Finite Group Databasebeta

← #6660666

Subgroups — #6660666

Order 512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic, complemented
2#35259C211yesmaximal, characteristic
3#32789C212yesmaximal, complemented
4#32770C212yesmaximal, complemented
5#30365C212yesmaximal, complemented
6#30356C212yesmaximal, complemented
7#29948C212yesmaximal, complemented
8#29898C214yesmaximal, complemented
9#29700C212yesmaximal, complemented
10#29340C212yesmaximal, complemented
11#29306C212yesmaximal, complemented
12#27677C212yesmaximal, complemented
13#24713C214yesmaximal, complemented
14#24304C212yesmaximal, complemented
15#20720C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#478C2214yes—
20C2×#433C2212yes—
21#2857C2212yes—
22#2857C2212yes—
23#2856C2211yescharacteristic
24#2854C2212yes—
25#2851C2211yescharacteristic
26#2629C2212yes—
27#2629C2211yescharacteristic
28#2629C2212yes—
29#2619C2211yescharacteristic
30#2615C2212yes—
31#2582—22no—
32#2538—22no—
33#2464—22no—
34#2458—22no—
35#2456—22no—
36#2456—22no—
37#2417—22no—
38#2417—22no—
39#2324C2212yes—
40#2316C2212yes—
41#2314C2211yescharacteristic
42#2311C2212yes—
43C4×#121C2212yes—
44C2×#384C2211yescharacteristic
45#2101C2214yes—
46#2099C2212yes—
47#2099C2212yes—
48#2093C2212yes—
49#2093C2212yes—
50#2083C2214yes—
51#2077C2214yes—
52#2076C2214yes—
53#2076C2214yes—
54#2075C2214yes—
55#2059C2214yes—
56#2058C2212yes—
57#2057C2212yes—
58#2041C2214yes—
59#2034C2214yes—
60#2033C2214yes—
61#2033C2214yes—
62#2033C2214yes—
63#2023C2212yes—
64#2023C2212yes—
65#1989C2214yes—
66#1947C2214yes—
67#1943C2214yes—
68#1942C2214yes—
69#1936C2214yes—
70#1934C2214yes—
71#1875C2214yes—
72#1874C2214yes—
73#1854C2214yes—
74#1852C2214yes—
75#1817C2212yes—
76#1802C2214yes—
77#1798C2212yes—
78#1797C2212yes—
79#1780C2212yes—
80#1750C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C22×#41C2311yescharacteristic
84#540—44no—
85#533—44no—
86C2×C4⋊Q8C2312yes—
87C2×#127C2311yescharacteristic
88C2×#125C2312yes—
89C2×#125C2311yescharacteristic
90C2×#123—22no—
91C2×#123—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×#121—22no—
96C2×#121—22no—
97C2×#121C2312yes—
98C2×#121C2312yes—
99C2×#121C2312yes—
100C4×#41—44no—
101C2×C4×Q8—22no—
102C2×C4×Q8D412yes—
103C2×C4×D4—22no—
104C2×C4×D4D412yes—
105C2×C4×D4D412yes—
106C2×C4×D4D412yes—
107C2×C4×D4C2312yes—
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C22×C4⋊C4—22no—
113C22×C4⋊C4C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#478—48no—
118#433—44no—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2314yes—
132C2×#97C2312yes—
133C2×C4×C8C2312yes—
134#398—24no—
135#398—24no—
136#398—22no—
137#396—24no—
138#396—24no—
139#396—22no—
140#394—22no—
141#393—22no—
142#392—22no—
143#392—22no—
144#392—22no—
145#392—22no—
146#391—22no—
147#391—22no—
148#391—22no—
149#390—22no—
150#390—22no—
151#390—22no—
152#390C2312yes—
153#390C2312yes—
154#389—24no—
155#389—22no—
156#389—22no—
157#389C2312yes—
158#389C2314yes—
159#389C2314yes—
160#389C2312yes—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#387—22no—
167#387—22no—
168#386—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#385—22no—
174#385—22no—
175#384—22no—
176#384C2312yes—
177#384C2311yescharacteristic
178#384C2312yes—
179#382C2311yescharacteristic
180#380—22no—
181#380—22no—
182#380—24no—
183#380C2314yes—
184#380C2314yes—
185#380C2312yes—
186#380C2312yes—
187#380C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C22⋊C4—22no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2312yes—
199#336C2312yes—
200#336C2314yes—
201C2×#41—24no—
202C2×#41—42no—
203C2×#41—42no—
204C22×Q8—22no—
205C22×Q8C2×D411yescharacteristic
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C2×D411yescharacteristic
211C22×D4C2×D411yescharacteristic
