459C42—24no—
460C42—22no—
461C42—22no—
462C42—41no—
463C42—41no—
464C42—44no—
465C42—44no—
466C42—84no—
467C42#13514yes—
468C42C2×#4112yes—
469C42H(5,𝔽2)12yes—
470C23—24no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23C23×D411yescharacteristic
485Q8—42no—
486Q8—44no—
487Q8—42no—
488Q8—42no—
489Q8—42no—
490Q8—44no—
491Q8—84no—
492D4—24no—
493D4—24no—
494D4—48no—
495D4—42no—
496D4—44no—
497D4—42no—
498D4—44no—
499D4—84no—
500D4—84no—
501D4—84no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
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—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—88no—
539C2×C4—88no—
540C2×C4—168no—
541C2×C4#55012yes—
542C2×C4#55012yes—
543C2×C4C2×#13512yes—
544C2×C4C2×#13512yes—
545C2×C4C2×#13512yes—
546C2×C4#57712yes—
547C2×C4C22×#4111yescharacteristic
548C2×C4C2×H(5,𝔽2)12yes—
549C2×C4C2×H(5,𝔽2)11yescharacteristic
550C2×C4#58512yes—
551C8—24no—
552C8—44no—
553C8—42no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22C2×#55011yescharacteristic
569C22#348411yescharacteristic
570C22#348711yescharacteristic
571C22C22×#13511yescharacteristic
572C22C2×#57711yescharacteristic
573C22#358511yescharacteristic
574C22#358611yescharacteristic
575C4—22no—
576C4—22no—
577C4—22no—
578C4—44no—
579C4—41no—
580C4—44no—
581C4—41no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4#339414yes—
589C4#349412yes—
590C4#354212yes—
591C2—84no—
592C2—162no—
593C2#4493311yescharacteristic
594C2C2×#339411yescharacteristic
595C2#6141511yescharacteristic
596C2#6156511yescharacteristic
597C2#6168511yescharacteristic
598C2#6188511yescharacteristic
599C2#6191611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28156C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59863C2semidirect
2. #59827C2semidirect
3. #32609C2semidirect
4. #30739C2semidirect
5. #28460C2semidirect
6. #28161C2semidirect
8. #27195C2semidirect
9. #25635C2semidirect
10. #24731C2semidirect
11. #24629C2semidirect
12. #23799C2semidirect
13. #22770C2semidirect
14. #22307C2semidirect
15. #22268C2semidirect

Maximal subgroups

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

subgroupindex
#598632
#598272
#326092
#307392
#284602
#281612
#281562
#271952
#256352
#247312
#246292
#237992
#227702
#223072
#222682