Finite Group Databasebeta

← #6488643

Subgroups — #6488643

Order 512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35268C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#33015C214yesmaximal, complemented
4#30080C212yesmaximal, complemented
5#30041C212yesmaximal, complemented
6#29500C212yesmaximal, complemented
7#29494C212yesmaximal, complemented
8#29264C212yesmaximal, complemented
9#29118C212yesmaximal, complemented
10#28323C212yesmaximal
11#28166C212yesmaximal, complemented
12#27650C212yesmaximal, complemented
13#24209C212yesmaximal, complemented
14#22162C214yesmaximal, complemented
15#20785C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#519C2212yes—
18C2×#482C2214yes—
19C2×#461C2214yes—
20C2×#448C2214yes—
21#2864C2211yescharacteristic
22#2860C2212yes—
23#2858C2211yescharacteristic
24#2628C2211yescharacteristic
25#2624C2212yes—
26#2619C2211yescharacteristic
27#2521—22no—
28#2516—22no—
29#2467—22no—
30#2459—22no—
31#2365—22no—
32#2365—22no—
33#2344—22no—
34#2344—22no—
35#2331C2211yescharacteristic
36#2328C2211yescharacteristic
37#2327C2212yes—
38#2327C2212yes—
39#2319C2212yes—
40#2310C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×C4⋊Q8C2212yes—
43C4×#121C2212yes—
44#2285C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2097C2214yes—
47#2084C2212yes—
48#2083C2212yes—
49#2074C2214yes—
50#2071C2212yes—
51#2071C2212yes—
52#2043C2212yes—
53#2043C2212yes—
54#2024C2214yes—
55#2014C2214yes—
56#2013C2212yes—
57#2013C2212yes—
58#1981C2214yes—
59#1973C2214yes—
60#1968C2214yes—
61#1949C2214yes—
62#1935C2214yes—
63#1933C2214yes—
64#1880C2214yes—
65#1880C2214yes—
66#1880C2214yes—
67#1880C2214yes—
68#1865C2214yes—
69#1865C2214yes—
70#1863C2214yes—
71#1863C2214yes—
72#1839C2212yes—
73#1832C2212yes—
74#1825C2214yes—
75#1818C2212yes—
76#1804C2214yes—
77#1797C2212yes—
78#1787C2214yes—
79#1768C2212yes—
80#1750C2212yes—
81#1745C2211yescharacteristic
82#1744C2211yescharacteristic
83C23×Q8C2312yes—
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#126—22no—
93C2×#124—22no—
94C2×#121C2312yes—
95#519—48no—
96C2×C4×Q8—24no—
97C2×C4×Q8C2314yes—
98C2×C4×Q8C2314yes—
99C2×C4×D4—22no—
100C2×C4×D4C2312yes—
101C2×#117—24no—
102C2×#117C2312yes—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C22×C4⋊C4—22no—
106C22×C42C2311yescharacteristic
107#482—48no—
108#461—216no—
109#448—48no—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×C4⋊C8C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×C22⋊C8C2314yes—
122C2×#97C2312yes—
123C2×C4×C8C2312yes—
124#400—21no—
125#399—22no—
126#399—21no—
127#397—22no—
128#395—21no—
129#394—21no—
130#394—21no—
131#394—22no—
132#394—22no—
133#393—21no—
134#392—21no—
135#392—21no—
136#391—22no—
137#391—22no—
138#391—22no—
139#391C2314yes—
140#391C2314yes—
141#390—22no—
142#390C2312yes—
143#390C2312yes—
144#390C2312yes—
145#390C2312yes—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389C2312yes—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#387—22no—
161#387—22no—
162#387—24no—
163#387—24no—
164#387—24no—
165#387—24no—
166#387—22no—
167#386—22no—
168#386—22no—
169#386—22no—
170#384C2312yes—
171#382C2312yes—
172#382C2311yescharacteristic
173#382C2311yescharacteristic
174#381—22no—
175#381—22no—
176#381—22no—
177#380—22no—
178#380—22no—
179#380C2312yes—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4D412yes—
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4D412yes—
190C4×C22⋊C4D412yes—
191#376C2311yescharacteristic
192#376C2311yescharacteristic
193C43C2311yescharacteristic
194C43C2311yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202#336C2312yes—
203#336C2314yes—
204#336C2314yes—
205C22×Q8—28no—
206C22×Q8—22no—
207C22×Q8—21no—
208C22×Q8—28no—
209C22×Q8—42no—
210C22×Q8—42no—
211C22×Q8C2414yes—
212C22×D4—22no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4C2412yes—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4C2×D412yes—
