Finite Group Databasebeta

← #6477117

Subgroups — #6477117

Order 512.

1723 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2#34203C211yesmaximal, characteristic
3#33157C212yesmaximal, complemented
4#32967C212yesmaximal, complemented
5#30182C214yesmaximal, complemented
6#29307C212yesmaximal, complemented
7#29117C212yesmaximal, complemented
8#27647C212yesmaximal, complemented
9#27602C212yesmaximal
10#24719C214yesmaximal, complemented
11#24307C212yesmaximal, complemented
12#24003C212yesmaximal
13#22084C212yesmaximal, complemented
14#22078C212yesmaximal, complemented
15#20716C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C2×#519C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C2×#483C2212yes—
21C2×#483C2212yes—
22C2×#479C2214yescomplemented
23C2×#433C2212yes—
24#2629—22no—
25#2629—22no—
26#2629C2212yes—
27#2629C2212yes—
28#2629C2212yes—
29#2629C2212yes—
30C2×#398—22no—
31C2×#396—22no—
32#2362—22no—
33#2360—22no—
34#2344—22no—
35#2342—22no—
36#2323C2212yes—
37#2323C2212yes—
38#2316C2211yescharacteristic
39#2310C2211yescharacteristic
40C4×C22⋊Q8C2212yes—
41C4×C22⋊Q8C2212yes—
42C2×#389C2212yes—
43C2×#384C2211yescharacteristic
44#2280C2211yescharacteristic
45C2×#380C2212yes—
46#2274C2211yescharacteristic
47#2084C2214yescomplemented
48#2077C2214yes—
49#2076C2214yes—
50#2072C2214yescomplemented
51#2059C2214yescomplemented
52#2035C2214yes—
53#2035C2214yes—
54#2026C2214yescomplemented
55#1989C2214yescomplemented
56#1948C2214yescomplemented
57#1944C2214yescomplemented
58#1941C2214yescomplemented
59#1935C2214yescomplemented
60#1933C2214yescomplemented
61#1920C2214yescomplemented
62#1919C2214yescomplemented
63#1872C2212yes—
64#1872C2212yes—
65#1865C2214yescomplemented
66#1863C2214yescomplemented
67#1854C2214yes—
68#1853C2212yes—
69#1853C2214yes—
70#1852C2212yes—
71#1825C2214yes—
72#1823C2212yes—
73#1823C2212yes—
74#1818C2212yes—
75#1803C2214yescomplemented
76#1798C2212yes—
77#1797C2212yes—
78#1787C2214yes—
79#1779C2212yes—
80#1750C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C23×Q8C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×#125—22no—
86C2×#125—44no—
87C2×#125C2312yes—
88C2×#125C2312yes—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94#519—44no—
95C2×C4×Q8—22no—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×#117—22no—
99C2×#117—44no—
100C2×#117D412yes—
101C2×#117D412yes—
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C22×C4⋊C4—21no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4D412yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—22no—
118C22×C22⋊C4D412yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#483—28no—
122#483—28no—
123#479—48no—
124#433—44no—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2312yes—
139C2×C4×C8C2312yes—
140#398—24no—
141#398—24no—
142#398—24no—
143#398—24no—
144#396—24no—
145#396—24no—
146#396—24no—
147#396—24no—
148#391—22no—
149#391C2312yes—
150#391C2312yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389C2312yes—
157#389C2312yes—
158#389C2314yes—
159#389C2314yes—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#385—22no—
169#385—22no—
170#384—22no—
171#384C2312yes—
172#384C2312yes—
173#381—21no—
174#381—22no—
175#380—22no—
176#380—22no—
177#380—22no—
178#380C2312yes—
179#380C2314yes—
180#380C2312yes—
181#380C2314yes—
182#379—21no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—21no—
191C4×C22⋊C4—21no—
192C4×C22⋊C4—22no—
193C2×#95—22no—
194C2×#95—22no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2312yes—
205#336C2312yes—
206C25C2×D411yescharacteristic
207C22×Q8—24no—
208C22×Q8—21no—
209C22×Q8—24no—
210C22×Q8—42no—
211C22×Q8C2412yes—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—42no—
219C23×C4C2×D412yes—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C22×C8C2414yes—
228C22×C8C2414yes—
229#125—28no—
230#125—42no—
231#125—42no—
232#125—48no—
233C22⋊Q8—28no—
