Finite Group Databasebeta

← #6495474

Subgroups — #6495474

Order 512.

1513 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2C4×#540C211yesmaximal, characteristic
3#33199C212yesmaximal, complemented
4#33197C212yesmaximal, complemented
5#29949C212yesmaximal, complemented
6#29700C212yesmaximal
7#29327C212yesmaximal, complemented
8#29275C212yesmaximal, complemented
9#29136C212yesmaximal, complemented
10#28202C212yesmaximal, complemented
11#27641C214yesmaximal, complemented
12#24708C212yesmaximal, complemented
13#24701C212yesmaximal, complemented
14#22122C214yesmaximal, complemented
15#20815C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C2×#478C2214yes—
18C2×#476C2212yes—
19C2×#476C2212yes—
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2647C2212yes—
26#2629C2212yes—
27#2507—22no—
28#2497—22no—
29#2446—22no—
30#2429—22no—
31#2335—22no—
32#2333—22no—
33#2330C2211yescharacteristic
34#2330C2212yes—
35#2328C2211yescharacteristic
36#2324C2212yes—
37#2319C2212yes—
38#2314C2211yescharacteristic
39#2314C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×#127C2211yescharacteristic
42C4×#125C2211yescharacteristic
43C4×#123—22no—
44C4×#121—22no—
45C4×#121C2212yes—
46C42×D4C2212yes—
47C4×#117C2211yescharacteristic
48#2101C2214yes—
49#2084C2214yes—
50#2077C2214yes—
51#2077C2214yes—
52#2059C2214yes—
53#2041C2214yes—
54#2034C2214yes—
55#2033C2214yes—
56#1989C2214yes—
57#1987C2212yes—
58#1987C2212yes—
59#1949C2214yes—
60#1948C2212yes—
61#1948C2214yes—
62#1948C2214yes—
63#1947C2212yes—
64#1943C2214yes—
65#1941C2214yes—
66#1936C2214yes—
67#1934C2214yes—
68#1926C2214yes—
69#1926C2214yes—
70#1920C2214yes—
71#1918C2214yes—
72#1842C2212yes—
73#1831C2212yes—
74#1804C2214yes—
75#1802C2214yes—
76#1802C2212yes—
77#1802C2212yes—
78#1802C2214yes—
79#1798C2212yes—
80#1798C2212yes—
81#1768C2212yes—
82#1768C2212yes—
83#540—28no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×#127C2311yescharacteristic
87C2×#125C2312yes—
88C2×#125C2311yescharacteristic
89C2×#123—22no—
90C2×C22⋊Q8—22no—
91C2×#121—22no—
92C2×#121C2312yes—
93C2×C4×Q8—22no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4C2312yes—
97C2×C4×D4C2312yes—
98C2×C4×D4C2312yes—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C42—22no—
102C22×C42C2311yescharacteristic
103#478—216no—
104#476—44no—
105#476—44no—
106#432—42no—
107#431—42no—
108C2×#107C2314yes—
109C2×#106C2314yes—
110C2×#106C2314yes—
111C2×#106C2314yes—
112C2×C4⋊C8C2314yes—
113C2×C4⋊C8C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×C22⋊C8C2312yes—
121C2×C22⋊C8C2312yes—
122C2×#97C2312yes—
123C2×C4×C8C2312yes—
124#400—24no—
125#398—24no—
126#398—24no—
127#396—24no—
128#391—22no—
129#391—22no—
130#390—22no—
131#390—22no—
132#390C2312yes—
133#390C2312yes—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—24no—
140#389C2312yes—
141#389C2314yes—
142#389C2312yes—
143#389C2314yes—
144#388—22no—
145#388—22no—
146#388—24no—
147#388—22no—
148#388—22no—
149#388—22no—
150#387—22no—
151#387—24no—
152#387—22no—
153#387—22no—
154#387—24no—
155#386—22no—
156#386—22no—
157#386—22no—
158#386—22no—
159#385—22no—
160#384—22no—
161#384C2311yescharacteristic
162#384C2312yes—
163#384C2312yes—
164#384C2312yes—
165#382C2311yescharacteristic
166#382C2312yes—
167#381—22no—
168#380—22no—
169#380—22no—
170#380C2312yes—
171#380C2314yes—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4D412yes—
185C4×C22⋊C4D412yes—
186#376C2311yescharacteristic
187#376C2311yescharacteristic
188C43C2311yescharacteristic
189C43C2311yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C22×Q8—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4C2412yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2411yescharacteristic
210C22×C8C2414yes—
211C22×C8C2414yes—
212C4⋊Q8—24no—
