Finite Group Databasebeta

← #6493785

Subgroups — #6493785

Order 512.

1551 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35309C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#33157C212yesmaximal, complemented
4#33117C212yesmaximal, complemented
5#29943C212yesmaximal, complemented
6#29922C212yesmaximal, complemented
7#27861C212yesmaximal, complemented
8#27855C212yesmaximal, complemented
9#27680C212yesmaximal, complemented
10#27652C212yesmaximal, complemented
11#27638C214yesmaximal, complemented
12#24569C214yesmaximal
13#21967C212yesmaximal, complemented
14#21898C212yesmaximal, complemented
15#20630C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#485C2214yes—
19C2×#424C2212yes—
20C2×#408C2212yes—
21#2857C2211yescharacteristic
22#2855C2212yes—
23#2851C2211yescharacteristic
24#2629C2211yescharacteristic
25#2621C2211yescharacteristic
26#2616C2212yes—
27#2468—22no—
28#2466—22no—
29#2462—22no—
30#2460—22no—
31#2355—22no—
32#2354—22no—
33#2352—22no—
34#2352—22no—
35#2323C2212yes—
36#2314C2212yes—
37#2314C2212yes—
38#2314C2211yescharacteristic
39#2311C2212yes—
40C4×C22⋊Q8C2212yes—
41#2285C2212yes—
42#2285C2212yes—
43#2285C2211yescharacteristic
44C2×#384C2211yescharacteristic
45#2274C2212yes—
46#2099C2212yes—
47#2099C2212yes—
48#2096C2212yes—
49#2096C2212yes—
50#2084C2212yes—
51#2083C2212yes—
52#2077C2214yes—
53#2076C2214yes—
54#2072C2212yes—
55#2072C2212yes—
56#2035C2214yes—
57#2035C2214yes—
58#1989C2214yes—
59#1981C2214yes—
60#1949C2214yes—
61#1949C2214yes—
62#1948C2214yes—
63#1948C2214yes—
64#1928C2214yes—
65#1927C2214yes—
66#1920C2214yes—
67#1919C2214yes—
68#1894C2214yes—
69#1892C2214yes—
70#1872C2214yes—
71#1853C2214yes—
72#1825C2214yes—
73#1809C2214yes—
74#1807C2214yes—
75#1805C2214yes—
76#1803C2214yes—
77#1798C2212yes—
78#1798C2212yes—
79#1797C2212yes—
80#1797C2212yes—
81#1728C2211yescharacteristic
82C2×#336C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C22×M16C2312yes—
85C23×C8C2312yes—
86#541—28no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#126—22no—
89C2×#125—22no—
90C2×#125C2311yescharacteristic
91C2×C22⋊Q8C2312yes—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C4×#41—28no—
95C2×C4×Q8D412yes—
96C2×C4×Q8C2312yes—
97C2×C4×D4D412yes—
98C2×C4×D4D412yes—
99C2×C4×D4D412yes—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117C2312yes—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C22×C4⋊C4—22no—
107C22×C4⋊C4C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111#485—48no—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×#106C2314yes—
115C2×#106C2314yes—
116#424—28no—
117C2×C4⋊C8C2314yes—
118C2×C4⋊C8C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123#408—44no—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2314yes—
126#398—22no—
127#398—22no—
128#397—22no—
129#396—22no—
130#395—22no—
131#394—22no—
132#394—22no—
133#394—22no—
134#391—22no—
135#391C2312yes—
136#391C2312yes—
137#390—22no—
138#389—22no—
139#389—22no—
140#389C2312yes—
141#389C2312yes—
142#388—24no—
143#388—24no—
144#388—22no—
145#388—22no—
146#388—22no—
147#387—24no—
148#387—24no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#386—22no—
156#386—22no—
157#385—22no—
158#385—22no—
159#385—22no—
160#384—22no—
161#384C2312yes—
162#384C2311yescharacteristic
163#384C2312yes—
164#384C2312yes—
165#382—22no—
166#382—22no—
167#382C2312yes—
168#382C2311yescharacteristic
169#381—22no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380—22no—
174#380C2314yes—
175#380C2314yes—
176#380C2312yes—
177#380C2312yes—
178#380C2312yes—
179C4×C4⋊C4—24no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2314yes—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—24no—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2312yes—
198#336C2312yes—
199#336C2312yes—
200#336C2312yes—
201C2×#41—24no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41—42no—
