Finite Group Databasebeta

← #7156384

Subgroups — #7156384

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59869C211yesmaximal, characteristic, complemented
2#59827C211yesmaximal, characteristic
3#32625C214yesmaximal, complemented
4#30757C214yesmaximal, complemented
5#28471C212yesmaximal, complemented
6#28161C211yesmaximal, characteristic, complemented
7#28157C211yesmaximal, characteristic
8#27210C214yesmaximal, complemented
9#25641C214yesmaximal
10#24730C212yesmaximal, complemented
11#24629C212yesmaximal, complemented
12#23799C211yesmaximal, characteristic
13#22762C212yesmaximal, complemented
14#22311C211yesmaximal, characteristic, complemented
15#22270C211yesmaximal, characteristic, complemented
16C2×#559C2212yes—
17C2×#497C2214yes—
18C2×#494C2211yescharacteristic
19C2×#462C2212yes—
20C2×#461C2214yes—
21C2×#461C2212yes—
22C2×C4×SD16C2212yes—
23C2×C4×SD16C2212yes—
24#2809—22no—
25#2724—22no—
26#2582C2212yes—
27#2571—21no—
28#2568C2214yes—
29#2547—21no—
30#2547C2212yes—
31#2546C2212yes—
32#2546C2212yes—
33#2508—22no—
34#2508—22no—
35#2486—22no—
36#2447—22no—
37#2447C2214yes—
38#2369—21no—
39#2355C2212yes—
40#2355C2212yes—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48#2096C2212yes—
49#2094C2212yes—
50#2072C2212yes—
51#2072C2212yes—
52#2066C2214yes—
53#2056C2214yes—
54#2043C2214yes—
55#2016C2212yes—
56#2016C2214yes—
57#2015C2214yes—
58#2015C2214yes—
59#2014C2212yes—
60#2014C2212yes—
61#2013C2212yes—
62#1985C2214yes—
63#1983C2214yes—
64#1971C2214yes—
65#1956C2211yescharacteristic
66#1956C2211yescharacteristic
67#1939C2212yes—
68#1935C2214yes—
69#1927C2214yes—
70#1920C2214yes—
71#1917C2214yes—
72#1880C2214yes—
73#1878C2214yes—
74#1878C2214yes—
75#1878C2214yes—
76#1877C2214yes—
77#1877C2214yes—
78#1834C2211yescharacteristic
79#1831C2211yescharacteristic
80#1803C2212yes—
81#1802C2212yes—
82#1770C2212yes—
83#1762C2214yes—
84#1744C2211yescharacteristic
85C22×Q16C2314yes—
86C22×Q16C2314yes—
87C22×SD16C2312yes—
88C22×SD16C2314yes—
89C22×SD16C2312yes—
90#559—48no—
91C2×C4⋊Q8D411yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2314yes—
96C2×#127C2311yescharacteristic
97C2×#124—22no—
98C2×#124C2314yes—
99C2×#123—24no—
100C2×C4×Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×C4×D4—22no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4C2312yes—
108C2×C4×D4C2314yes—
109C22×C4⋊C4—22no—
110#497—48no—
111#494—24no—
112#462—44no—
113#461—44no—
114#461—48no—
115C4×SD16—28no—
116C4×SD16—44no—
117C2×#106C2312yes—
118C2×#106C2312yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2314yes—
121C2×#103C2312yes—
122C2×#103C2312yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#102C2312yes—
129C2×#102C2314yes—
130C2×#102C2312yes—
131C2×#97C2314yes—
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134#400—24no—
135#400—24no—
136#398—24no—
137#398—24no—
138#398—24no—
139#397—24no—
140#397—24no—
141#397—24no—
142#391—24no—
143#391—22no—
144#391D412yes—
145#391C2312yes—
146#391C2312yes—
147#391C2312yes—
148#391C2314yes—
149#391C2314yes—
150#391C2314yes—
151#390—21no—
152#390—21no—
153#390C2312yes—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389D412yes—
166#389D412yes—
167#389C2314yes—
168#387—24no—
169#387—24no—
170#385—24no—
171#385—24no—
172#384D411yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#382—24no—
178#382—22no—
179#382—22no—
180#381—24no—
181#380—22no—
182#380—22no—
183#380—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
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×Q8C2412yes—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—24no—
214C2×Q16—28no—
215C2×Q16—44no—
216C2×Q16—44no—
217C2×Q16—44no—
218C2×Q16—44no—
219C2×Q16—44no—
220C2×SD16—24no—
221C2×SD16—24no—
222C2×SD16—28no—
223C2×SD16—24no—
224C2×SD16—42no—
225C2×SD16—42no—
226C2×SD16—42no—
227C2×SD16—44no—
228C2×SD16—44no—
229C22×C8C2414yes—
230C22×C8C2412yes—
