Finite Group Databasebeta

← #6660964

Subgroups — #6660964

Order 512.

1501 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#35307C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#32755C214yesmaximal, complemented
4#30366C212yesmaximal, complemented
5#30359C212yesmaximal, complemented
6#29971C212yesmaximal, complemented
7#29951C212yesmaximal, complemented
8#29908C212yesmaximal, complemented
9#29645C212yesmaximal, complemented
10#29307C212yesmaximal, complemented
11#29117C212yesmaximal, complemented
12#27609C212yesmaximal
13#24619C214yesmaximal, complemented
14#24016C212yesmaximal
15#20720C211yesmaximal, characteristic
16C2×#536C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#485C2214yes—
20C2×#433C2212yes—
21#2857C2212yes—
22#2857C2212yes—
23#2855C2212yes—
24#2855C2212yes—
25#2629C2212yes—
26#2629C2212yes—
27#2625C2212yes—
28#2625C2212yes—
29#2582—22no—
30#2538—22no—
31#2447—22no—
32#2446—22no—
33#2419—22no—
34#2419—22no—
35#2417—22no—
36#2417—22no—
37#2316C2212yes—
38#2316C2211yescharacteristic
39#2311C2212yes—
40#2311C2211yescharacteristic
41#2310C2211yescharacteristic
42#2296C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#384C2211yescharacteristic
45#2099C2212yes—
46#2099C2212yes—
47#2096C2212yes—
48#2096C2212yes—
49#2088C2214yes—
50#2087C2214yes—
51#2084C2214yes—
52#2077C2214yes—
53#2077C2214yes—
54#2072C2214yes—
55#2066C2214yes—
56#2066C2214yes—
57#2059C2214yes—
58#2058C2212yes—
59#2057C2212yes—
60#2036C2214yes—
61#2035C2214yes—
62#2026C2214yes—
63#2024C2212yes—
64#2024C2212yes—
65#1944C2214yes—
66#1941C2214yes—
67#1935C2214yes—
68#1933C2214yes—
69#1872C2214yes—
70#1854C2214yes—
71#1853C2214yes—
72#1853C2214yes—
73#1825C2214yes—
74#1825C2214yes—
75#1817C2212yes—
76#1798C2212yes—
77#1797C2212yes—
78#1787C2214yes—
79#1780C2212yes—
80#1750C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85#536—44no—
86#533—44no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2312yes—
89C2×#125C2312yes—
90C2×#124—21no—
91C2×#123—21no—
92C2×#123—21no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C4×#41—44no—
99C2×C4×Q8—22no—
100C2×C4×Q8D412yes—
101C2×C4×D4D412yes—
102C2×C4×D4D412yes—
103C2×C4×D4D412yes—
104C2×#117C2311yescharacteristic
105C2×#117C2312yes—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C22×C4⋊C4—22no—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#485—48no—
117#433—44no—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×#97C2312yes—
132C2×C4×C8C2312yes—
133#400—22no—
134#400—22no—
135#398—24no—
136#398—24no—
137#398—22no—
138#398—22no—
139#397—22no—
140#397—22no—
141#397—22no—
142#397—22no—
143#396—24no—
144#396—24no—
145#391—22no—
146#391—21no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#391C2312yes—
155#390—21no—
156#390—22no—
157#389—24no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#389C2314yes—
164#389C2312yes—
165#389C2314yes—
166#389C2312yes—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#386—22no—
172#386—22no—
173#386—21no—
174#386—21no—
175#385—22no—
176#385—22no—
177#385—22no—
178#384—22no—
179#384C2312yes—
180#384C2312yes—
181#381—22no—
182#381—22no—
183#381—22no—
184#380—22no—
185#380—22no—
186#380—24no—
187#380—22no—
188#380C2312yes—
189#380C2314yes—
190#380C2312yes—
191#380C2314yes—
192C4×C4⋊C4—22no—
193C2×#95—22no—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#336C2312yes—
203#336C2312yes—
204#336C2314yes—
205C2×#41—24no—
206C2×#41—24no—
207C2×#41—42no—
208C22×Q8—22no—
209C22×Q8—21no—
210C22×Q8—24no—
211C22×Q8—42no—
