Finite Group Databasebeta

← #6651610

Subgroups — #6651610

Order 512.

1631 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#35111C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3#33098C211yesmaximal, characteristic, complemented
4#33094C211yesmaximal, characteristic
5#32690C214yesmaximal, complemented
6#32639C214yesmaximal, complemented
7#32586C214yesmaximal, complemented
8#32584C214yesmaximal, complemented
9#29641C211yesmaximal, characteristic, complemented
10#29638C211yesmaximal, characteristic
11#27616C212yesmaximal, complemented
12#24609C212yesmaximal, complemented
13#24608C212yesmaximal
14#24066C212yesmaximal
15#21822C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C2×#529C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#483C2212yes—
21C2×#480C2212yes—
22C2×#432C2212yes—
23#2855C2211yescharacteristic
24#2855C2211yescharacteristic
25#2854C2211yescharacteristic
26#2854C2211yescharacteristic
27#2623—21no—
28#2614—21no—
29#2604—22no—
30#2599—22no—
31#2592C2212yes—
32#2584C2212yes—
33#2581C2212yes—
34#2580C2212yes—
35#2546—22no—
36#2546—22no—
37#2546C2214yes—
38#2546C2214yes—
39C2×#398—21no—
40C2×#398—21no—
41#2322—21no—
42#2322—21no—
43C2×#391—21no—
44C2×#389C2212yes—
45C2×#389C2212yes—
46#2283C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48#2101C2214yes—
49#2088C2214yes—
50#2087C2214yes—
51#2085C2214yes—
52#2068C2214yes—
53#2065C2214yes—
54#2059C2214yes—
55#2058C2214yes—
56#2057C2214yes—
57#2042C2214yes—
58#2024C2214yes—
59#2023C2214yes—
60#1985C2214yes—
61#1985C2214yes—
62#1944C2214yes—
63#1943C2214yes—
64#1936C2214yes—
65#1935C2214yes—
66#1927C2214yes—
67#1925C2214yes—
68#1921C2214yes—
69#1920C2214yes—
70#1920C2214yes—
71#1918C2214yes—
72#1874C2212yes—
73#1872C2212yes—
74#1864C2214yes—
75#1863C2214yes—
76#1853C2212yes—
77#1852C2212yes—
78#1823C2212yes—
79#1823C2212yes—
80#1817C2212yes—
81#1817C2211yescharacteristic
82#1817C2211yescharacteristic
83C23×Q8—21no—
84C23×Q8D411yescharacteristic
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—44no—
89C2×C4⋊Q8—48no—
90C2×C4⋊Q8C2314yes—
91#529—44no—
92C2×#126—21no—
93C2×#125C2311yescharacteristic
94C2×#124C2311yescharacteristic
95C2×#123—21no—
96C2×#123—21no—
97C2×#123—21no—
98C2×C22⋊Q8C2311yescharacteristic
99C2×#121C2311yescharacteristic
100C2×C4×Q8—44no—
101C2×C4×Q8D412yes—
102C2×C4×Q8D412yes—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—21no—
107C2×C4×D4C2311yescharacteristic
108C2×#117—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#483—44no—
122#480—44no—
123#432—44no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×#97C2314yes—
136#400—22no—
137#399—22no—
138#398—22no—
139#398—24no—
140#398—24no—
141#398—24no—
142#398—24no—
143#397—22no—
144#395—22no—
145#395—22no—
146#393—22no—
147#393—22no—
148#391—24no—
149#391—24no—
150#391C2312yes—
151#391C2312yes—
152#390C2312yes—
153#390C2312yes—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2312yes—
161#389C2312yes—
162#389C2314yes—
163#389C2314yes—
164#389C2314yes—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#386—21no—
177#386—21no—
178#385—21no—
179#385—21no—
180#385—22no—
181#385—22no—
182#383—21no—
183#382C2312yes—
184#381—21no—
185#381—21no—
186#381—21no—
187#380C2311yescharacteristic
188#380C2311yescharacteristic
189C4×C4⋊C4—22no—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2314yes—
192C4×C22⋊C4—21no—
193#376—21no—
194C2×#95—21no—
195C2×#95—22no—
196C2×#95—22no—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#336C2314yes—
206#336C2314yes—
207#336C2314yes—
208#336C2314yes—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—42no—
212C22×Q8—42no—
213C22×Q8—44no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×Q8C2×D412yes—
217C22×Q8C2×D412yes—
