Finite Group Databasebeta

← #6488644

Subgroups — #6488644

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35314C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic
3#33019C214yesmaximal
4#30080C212yesmaximal, complemented
5#30042C212yesmaximal, complemented
6#29501C212yesmaximal, complemented
7#29495C212yesmaximal, complemented
8#29264C212yesmaximal, complemented
9#29118C212yesmaximal
10#28322C212yesmaximal
11#28170C212yesmaximal, complemented
12#27652C212yesmaximal
13#24212C212yesmaximal
14#22164C214yesmaximal, complemented
15#20785C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#519C2212yes—
18C2×#485C2214yes—
19C2×#462C2214yes—
20C2×#451C2214yes—
21#2864C2211yescharacteristic
22#2863C2212yes—
23#2862C2211yescharacteristic
24#2628C2211yescharacteristic
25#2625C2212yes—
26#2621C2211yescharacteristic
27#2524—22no—
28#2523—22no—
29#2520—22no—
30#2517—22no—
31#2365—22no—
32#2365—22no—
33#2344—22no—
34#2344—22no—
35#2331C2211yescharacteristic
36#2327C2211yescharacteristic
37#2327C2212yes—
38#2327C2212yes—
39#2320C2212yes—
40#2310C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×C4⋊Q8C2212yes—
43C4×C22⋊Q8C2212yes—
44#2285C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2097C2214yes—
47#2084C2212yes—
48#2083C2212yes—
49#2074C2214yes—
50#2072C2212yes—
51#2072C2212yes—
52#2043C2212yes—
53#2043C2212yes—
54#2024C2214yes—
55#2016C2214yes—
56#2016C2212yes—
57#2016C2212yes—
58#1981C2214yes—
59#1971C2214yes—
60#1969C2214yes—
61#1949C2214yes—
62#1935C2214yes—
63#1933C2214yes—
64#1878C2214yes—
65#1878C2214yes—
66#1878C2214yes—
67#1878C2214yes—
68#1865C2214yes—
69#1865C2214yes—
70#1863C2214yes—
71#1863C2214yes—
72#1839C2212yes—
73#1832C2212yes—
74#1825C2214yes—
75#1818C2212yes—
76#1805C2214yes—
77#1797C2212yes—
78#1787C2214yes—
79#1768C2212yes—
80#1750C2212yes—
81#1745C2211yescharacteristic
82#1744C2211yescharacteristic
83C23×Q8C2312yes—
84C22×Q16C2314yes—
85C22×Q16C2314yes—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C22⋊Q8C2312yes—
93#519—48no—
94C2×C4×Q8—24no—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2314yes—
97C2×C4×Q8C2314yes—
98C2×#117—24no—
99C2×#117C2312yes—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—22no—
103C22×C42C2311yescharacteristic
104#485—48no—
105#462—216no—
106#451—48no—
107C2×#107C2314yes—
108C2×#106C2314yes—
109C2×C4⋊C8C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×C22⋊C8C2314yes—
119C2×#97C2312yes—
120C2×C4×C8C2312yes—
121#400—22no—
122#400—21no—
123#399—21no—
124#398—21no—
125#398—21no—
126#398—22no—
127#397—21no—
128#397—21no—
129#397—22no—
130#397—22no—
131#395—21no—
132#393—21no—
133#391—22no—
134#391—22no—
135#391—22no—
136#391—22no—
137#391C2314yes—
138#391C2314yes—
139#391C2312yes—
140#391C2312yes—
141#391C2312yes—
142#391C2312yes—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—24no—
160#387—24no—
161#387—24no—
162#387—24no—
163#387—22no—
164#387—22no—
165#387—22no—
166#385—22no—
167#384C2312yes—
168#383—22no—
169#383—22no—
170#382—22no—
171#382—22no—
172#382C2312yes—
173#382C2311yescharacteristic
174#382C2311yescharacteristic
175#381—22no—
176#381—22no—
177#380—22no—
178#380C2312yes—
179C4×C4⋊C4—22no—
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×Q8C2412yes—
212C22×Q8C2414yes—
213C23×C4—22no—
214C23×C4C2×D412yes—
215C23×C4C2411yescharacteristic
216C2×Q16—28no—
217C2×Q16—28no—
218C2×Q16—44no—
219C2×Q16—44no—
220C2×Q16—44no—
221C2×Q16—44no—
222C22×C8C2414yes—
223C22×C8C2414yes—
224C4⋊Q8—216no—
225C4⋊Q8—28no—
226C4⋊Q8—24no—
227C4⋊Q8—42no—
228C4⋊Q8—42no—
229C4⋊Q8#4118yes—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C4×Q8—28no—
233C4×Q8—44no—
234C4×Q8—44no—
235C4×Q8—44no—
236C4×Q8—44no—
237C4×Q8—416no—
238#117—24no—
239#117—28no—
240#117—42no—
241#117—416no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4C2×D412yes—
270C2×C22⋊C4C2×D411yescharacteristic
271C2×C22⋊C4C2×D411yescharacteristic
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289#107—44no—
290#106—44no—
291C4⋊C8—28no—
292#103—28no—
293#103—28no—
294#103—44no—
295#103—44no—
296#103—44no—
297#103—44no—
298#103—44no—
299#103—44no—
300C22⋊C8—28no—
301#97—42no—
302C4×C8—42no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95C2×D412yes—
327#95C2×D412yes—
328#95C2×D412yes—
329#95C2×D412yes—
330C24C22×D411yescharacteristic
331C2×Q8—28no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—22no—
337C2×Q8—22no—
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—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D412yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4H(5,𝔽2)14yes—
372C22×C4C2511yescharacteristic
373Q16—88no—
374Q16—88no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—216no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—44no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C22⋊C4—24no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—82no—
412C42—24no—
413C42—24no—
414C42—24no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—42no—
421C42—42no—
422C42—48no—
423C42—48no—
424C42—44no—
425C42—44no—
426C42—82no—
427C42C2×#4114yes—
428C23—22no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23#53512yes—
433C23C23×D411yescharacteristic
434Q8—28no—
435Q8—28no—
436Q8—416no—
437Q8—44no—
438Q8—44no—
439Q8—44no—
440Q8—44no—
441Q8—162no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—28no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—28no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—48no—
465C2×C4—416no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4#52514yes—
484C2×C4#53512yes—
485C2×C4#57814yes—
486C2×C4C22×#4112yes—
487C2×C4C2×H(5,𝔽2)12yes—
488C2×C4#58512yes—
489C8—44no—
490C8—44no—
491C22—22no—
492C22—24no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22C2×#52511yescharacteristic
497C22C2×#53511yescharacteristic
498C22#344911yescharacteristic
499C22C2×#57812yes—
500C22#358612yes—
501C4—22no—
502C4—24no—
503C4—24no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—48no—
509C4—42no—
510C4—42no—
511C4—82no—
512C4—82no—
513C4—82no—
514C4—84no—
515C4—82no—
516C4#351314yes—
517C2—42no—
518C2#3523211yescharacteristic
519C2#6058312yes—
520C2#6083712yes—
521C2#6101311yescharacteristic
522C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
4. #30080C2semidirect
5. #30042C2semidirect
6. #29501C2semidirect
7. #29495C2semidirect
8. #29264C2semidirect
11. #28170C2semidirect
14. #22164C2semidirect

Maximal subgroups

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

subgroupindex
#353142
#342092
#330192
#300802
#300422
#295012
#294952
#292642
#291182
#283222
#281702
#276522
#242122
#221642
#207852