205C22×Q8C2×D411yescharacteristic
206C22×Q8C2412yes—
207C22×D4C2×D411yescharacteristic
208C22×D4C2×D411yescharacteristic
209C22×D4C2×D411yescharacteristic
210C23×C4—22no—
211C23×C4C2×D412yes—
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C2×M16—24no—
218C2×M16—42no—
219C2×M16—42no—
220C22×C8—24no—
221C22×C8—42no—
222C22×C8—42no—
223C22×C8C2414yes—
224C22×C8C2414yes—
225C4⋊Q8—24no—
226#126—48no—
227#125—24no—
228#125—48no—
229C22⋊Q8—28no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—44no—
233C4×Q8—28no—
234C4×Q8—28no—
235C4×D4—28no—
236C4×D4—28no—
237C4×D4—28no—
238#117—24no—
239#117—24no—
240#117—24no—
241#117—24no—
242#117—84no—
243#117—84no—
244#117#4118yes—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—84no—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4C2×D412yes—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42C2×D412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D412yes—
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297#107—44no—
298#106—44no—
299#106—44no—
300#106—44no—
301C4⋊C8—28no—
302C4⋊C8—28no—
303#103—44no—
304#103—44no—
305#103—44no—
306#103—44no—
307C22⋊C8—28no—
308C22⋊C8—44no—
309#95—24no—
310#95—24no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325C24C22×D411yescharacteristic
326C24C22×D411yescharacteristic
327C24C22×D411yescharacteristic
328#41—48no—
329#41—48no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—41no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—84no—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4C22×D412yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D412yes—
379C22×C4C2×#4114yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+412yes—
382C22×C4C2511yescharacteristic
383M16—84no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—84no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—84no—
416C4⋊C4#12318yes—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C22⋊C4—84no—
427C22⋊C4—84no—
428C22⋊C4#12318yes—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—22no—
433C42—22no—
434C42—44no—
435C42—44no—
436C42—82no—
437C42—84no—
438C42—84no—
439C42C2×#4114yes—
440C42C2×#4114yes—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23C2×#12312yes—
455C23#53712yes—
456C23C23×D411yescharacteristic
457Q8—44no—
458Q8—44no—
459Q8—44no—
460D4—44no—
461D4—44no—
462D4—44no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—164no—
507C2×C4C2×#12312yes—
508C2×C4C2×#12314yes—
509C2×C4C2×#12314yes—
510C2×C4#53712yes—
511C2×C4#57714yes—
512C2×C4C22×#4112yes—
513C2×C4C2×H(5,𝔽2)11yescharacteristic
514C2×C4C2×2−1+411yescharacteristic
515C2×C4#58512yes—
516C8—44no—
517C8—44no—
518C22—24no—
519C22—24no—
520C22—22no—
521C22—22no—
522C22—42no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—42no—
528C22—42no—
529C22—41no—
530C22—41no—
531C22—42no—
532C22—41no—
533C22C22×#12311yescharacteristic
534C22C2×#53711yescharacteristic
535C22#344911yescharacteristic
536C22C2×#57712yes—
537C22#358612yes—
538C4—24no—
539C4—24no—
540C4—24no—
541C4—22no—
542C4—22no—
543C4—42no—
544C4—41no—
545C4—82no—
546C4—84no—
547C4—84no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4#349412yes—
552C4#349512yes—
553C2—42no—
554C2—42no—
555C2—81no—
556C2#3526311yescharacteristic
557C2#6052212yes—
558C2#6093112yes—
559C2#6101211yescharacteristic
560C2#6102011yescharacteristic
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. #33157C2semidirect
4. #33117C2semidirect
5. #29943C2semidirect
6. #29922C2semidirect
7. #27861C2semidirect
8. #27855C2semidirect
9. #27680C2semidirect
10. #27652C2semidirect
11. #27638C2semidirect
13. #21967C2semidirect
14. #21898C2semidirect

Maximal subgroups

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

subgroupindex
#353092
#341882
#331572
#331172
#299432
#299222
#278612
#278552
#276802
#276522
#276382
#245692
#219672
#218982
#206302