218C22×Q8C2411yescharacteristic
219C22×D4—21no—
220C22×D4C2411yescharacteristic
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2411yescharacteristic
231C22×C8C2414yes—
232C22×C8C2414yes—
233C4⋊Q8—44no—
234C4⋊Q8—44no—
235C4⋊Q8—48no—
236C4⋊Q8—416no—
237C4⋊Q8—416no—
238#126—44no—
239#125—42no—
240#124—42no—
241#123—44no—
242#123—44no—
243#123—44no—
244C22⋊Q8—24no—
245#121—24no—
246C4×Q8—44no—
247C4×Q8—44no—
248C4×Q8—42no—
249C4×Q8—42no—
250C4×Q8—42no—
251C4×Q8—42no—
252C4×Q8—42no—
253C4×D4—42no—
254C4×D4—44no—
255#117—44no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2411yescharacteristic
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2414yes—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—21no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—44no—
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2412yes—
317C2×C42C2412yes—
318#107—28no—
319#107—44no—
320#106—28no—
321#106—44no—
322C4⋊C8—44no—
323#103—28no—
324#103—44no—
325#102—28no—
326#102—44no—
327C22⋊C8—24no—
328C22⋊C8—24no—
329#97—44no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—21no—
336#95—21no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348C24—21no—
349C24C22×D411yescharacteristic
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—22no—
353C2×Q8—24no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—416no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—816no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—82no—
398C22×C4AGL(1,ℤ/8)14yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4H(5,𝔽2)14yes—
409C22×C4C2511yescharacteristic
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C4⋊C4—28no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C42−1+418yes—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—44no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—44no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C42—22no—
457C42—24no—
458C42—22no—
459C42—24no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—42no—
467C42—48no—
468C42—82no—
469C23—22no—
470C23—22no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23#53412yes—
477C23C2×AGL(1,ℤ/8)12yes—
478C23#57512yes—
479C23C23×D411yescharacteristic
480Q8—48no—
481Q8—48no—
482Q8—44no—
483Q8—816no—
484D4—44no—
485D4—84no—
486C2×C4—22no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—28no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—44no—
512C2×C4—416no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4—88no—
537C2×C4#53412yes—
538C2×C4C2×AGL(1,ℤ/8)14yes—
539C2×C4C2×H(5,𝔽2)12yes—
540C2×C4C2×2−1+414yes—
541C8—44no—
542C8—44no—
543C22—22no—
544C22—24no—
545C22—24no—
546C22—22no—
547C22—81no—
548C22—81no—
549C22—82no—
550C22—82no—
551C22—81no—
552C22#319414yes—
553C22C2×#53411yescharacteristic
554C22#348612yes—
555C22C22×AGL(1,ℤ/8)11yescharacteristic
556C22C2×#57511yescharacteristic
557C22#358612yes—
558C4—24no—
559C4—22no—
560C4—41no—
561C4—41no—
562C4—42no—
563C4—42no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—48no—
568C4—84no—
569C4—81no—
570C4—81no—
571C4—82no—
572C4—82no—
573C4—81no—
574C4#354414yes—
575C2—24no—
576C2—161no—
577C2#3725811yescharacteristic
578C2C2×#319411yescharacteristic
579C2#6075411yescharacteristic
580C2#6182712yes—
581C2#6183212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2546C2semidirect
3. #33098C2semidirect
5. #32690C2semidirect
6. #32639C2semidirect
7. #32586C2semidirect
8. #32584C2semidirect
9. #29641C2semidirect
11. #27616C2semidirect
12. #24609C2semidirect
15. #21822C2semidirect

Maximal subgroups

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

subgroupindex
#351112
C2×#25462
#330982
#330942
#326902
#326392
#325862
#325842
#296412
#296382
#276162
#246092
#246082
#240662
#218222