Finite Group Databasebeta

← #7187373

Subgroups — #7187373

Order 512.

1608 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, complemented
2#59827C211yesmaximal, characteristic, complemented
3#32552C214yesmaximal, complemented
4#31645C214yesmaximal, complemented
5#29259C211yesmaximal, characteristic, complemented
6#28156C211yesmaximal, characteristic
7#27748C212yesmaximal, complemented
8#26164C214yesmaximal, complemented
9#26111C212yesmaximal, complemented
10#25367C212yesmaximal, complemented
11#24192C214yesmaximal, complemented
12#23815C211yesmaximal, characteristic, complemented
13#22995C212yesmaximal, complemented
14#22307C211yesmaximal, characteristic, complemented
15#22268C211yesmaximal, characteristic, complemented
16C2×#559C2212yes—
17C2×#492C2211yescharacteristic
18C2×#463C2214yes—
19C2×#459C2212yes—
20C2×C4×D8C2214yes—
21#2804—22no—
22#2724—22no—
23#2582C2212yes—
24#2580—22no—
25#2580C2214yes—
26#2571C2212yes—
27#2563—21no—
28#2563C2212yes—
29#2547—21no—
30#2547C2212yes—
31#2487—22no—
32#2486—22no—
33#2447—22no—
34#2447C2214yes—
35#2366C2212yes—
36D4×C4⋊C4—21no—
37D4×C4⋊C4—21no—
38D4×C4⋊C4C2212yes—
39D4×C4⋊C4C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×#127C2211yescharacteristic
44C4×#127C2211yescharacteristic
45#2101C2214yes—
46#2099C2212yes—
47#2095C2212yes—
48#2087C2214yes—
49#2074C2214yes—
50#2071C2212yes—
51#2071C2212yes—
52#2055C2214yes—
53#2042C2214yes—
54#2041C2212yes—
55#2024C2212yes—
56#2023C2212yes—
57#2012C2214yes—
58#2011C2212yes—
59#1984C2214yes—
60#1981C2212yes—
61#1958C2214yes—
62#1957C2211yescharacteristic
63#1956C2211yescharacteristic
64#1947C2212yes—
65#1939C2212yes—
66#1934C2214yes—
67#1933C2214yes—
68#1916C2214yes—
69#1880C2214yes—
70#1879C2214yes—
71#1879C2214yes—
72#1877C2214yes—
73#1877C2214yes—
74#1863C2214yes—
75#1862C2214yes—
76#1862C2214yes—
77#1834C2211yescharacteristic
78#1831C2211yescharacteristic
79#1804C2214yes—
80#1802C2212yes—
81C8×C4⋊C4C2212yes—
82C4×#102C2212yes—
83C4×#102C2212yes—
84#1744C2211yescharacteristic
85C22×SD16C2314yes—
86C22×D8C2314yes—
87#559—48no—
88C2×C4⋊Q8D411yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#124—22no—
94C2×#124C2314yes—
95C2×#123—24no—
96C2×C4×Q8—21no—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2312yes—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4C2312yes—
106C2×C4×D4C2314yes—
107C2×C4×D4C2312yes—
108C2×C4×D4C2314yes—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—22no—
111#492—24no—
112#463—48no—
113#459—28no—
114C4×D8—48no—
115C2×#107C2314yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#102C2314yes—
122C2×#102C2312yes—
123C2×#102C2312yes—
124C2×#102C2314yes—
125C2×#102C2312yes—
126C2×#102C2314yes—
127C2×#102C2312yes—
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2314yes—
131#400—24no—
132#399—24no—
133#398—24no—
134#397—24no—
135#397—24no—
136#397—24no—
137#393—24no—
138#393—24no—
139#391—22no—
140#391—21no—
141#391D412yes—
142#391C2312yes—
143#391C2314yes—
144#390—22no—
145#390—21no—
146#390—21no—
147#390—21no—
148#390—21no—
149#390C2312yes—
150#390C2312yes—
151#390C2314yes—
152#389—22no—
153#389—22no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389D412yes—
158#389D412yes—
159#389C2312yes—
160#389C2314yes—
161#389C2314yes—
162#386—24no—
163#385—24no—
164#385—24no—
165#385—24no—
166#385—24no—
167#385—24no—
168#384D411yescharacteristic
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#382—22no—
174#382C2314yes—
175#381—24no—
176#380—22no—
177#380—22no—
178#380—22no—
179#380—22no—
180#380—22no—
181#380—22no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C43C2311yescharacteristic
189#340C2312yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2312yes—
194#340C2312yes—
195#340C2312yes—
196#336C2312yes—
197#336C2312yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202C22×Q8—21no—
203C22×Q8C2×D412yes—
204C22×Q8C2412yes—
205C22×D4—21no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C2412yes—
