Finite Group Databasebeta

← #6435355

Subgroups — #6435355

Order 512.

1651 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35322C211yesmaximal, characteristic, complemented
2#34280C211yesmaximal, characteristic
3#30120C214yesmaximal, complemented
4#30067C212yesmaximal, complemented
5#30039C212yesmaximal, complemented
6#29982C212yesmaximal, complemented
7#29930C212yesmaximal, complemented
8#29798C212yesmaximal, complemented
9#29673C212yesmaximal
10#23831C212yesmaximal, complemented
11#23811C212yesmaximal, complemented
12#23574C212yesmaximal, complemented
13#23568C212yesmaximal, complemented
14#21514C214yesmaximal, complemented
15#20784C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C2×#533C2212yes—
18C2×#461C2214yes—
19C2×#440C2214yes—
20C2×C4×SD16C2214yes—
21#2868C2211yescharacteristic
22#2862C2212yes—
23#2862C2211yescharacteristic
24#2621C2212yes—
25#2621C2212yes—
26#2565—22no—
27#2565—22no—
28#2499—22no—
29#2499—22no—
30#2374—22no—
31#2364—22no—
32#2363—22no—
33#2359—22no—
34#2330C2211yescharacteristic
35#2328C2211yescharacteristic
36#2327C2212yes—
37#2327C2212yes—
38#2324C2212yes—
39#2314C2211yescharacteristic
40C4×#127C2211yescharacteristic
41C4×#125C2211yescharacteristic
42C4×#121C2212yes—
43#2293C2212yes—
44#2275C2212yes—
45C4×#117C2211yescharacteristic
46#2101C2214yes—
47#2097C2214yes—
48#2092C2214yes—
49#2072C2214yes—
50#2058C2212yes—
51#2057C2212yes—
52#2043C2212yes—
53#2043C2212yes—
54#2043C2214yes—
55#2036C2214yes—
56#2035C2214yes—
57#2026C2214yes—
58#2023C2212yes—
59#2023C2212yes—
60#2014C2214yes—
61#2013C2214yes—
62#2013C2212yes—
63#2013C2212yes—
64#2000C2214yes—
65#2000C2214yes—
66#1981C2214yes—
67#1949C2214yes—
68#1928C2214yes—
69#1927C2214yes—
70#1877C2214yes—
71#1877C2214yes—
72#1839C2212yes—
73#1831C2212yes—
74#1818C2212yes—
75#1804C2214yes—
76#1798C2212yes—
77#1768C2212yes—
78#1763C2214yes—
79C4×#103C2214yes—
80#1749C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1746C2212yes—
83C22×SD16C2314yes—
84C22×SD16C2314yes—
85#540—28no—
86#533—48no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#127C2311yescharacteristic
92C2×#125—22no—
93C2×#125C2311yescharacteristic
94C2×#123—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8C2314yes—
97C2×#121—22no—
98C2×#121C2312yes—
99C2×C4×Q8—22no—
100C2×C4×Q8C2314yes—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C22×C42C2311yescharacteristic
111#461—216no—
112#440—48no—
113C4×SD16—48no—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2312yes—
127C2×#97C2312yes—
128C2×C4×C8C2312yes—
129#398—24no—
130#398—22no—
131#398—22no—
132#397—24no—
133#397—22no—
134#397—22no—
135#391—22no—
136#391—22no—
137#391—22no—
138#391—22no—
139#391C2314yes—
140#391C2314yes—
141#390—22no—
142#390—22no—
143#390C2312yes—
144#390C2312yes—
145#389—22no—
146#389C2312yes—
147#389C2312yes—
148#388—22no—
149#388—22no—
150#388—22no—
151#388—22no—
152#388—22no—
153#387—22no—
154#387—22no—
155#387—24no—
156#387—24no—
157#387—24no—
158#387—24no—
159#387—22no—
160#386—22no—
161#386—22no—
162#386—22no—
163#386—22no—
164#385—22no—
165#384—22no—
166#384—22no—
167#384C2311yescharacteristic
168#382—22no—
169#382—22no—
170#382C2311yescharacteristic
171#382C2312yes—
172#382C2312yes—
173#380C2312yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4D412yes—
187#376C2311yescharacteristic
188#376C2311yescharacteristic
189C2×#95—22no—
190C43C2311yescharacteristic
191C43C2311yescharacteristic
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2312yes—
198#336C2312yes—
199#336C2314yes—
200#336C2314yes—
201C22×Q8—22no—
202C22×Q8C2414yes—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4C2412yes—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4C2×D412yes—
210C23×C4C2411yescharacteristic
