Finite Group Databasebeta

← #6480289

Subgroups — #6480289

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic
2#34281C211yesmaximal, characteristic
3#33041C214yesmaximal, complemented
4#30067C212yesmaximal, complemented
5#30042C212yesmaximal, complemented
6#29559C212yesmaximal, complemented
7#29553C212yesmaximal, complemented
8#29361C212yesmaximal, complemented
9#28391C212yesmaximal, complemented
10#27592C212yesmaximal
11#24306C212yesmaximal
12#24212C212yesmaximal
13#23943C212yesmaximal
14#22054C214yesmaximal
15#20785C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#479C2214yes—
20C2×#462C2214yes—
21C2×#451C2214yes—
22#2862C2212yes—
23#2862C2212yes—
24#2642C2212yes—
25#2642C2212yes—
26#2568—22no—
27#2568—22no—
28#2503—22no—
29#2503—22no—
30#2373—22no—
31#2373—22no—
32#2371—22no—
33#2370—22no—
34#2330C2211yescharacteristic
35#2327C2211yescharacteristic
36#2323C2212yes—
37#2314C2211yescharacteristic
38#2310C2212yes—
39#2310C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×#125C2211yescharacteristic
42C4×C22⋊Q8C2212yes—
43#2295C2212yes—
44#2276C2212yes—
45C4×#117C2211yescharacteristic
46#2097C2214yes—
47#2072C2214yes—
48#2058C2212yes—
49#2057C2212yes—
50#2043C2212yes—
51#2043C2212yes—
52#2026C2214yes—
53#2024C2212yes—
54#2024C2212yes—
55#2016C2212yes—
56#2016C2212yes—
57#2016C2214yes—
58#1989C2214yes—
59#1981C2214yes—
60#1971C2214yes—
61#1969C2214yes—
62#1935C2214yes—
63#1933C2214yes—
64#1928C2214yes—
65#1928C2214yes—
66#1927C2214yes—
67#1927C2214yes—
68#1878C2214yes—
69#1878C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1873C2214yes—
73#1854C2214yes—
74#1839C2212yes—
75#1832C2212yes—
76#1818C2212yes—
77#1797C2212yes—
78#1787C2214yes—
79#1768C2212yes—
80#1750C2212yes—
81#1745C2211yescharacteristic
82#1744C2211yescharacteristic
83C23×Q8C2312yes—
84C22×Q16C2314yes—
85C22×Q16C2314yes—
86#541—28no—
87#533—44no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—44no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×C4⋊Q8C2312yes—
94C2×#126—22no—
95C2×#125—22no—
96C2×#125C2311yescharacteristic
97C2×#125C2312yes—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8C2312yes—
100C2×C4×Q8—22no—
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×#117C2312yes—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C4⋊C4—22no—
108C22×C42C2311yescharacteristic
109#479—48no—
110#462—48no—
111#451—216no—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×C22⋊C8C2314yes—
124C2×#97C2312yes—
125C2×C4×C8C2312yes—
126#400—24no—
127#400—24no—
128#400—24no—
129#400—24no—
130#391—24no—
131#391—22no—
132#391—22no—
133#391—22no—
134#391—22no—
135#391—22no—
136#391—22no—
137#391C2312yes—
138#391C2314yes—
139#391C2314yes—
140#391C2314yes—
141#391C2314yes—
142#391C2312yes—
143#389—22no—
144#389—22no—
145#389—22no—
146#389C2312yes—
147#389C2312yes—
148#388—22no—
149#388—22no—
150#388—22no—
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—22no—
160#385—22no—
161#384C2311yescharacteristic
162#382—22no—
163#382—22no—
164#382—22no—
165#382—22no—
166#382C2311yescharacteristic
167#381—24no—
168#381—22no—
169#381—22no—
170#380C2312yes—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4D412yes—
179C4×C4⋊C4D412yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183C4×C22⋊C4—22no—
184C4×C22⋊C4D412yes—
185C4×C22⋊C4D412yes—
186#376—22no—
187#376—22no—
188#376C2311yescharacteristic
189#376C2311yescharacteristic
190C43C2311yescharacteristic
191C43C2311yescharacteristic