210C22×D4C2412yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C2×SD16—28no—
216C2×SD16—44no—
217C2×SD16—44no—
218C2×D8—28no—
219C2×D8—28no—
220C2×D8—44no—
221C22×C8C2414yes—
222C22×C8C2412yes—
223C22×C8C2412yes—
224C4⋊Q8—24no—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227C4⋊Q8—42no—
228#127—24no—
229#127—22no—
230#127#4114yes—
231#124—28no—
232#124—44no—
233#124—48no—
234#123—416no—
235C4×Q8—44no—
236C4×Q8—44no—
237C4×Q8—82no—
238C4×D4—28no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—44no—
242C4×D4—44no—
243C4×D4—44no—
244C4×D4—44no—
245C4×D4—48no—
246C4×D4—44no—
247C4×D4—44no—
248C4×D4—82no—
249C4×D4—84no—
250C4×D4—82no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—84no—
267C2×C4⋊C4—84no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2414yes—
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301#107—44no—
302C4⋊C8—24no—
303C4⋊C8—44no—
304#103—44no—
305#103—44no—
306#103—44no—
307#102—28no—
308#102—24no—
309#102—24no—
310#102—42no—
311#102—42no—
312#102—44no—
313#102—44no—
314C4×C8—22no—
315C4×C8—22no—
316C4×C8—44no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95C2×D414yes—
328#95C2×D414yes—
329C24—22no—
330C24—22no—
331C2×Q8—24no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—81no—
340C2×Q8—81no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—81no—
358C2×D4—81no—
359C2×D4—81no—
360C2×D4—81no—
361C2×D4C2×#4114yes—
362C2×D4H(5,𝔽2)14yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4C22×D414yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D414yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C2511yescharacteristic
390SD16—88no—
391D8—416no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—81no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—82no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—84no—
442C42—22no—
443C42—24no—
444C42—24no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—41no—
449C42—44no—
450C42—44no—
451C42#13514yes—
452C42C2×#4112yes—
453C422−1+412yes—
454C23—24no—
455C23—24no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23C23×D411yescharacteristic
469Q8—44no—
470Q8—44no—
471Q8—84no—
472Q8—84no—
473D4—24no—
474D4—28no—
475D4—24no—
476D4—48no—
477D4—48no—
478D4—42no—
479D4—44no—
480D4—42no—
481D4—84no—
482D4—84no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—24no—
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—88no—
521C2×C4—168no—
522C2×C4—168no—
523C2×C4#54812yes—
524C2×C4#54812yes—
525C2×C4C2×#13512yes—
526C2×C4C2×#13512yes—
527C2×C4C2×#13512yes—
528C2×C4#57612yes—
529C2×C4C22×#4111yescharacteristic
530C2×C4C2×H(5,𝔽2)12yes—
531C2×C4C2×2−1+411yescharacteristic
532C2×C4#58512yes—
533C8—44no—
534C8—42no—
535C8—42no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22C2×#54811yescharacteristic
551C22#348411yescharacteristic
552C22#348711yescharacteristic
553C22C22×#13511yescharacteristic
554C22C2×#57611yescharacteristic
555C22#358511yescharacteristic
556C22#358611yescharacteristic
557C4—22no—
558C4—22no—
559C4—22no—
560C4—44no—
561C4—41no—
562C4—44no—
563C4—41no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4#336114yes—
571C4#351312yes—
572C4#355912yes—
573C2—84no—
574C2—84no—
575C2#4527111yescharacteristic
576C2C2×#336111yescharacteristic
577C2#6133311yescharacteristic
578C2#6156511yescharacteristic
579C2#6167911yescharacteristic
580C2#6188511yescharacteristic
581C2#6191211yescharacteristic
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. #59864C2semidirect
2. #59827C2semidirect
3. #32552C2semidirect
4. #31645C2semidirect
5. #29259C2semidirect
7. #27748C2semidirect
8. #26164C2semidirect
9. #26111C2semidirect
10. #25367C2semidirect
11. #24192C2semidirect
12. #23815C2semidirect
13. #22995C2semidirect
14. #22307C2semidirect
15. #22268C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#598272
#325522
#316452
#292592
#281562
#277482
#261642
#261112
#253672
#241922
#238152
#229952
#223072
#222682