234C22⋊Q8—28no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C22⋊Q8—24no—
238C22⋊Q8#4118yes—
239C4×Q8—28no—
240C4×Q8—44no—
241C4×Q8—48no—
242#117—24no—
243#117—24no—
244#117—24no—
245#117—42no—
246#117—42no—
247#117—44no—
248#117—44no—
249#117—48no—
250#117—42no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—84no—
274C2×C4⋊C4—84no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—84no—
295C2×C22⋊C4C2×D412yes—
296C2×C22⋊C4C2×D412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D412yes—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—42no—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#107—44no—
314#106—28no—
315#106—28no—
316#106—44no—
317C4⋊C8—28no—
318#103—28no—
319#103—28no—
320#103—44no—
321#103—44no—
322C22⋊C8—24no—
323C22⋊C8—24no—
324C22⋊C8—28no—
325C22⋊C8—44no—
326#97—42no—
327C4×C8—42no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—41no—
346C24—42no—
347C24C22×D411yescharacteristic
348C24C22×D411yescharacteristic
349C24C22×D411yescharacteristic
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—416no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8C2×#4114yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—416no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—82no—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4114yes—
406C22×C42−1+414yes—
407C22×C4C2511yescharacteristic
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C4⋊C4—24no—
415C4⋊C4—28no—
416C4⋊C4—28no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4C2×#4114yes—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—82no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C42—24no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—42no—
458C42—42no—
459C42—44no—
460C42—44no—
461C42—42no—
462C42—82no—
463C42—82no—
464C42H(5,𝔽2)14yes—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—22no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—44no—
478C23—48no—
479C23—44no—
480C23—84no—
481C23—82no—
482C23#52512yes—
483C23#53612yes—
484C23#57714yes—
485C23C23×D411yescharacteristic
486Q8—28no—
487Q8—48no—
488Q8—44no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—44no—
512C2×C4—48no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4—88no—
536C2×C4—88no—
537C2×C4#52512yes—
538C2×C4#53414yes—
539C2×C4#53612yes—
540C2×C4C22×#4112yes—
541C2×C4C2×H(5,𝔽2)12yes—
542C2×C4C2×2−1+412yes—
543C8—44no—
544C8—44no—
545C22—22no—
546C22—24no—
547C22—22no—
548C22—24no—
549C22—22no—
550C22—24no—
551C22—42no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—42no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—88no—
560C22#312614yes—
561C22C2×#52511yescharacteristic
562C22C2×#53411yescharacteristic
563C22C2×#53611yescharacteristic
564C22C2×#57712yes—
565C22#358612yes—
566C4—22no—
567C4—22no—
568C4—24no—
569C4—41no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—44no—
575C4—82no—
576C4—84no—
577C4—84no—
578C4—82no—
579C4#352014yes—
580C2—24no—
581C2—42no—
582C2—81no—
583C2#3508411yescharacteristic
584C2C2×#312611yescharacteristic
585C2#6056911yescharacteristic
586C2#6076312yes—
587C2#6087212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33157C2semidirect
4. #32967C2semidirect
5. #30182C2semidirect
6. #29307C2semidirect
7. #29117C2semidirect
8. #27647C2semidirect
10. #24719C2semidirect
11. #24307C2semidirect
13. #22084C2semidirect
14. #22078C2semidirect
22. C2×#479C22semidirect
47. #2084C22semidirect
50. #2072C22semidirect
51. #2059C22semidirect
54. #2026C22semidirect
55. #1989C22semidirect
56. #1948C22semidirect
57. #1944C22semidirect
58. #1941C22semidirect
59. #1935C22semidirect
60. #1933C22semidirect
61. #1920C22semidirect
62. #1919C22semidirect
65. #1865C22semidirect
66. #1863C22semidirect
75. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
#342032
#331572
#329672
#301822
#293072
#291172
#276472
#276022
#247192
#243072
#240032
#220842
#220782
#207162