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2312yes—
197#336C2312yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—28no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8—44no—
208C22×Q8C2414yes—
209C22×Q8C2412yes—
210C23×C4—22no—
211C23×C4C2×D412yes—
212C23×C4C2411yescharacteristic
213C2×Q16—28no—
214C2×Q16—28no—
215C2×Q16—44no—
216C2×Q16—44no—
217C2×Q16—44no—
218C2×Q16—44no—
219C22×C8C2414yes—
220C22×C8C2414yes—
221C4⋊Q8—24no—
222C4⋊Q8—28no—
223C4⋊Q8—44no—
224C4⋊Q8—44no—
225C4⋊Q8—48no—
226#126—216no—
227#125—216no—
228#125—24no—
229#125—28no—
230C22⋊Q8—24no—
231C22⋊Q8—48no—
232C22⋊Q8#4118yes—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—28no—
236C4×Q8—44no—
237C4×Q8—48no—
238#117—24no—
239#117—42no—
240#117—42no—
241#117—42no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
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—48no—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4C2×D412yes—
268C2×C22⋊C4C2×D411yescharacteristic
269C2×C22⋊C4C2×D411yescharacteristic
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
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—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95C2×D412yes—
323#95C2×D412yes—
324#95C2×D412yes—
325#95C2×D412yes—
326C24C22×D411yescharacteristic
327C2×Q8—22no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—88no—
343C2×Q8—88no—
344C2×Q8C2×#4114yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C2×#4114yes—
369C22×C4C2511yescharacteristic
370Q16—88no—
371Q16—88no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C4⋊C4—48no—
397C4⋊C4—44no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4C2×#4114yes—
401C22⋊C4—28no—
402C22⋊C4—24no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—44no—
406C22⋊C4—82no—
407C42—28no—
408C42—24no—
409C42—24no—
410C42—24no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—42no—
417C42—42no—
418C42—44no—
419C42—44no—
420C42—44no—
421C42—82no—
422C42—82no—
423C42—82no—
424C42H(5,𝔽2)14yes—
425C23—22no—
426C23—21no—
427C23—21no—
428C23—22no—
429C23#52512yes—
430C23C23×D411yescharacteristic
431Q8—28no—
432Q8—44no—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—44no—
437Q8—88no—
438Q8—88no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—28no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—48no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—88no—
480C2×C4#52512yes—
481C2×C4#53514yes—
482C2×C4#57814yes—
483C2×C4C22×#4112yes—
484C2×C4C2×H(5,𝔽2)12yes—
485C2×C4#58512yes—
486C8—44no—
487C8—44no—
488C22—22no—
489C22—24no—
490C22—41no—
491C22—41no—
492C22—42no—
493C22C2×#52511yescharacteristic
494C22C2×#53511yescharacteristic
495C22#344911yescharacteristic
496C22C2×#57812yes—
497C22#358612yes—
498C4—22no—
499C4—24no—
500C4—24no—
501C4—44no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—42no—
507C4—42no—
508C4—44no—
509C4—84no—
510C4—82no—
511C4—82no—
512C4—84no—
513C4#351414yes—
514C2—42no—
515C2#3512911yescharacteristic
516C2#6059412yes—
517C2#6081412yes—
518C2#6101311yescharacteristic
519C2#6102011yescharacteristic
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
3. #33041C2semidirect
4. #30067C2semidirect
5. #30042C2semidirect
6. #29559C2semidirect
7. #29553C2semidirect
8. #29361C2semidirect
9. #28391C2semidirect

Maximal subgroups

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

subgroupindex
#353202
#342812
#330412
#300672
#300422
#295592
#295532
#293612
#283912
#275922
#243062
#242122
#239432
#220542
#207852