212C22×Q8—42no—
213C22×Q8C2×D411yescharacteristic
214C22×Q8C2412yes—
215C22×D4C2×D411yescharacteristic
216C22×D4C2×D411yescharacteristic
217C22×D4C2×D411yescharacteristic
218C23×C4—21no—
219C23×C4—22no—
220C23×C4—21no—
221C23×C4C2×D412yes—
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C22×C8C2414yes—
228C22×C8C2414yes—
229C4⋊Q8—44no—
230C4⋊Q8—44no—
231#125—42no—
232#125—42no—
233#124—44no—
234#123—44no—
235#123—44no—
236C22⋊Q8—28no—
237C22⋊Q8—44no—
238C22⋊Q8—48no—
239C22⋊Q8—84no—
240C22⋊Q8—84no—
241C4×Q8—44no—
242C4×Q8—48no—
243C4×D4—44no—
244C4×D4—44no—
245C4×D4—44no—
246#117—24no—
247#117—28no—
248#117—24no—
249#117—42no—
250#117—42no—
251#117—42no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—84no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2412yes—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304#107—44no—
305#106—44no—
306#106—44no—
307#106—44no—
308C4⋊C8—44no—
309#103—44no—
310#103—44no—
311#103—44no—
312#103—44no—
313C22⋊C8—28no—
314C22⋊C8—28no—
315C22⋊C8—42no—
316C22⋊C8—42no—
317#97—42no—
318C4×C8—42no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337C24C22×D411yescharacteristic
338C24C22×D411yescharacteristic
339C24C22×D411yescharacteristic
340#41—416no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—41no—
347C2×Q8—44no—
348C2×Q8—42no—
349C2×Q8—44no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—44no—
353C2×Q8—88no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—84no—
391C22×C4—82no—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4H(5,𝔽2)14yes—
400C22×C42−1+414yes—
401C22×C4C2511yescharacteristic
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—42no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C22⋊C4—24no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—42no—
441C22⋊C4—82no—
442C22⋊C4—82no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C42—24no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—44no—
454C42—42no—
455C23—22no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23#53412yes—
469C23#53612yes—
470C23C23×D411yescharacteristic
471Q8—44no—
472Q8—48no—
473Q8—88no—
474Q8—84no—
475D4—44no—
476D4—44no—
477D4—44no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—164no—
522C2×C4—162no—
523C2×C4#53412yes—
524C2×C4#53612yes—
525C2×C4#57714yes—
526C2×C4C2×H(5,𝔽2)12yes—
527C2×C4C2×2−1+412yes—
528C2×C4#58512yes—
529C8—44no—
530C8—44no—
531C22—22no—
532C22—22no—
533C22—24no—
534C22—24no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—42no—
540C22—42no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—42no—
545C22—42no—
546C22C2×#53411yescharacteristic
547C22C2×#53611yescharacteristic
548C22#344911yescharacteristic
549C22C2×#57712yes—
550C22#358612yes—
551C4—22no—
552C4—22no—
553C4—24no—
554C4—41no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4—84no—
560C4—82no—
561C4—82no—
562C4—84no—
563C4—82no—
564C4#353514yes—
565C2—42no—
566C2—42no—
567C2—81no—
568C2#3744111yescharacteristic
569C2#6076312yes—
570C2#6088912yes—
571C2#6101211yescharacteristic
572C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
3. #32755C2semidirect
4. #30366C2semidirect
5. #30359C2semidirect
6. #29971C2semidirect
7. #29951C2semidirect
8. #29908C2semidirect
9. #29645C2semidirect
10. #29307C2semidirect
11. #29117C2semidirect
13. #24619C2semidirect

Maximal subgroups

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

subgroupindex
#353072
#352802
#327552
#303662
#303592
#299712
#299512
#299082
#296452
#293072
#291172
#276092
#246192
#240162
#207202