219C23×C4C2411yescharacteristic
220C2×SD16—28no—
221C2×SD16—28no—
222C2×SD16—44no—
223C2×SD16—44no—
224C2×SD16—44no—
225C2×SD16—44no—
226C22×C8C2414yes—
227C22×C8C2414yes—
228C4⋊Q8—216no—
229C4⋊Q8—28no—
230C4⋊Q8—24no—
231C4⋊Q8—42no—
232C4⋊Q8#4118yes—
233#127—42no—
234#126—84no—
235#124—84no—
236#121—24no—
237#121—24no—
238C4×Q8—44no—
239C4×Q8—44no—
240C4×Q8—44no—
241C4×Q8—44no—
242C4×Q8—416no—
243C4×D4—28no—
244C4×D4—84no—
245#117—24no—
246#117—28no—
247#117—42no—
248#117—416no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
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⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300#107—44no—
301#106—44no—
302C4⋊C8—28no—
303#103—28no—
304#103—28no—
305#103—44no—
306#103—44no—
307#103—44no—
308#103—44no—
309#102—44no—
310#102—44no—
311C22⋊C8—28no—
312#97—42no—
313C4×C8—42no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—22no—
329#95C2×D412yes—
330#95C2×D412yes—
331#95C2×D412yes—
332#95C2×D412yes—
333C24—22no—
334C24C22×D411yescharacteristic
335C2×Q8—28no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—416no—
342C2×Q8—416no—
343C2×Q8—44no—
344C2×Q8—42no—
345C2×Q8—44no—
346C2×Q8—82no—
347C2×Q8C2×#4118yes—
348C2×D4—24no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—24no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—44no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4—82no—
358C2×D4—82no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—28no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D412yes—
384C22×C4H(5,𝔽2)14yes—
385C22×C4C2511yescharacteristic
386SD16—88no—
387SD16—88no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C4⋊C4—28no—
395C4⋊C4—28no—
396C4⋊C4—216no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—44no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C22⋊C4—24no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C22⋊C4—82no—
430C42—24no—
431C42—24no—
432C42—24no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—42no—
439C42—42no—
440C42—48no—
441C42—48no—
442C42—44no—
443C42—82no—
444C42C2×#4114yes—
445C23—22no—
446C23—21no—
447C23—21no—
448C23—22no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23#53512yes—
457C23C23×D411yescharacteristic
458Q8—28no—
459Q8—28no—
460Q8—416no—
461Q8—44no—
462Q8—44no—
463Q8—162no—
464D4—44no—
465D4—44no—
466D4—88no—
467D4—162no—
468D4—162no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—28no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—28no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—48no—
492C2×C4—416no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—164no—
508C2×C4#52514yes—
509C2×C4#53512yes—
510C2×C4#57714yes—
511C2×C4C22×#4112yes—
512C2×C4C2×H(5,𝔽2)12yes—
513C2×C4#58512yes—
514C8—44no—
515C8—44no—
516C22—22no—
517C22—24no—
518C22—41no—
519C22—41no—
520C22—42no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22C2×#52511yescharacteristic
529C22C2×#53511yescharacteristic
530C22#344911yescharacteristic
531C22C2×#57712yes—
532C22#358612yes—
533C4—22no—
534C4—24no—
535C4—24no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—48no—
541C4—42no—
542C4—42no—
543C4—82no—
544C4—84no—
545C4—82no—
546C4—82no—
547C4#351314yes—
548C2—42no—
549C2—162no—
550C2#3523211yescharacteristic
551C2#6058212yes—
552C2#6083612yes—
553C2#6101211yescharacteristic
554C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
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
2. #34209C2semidirect
3. #33015C2semidirect
4. #30080C2semidirect
5. #30041C2semidirect
6. #29500C2semidirect
7. #29494C2semidirect
8. #29264C2semidirect
9. #29118C2semidirect
11. #28166C2semidirect
12. #27650C2semidirect
13. #24209C2semidirect
14. #22162C2semidirect
15. #20785C2semidirect

Maximal subgroups

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

subgroupindex
#352682
#342092
#330152
#300802
#300412
#295002
#294942
#292642
#291182
#283232
#281662
#276502
#242092
#221622
#207852