212C22×D4C2×D411yescharacteristic
213C22×D4C2412yes—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×D412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C22×C8C2414yes—
223C22×C8C2414yes—
224C4⋊Q8—44no—
225#127—42no—
226#125—28no—
227#125—42no—
228#123—48no—
229#123—84no—
230C22⋊Q8—48no—
231C22⋊Q8—84no—
232C22⋊Q8—84no—
233#121—28no—
234#121—48no—
235#121—44no—
236#121—44no—
237#121—84no—
238C4×Q8—44no—
239C4×Q8—48no—
240C4×D4—28no—
241C4×D4—44no—
242C4×D4—44no—
243C4×D4—44no—
244C4×D4—84no—
245#117—24no—
246#117—24no—
247#117—42no—
248#117—42no—
249#117—42no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—84no—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D412yes—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301#107—44no—
302#107—44no—
303#107—44no—
304#106—44no—
305C4⋊C8—28no—
306#102—44no—
307#102—44no—
308#102—44no—
309#102—44no—
310C22⋊C8—28no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313C22⋊C8—44no—
314#97—42no—
315C4×C8—42no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—22no—
327C24—22no—
328C24C22×D411yescharacteristic
329C24C22×D411yescharacteristic
330C24C22×D411yescharacteristic
331#41—416no—
332C2×Q8—22no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—82no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4—82no—
360C2×D4—82no—
361C2×D4—82no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—84no—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C42−1+414yes—
397C22×C4C2511yescharacteristic
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—84no—
422C4⋊C4—82no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—82no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—42no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C22⋊C4—82no—
443C42—24no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—42no—
448C42—42no—
449C42—44no—
450C42—82no—
451C42H(5,𝔽2)14yes—
452C23—22no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—22no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23#53612yes—
474C23C23×D411yescharacteristic
475Q8—44no—
476Q8—88no—
477D4—44no—
478D4—44no—
479D4—44no—
480D4—48no—
481D4—88no—
482D4—164no—
483D4—164no—
484D4—164no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—28no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—164no—
525C2×C4—164no—
526C2×C4#53414yes—
527C2×C4#53612yes—
528C2×C4#57614yes—
529C2×C4C2×H(5,𝔽2)12yes—
530C2×C4C2×2−1+412yes—
531C2×C4#58512yes—
532C8—44no—
533C8—44no—
534C22—22no—
535C22—24no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22—42no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—42no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22C2×#53411yescharacteristic
557C22C2×#53611yescharacteristic
558C22#344911yescharacteristic
559C22C2×#57612yes—
560C22#358612yes—
561C4—22no—
562C4—24no—
563C4—24no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4—84no—
569C4—82no—
570C4—82no—
571C4—84no—
572C4—82no—
573C4#354114yes—
574C2—42no—
575C2—81no—
576C2—81no—
577C2—162no—
578C2#3743811yescharacteristic
579C2#6076212yes—
580C2#6088812yes—
581C2#6101111yescharacteristic
582C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35259C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35280C2semidirect
3. #32789C2semidirect
4. #32770C2semidirect
5. #30365C2semidirect
6. #30356C2semidirect
7. #29948C2semidirect
8. #29898C2semidirect
9. #29700C2semidirect
10. #29340C2semidirect
11. #29306C2semidirect
12. #27677C2semidirect
13. #24713C2semidirect
14. #24304C2semidirect
15. #20720C2semidirect

Maximal subgroups

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

subgroupindex
#352802
#352592
#327892
#327702
#303652
#303562
#299482
#298982
#297002
#293402
#293062
#276772
#247132
#243042
#207202