213C4⋊Q8—24no—
214C4⋊Q8—84no—
215#127—24no—
216#125—24no—
217#125—28no—
218#123—216no—
219C22⋊Q8—84no—
220#121—216no—
221#121—24no—
222#121—24no—
223C4×Q8—84no—
224C4×D4—24no—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—84no—
230C4×D4—84no—
231C4×D4#4118yes—
232#117—24no—
233#117—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2414yes—
251C2×C4⋊C4C2414yes—
252C2×C4⋊C4C2412yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4C2×D412yes—
264C2×C22⋊C4C2×D412yes—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—84no—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282C2×C42C2411yescharacteristic
283C2×C42C2411yescharacteristic
284#107—28no—
285#106—28no—
286#106—44no—
287#106—44no—
288C4⋊C8—28no—
289C4⋊C8—24no—
290C4⋊C8—24no—
291C4⋊C8—28no—
292#102—28no—
293#102—28no—
294#102—44no—
295#102—44no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298#97—42no—
299C4×C8—42no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95C2×D412yes—
313#95C2×D412yes—
314#95C2×D412yes—
315#95C2×D412yes—
316C24—22no—
317C24C22×D411yescharacteristic
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—82no—
329C2×D4—82no—
330C2×D4—82no—
331C2×D4—82no—
332C2×D4C2×#4114yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—28no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2511yescharacteristic
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C4⋊C4—28no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—44no—
384C4⋊C4—48no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4C2×#4114yes—
395C22⋊C4—28no—
396C22⋊C4—28no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C22⋊C4—84no—
407C42—22no—
408C42—22no—
409C42—24no—
410C42—24no—
411C42—22no—
412C42—21no—
413C42—21no—
414C42—22no—
415C42—42no—
416C42—42no—
417C42—44no—
418C42—82no—
419C42—82no—
420C42—84no—
421C42C2×#4114yes—
422C42H(5,𝔽2)12yes—
423C422−1+414yes—
424C422−1+412yes—
425C23—22no—
426C23—21no—
427C23—24no—
428C23—21no—
429C23—21no—
430C23—22no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23C23×D411yescharacteristic
438Q8—88no—
439D4—24no—
440D4—24no—
441D4—42no—
442D4—42no—
443D4—88no—
444D4—88no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—82no—
482C2×C4—88no—
483C2×C4—88no—
484C2×C4—88no—
485C2×C4#52514yes—
486C2×C4#53614yes—
487C2×C4#53712yes—
488C2×C4#53712yes—
489C2×C4#57714yes—
490C2×C4C22×#4112yes—
491C2×C4C2×H(5,𝔽2)11yescharacteristic
492C2×C4C2×2−1+411yescharacteristic
493C2×C4C2×2−1+412yes—
494C8—44no—
495C8—44no—
496C22—44no—
497C22—42no—
498C22—44no—
499C22—42no—
500C22—41no—
501C22—44no—
502C22—41no—
503C22—41no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22C2×#52511yescharacteristic
509C22C2×#53611yescharacteristic
510C22C2×#53711yescharacteristic
511C22C2×#57712yes—
512C22#358512yes—
513C4—24no—
514C4—24no—
515C4—22no—
516C4—42no—
517C4—44no—
518C4—42no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—84no—
523C4—82no—
524C4—82no—
525C4—84no—
526C4#326014yes—
527C4#350912yes—
528C4#351112yes—
529C2—81no—
530C2—84no—
531C2#3529711yescharacteristic
532C2#6059712yes—
533C2#6089212yes—
534C2C2×#326011yescharacteristic
535C2#6092011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
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
1. #35284C2semidirect
3. #33199C2semidirect
4. #33197C2semidirect
5. #29949C2semidirect
7. #29327C2semidirect
8. #29275C2semidirect
9. #29136C2semidirect
10. #28202C2semidirect
11. #27641C2semidirect
12. #24708C2semidirect
13. #24701C2semidirect
14. #22122C2semidirect
15. #20815C2semidirect

Maximal subgroups

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

subgroupindex
#352842
C4×#5402
#331992
#331972
#299492
#297002
#293272
#292752
#291362
#282022
#276412
#247082
#247012
#221222
#208152