231C22×C8C2412yes—
232C4⋊Q8—24no—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—42no—
238C4⋊Q8—44no—
239C4⋊Q8—44no—
240#127—24no—
241#124—28no—
242#124—44no—
243#124—48no—
244#123—416no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—44no—
248C4×Q8—44no—
249C4×Q8—44no—
250C4×D4—28no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—44no—
254C4×D4—48no—
255C4×D4—44no—
256C4×D4—82no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
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—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—84no—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2414yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306#106—24no—
307#106—42no—
308C4⋊C8—24no—
309C4⋊C8—44no—
310#103—24no—
311#103—44no—
312#103—42no—
313#103—44no—
314#103—44no—
315#103—44no—
316#103—44no—
317#102—24no—
318#102—44no—
319#102—42no—
320#97—44no—
321C4×C8—22no—
322C4×C8#4114yes—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95C2×D414yes—
338#95C2×D414yes—
339C24—22no—
340C2×Q8—24no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—24no—
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—42no—
365C2×D4—42no—
366C2×D4—81no—
367C2×D4—81no—
368C2×D4H(5,𝔽2)14yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—48no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—84no—
381C22×C4—84no—
382C22×C4C22×D414yes—
383C22×C4C22×D414yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C2511yescharacteristic
391Q16—88no—
392Q16—88no—
393SD16—48no—
394SD16—48no—
395SD16—88no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8C2×#4114yes—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—84no—
434C4⋊C4—81no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—82no—
443C42—22no—
444C42—24no—
445C42—24no—
446C42—22no—
447C42—22no—
448C42—41no—
449C42—41no—
450C42—44no—
451C42—44no—
452C42—44no—
453C42—84no—
454C42#13714yes—
455C42C2×#4112yes—
456C42H(5,𝔽2)12yes—
457C23—24no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C23×D411yescharacteristic
465Q8—42no—
466Q8—44no—
467Q8—44no—
468Q8—42no—
469Q8—42no—
470Q8—44no—
471Q8—42no—
472Q8—44no—
473Q8—84no—
474D4—24no—
475D4—24no—
476D4—48no—
477D4—42no—
478D4—42no—
479D4—84no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
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—88no—
520C2×C4—168no—
521C2×C4#55012yes—
522C2×C4#55012yes—
523C2×C4C2×#13712yes—
524C2×C4C2×#13712yes—
525C2×C4C2×#13712yes—
526C2×C4#57712yes—
527C2×C4C22×#4111yescharacteristic
528C2×C4C2×H(5,𝔽2)12yes—
529C2×C4C2×H(5,𝔽2)11yescharacteristic
530C2×C4#58512yes—
531C8—24no—
532C8—42no—
533C8—44no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22C2×#55011yescharacteristic
542C22#348411yescharacteristic
543C22#348711yescharacteristic
544C22C22×#13711yescharacteristic
545C22C2×#57711yescharacteristic
546C22#358611yescharacteristic
547C22#358611yescharacteristic
548C4—22no—
549C4—22no—
550C4—22no—
551C4—44no—
552C4—41no—
553C4—44no—
554C4—41no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4#340014yes—
563C4#349412yes—
564C4#354212yes—
565C2—84no—
566C2#4493311yescharacteristic
567C2C2×#340011yescharacteristic
568C2#6141611yescharacteristic
569C2#6158611yescharacteristic
570C2#6168011yescharacteristic
571C2#6189311yescharacteristic
572C2#6191611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
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. #59869C2semidirect
3. #32625C2semidirect
4. #30757C2semidirect
5. #28471C2semidirect
6. #28161C2semidirect
8. #27210C2semidirect
10. #24730C2semidirect
11. #24629C2semidirect
13. #22762C2semidirect
14. #22311C2semidirect
15. #22270C2semidirect

Maximal subgroups

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

subgroupindex
#598692
#598272
#326252
#307572
#284712
#281612
#281572
#272102
#256412
#247302
#246292
#237992
#227622
#223112
#222702