211C2×SD16—28no—
212C2×SD16—28no—
213C2×SD16—44no—
214C2×SD16—44no—
215C2×SD16—44no—
216C2×SD16—44no—
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—28no—
220C4⋊Q8—44no—
221C4⋊Q8—44no—
222C4⋊Q8—44no—
223#127—24no—
224#125—24no—
225#125—48no—
226#123—216no—
227C22⋊Q8—44no—
228C22⋊Q8—44no—
229C22⋊Q8—48no—
230#121—216no—
231#121—24no—
232#121—24no—
233C4×Q8—28no—
234C4×Q8—28no—
235C4×Q8—48no—
236C4×D4—24no—
237C4×D4—28no—
238C4×D4—44no—
239C4×D4—84no—
240C4×D4—84no—
241C4×D4#4118yes—
242#117—24no—
243#117—42no—
244#117—48no—
245#117—84no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2412yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4C2×D412yes—
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C22⋊C4C2×D411yescharacteristic
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294#107—44no—
295#106—44no—
296C4⋊C8—28no—
297#103—28no—
298#103—28no—
299#103—44no—
300#103—44no—
301#103—44no—
302#103—44no—
303#102—44no—
304#102—44no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307#97—24no—
308C4×C8—24no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95—24no—
323#95C2×D412yes—
324#95C2×D412yes—
325#95C2×D412yes—
326#95C2×D412yes—
327C24—22no—
328C24C22×D411yescharacteristic
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—82no—
344C2×D4—82no—
345C2×D4—82no—
346C2×D4—82no—
347C2×D4C2×#4114yes—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D412yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×D412yes—
377C22×C4C2511yescharacteristic
378SD16—416no—
379SD16—416no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—44no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4C2×#4114yes—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4—24no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—44no—
413C22⋊C4—48no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C42—24no—
421C42—24no—
422C42—24no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—42no—
428C42—42no—
429C42—48no—
430C42—82no—
431C42—82no—
432C42—82no—
433C42C2×#4114yes—
434C42H(5,𝔽2)14yes—
435C23—22no—
436C23—21no—
437C23—24no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23C23×D411yescharacteristic
448Q8—44no—
449Q8—44no—
450Q8—44no—
451Q8—44no—
452Q8—88no—
453D4—28no—
454D4—44no—
455D4—88no—
456D4—88no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—48no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—88no—
495C2×C4—164no—
496C2×C4—164no—
497C2×C4#52514yes—
498C2×C4#53514yes—
499C2×C4#57714yes—
500C2×C4C22×#4112yes—
501C2×C4C2×H(5,𝔽2)12yes—
502C2×C4#58512yes—
503C8—44no—
504C8—44no—
505C22—41no—
506C22—44no—
507C22—41no—
508C22—41no—
509C22—42no—
510C22—44no—
511C22—42no—
512C22—44no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22C2×#52511yescharacteristic
518C22C2×#53511yescharacteristic
519C22#344911yescharacteristic
520C22C2×#57712yes—
521C22#358612yes—
522C4—24no—
523C4—24no—
524C4—24no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—42no—
531C4—44no—
532C4—84no—
533C4—82no—
534C4—82no—
535C4—84no—
536C4#346014yes—
537C2—81no—
538C2—84no—
539C2#3460911yescharacteristic
540C2#6059112yes—
541C2#6081312yes—
542C2#6101211yescharacteristic
543C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35322C2semidirect
3. #30120C2semidirect
4. #30067C2semidirect
5. #30039C2semidirect
6. #29982C2semidirect
7. #29930C2semidirect
8. #29798C2semidirect
10. #23831C2semidirect
11. #23811C2semidirect
12. #23574C2semidirect
13. #23568C2semidirect
14. #21514C2semidirect
15. #20784C2semidirect

Maximal subgroups

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

subgroupindex
#353222
#342802
#301202
#300672
#300392
#299822
#299302
#297982
#296732
#238312
#238112
#235742
#235682
#215142
#207842