Finite Group Databasebeta

← #7459842

Subgroups — #7459842

Order 512.

1519 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46147C211yesmaximal, characteristic, complemented
2#38955C211yesmaximal, characteristic
3#32693C212yesmaximal, complemented
4#32259C212yesmaximal, complemented
5#31714C214yesmaximal, complemented
6#31104C214yesmaximal, complemented
7#30310C212yesmaximal, complemented
8#29580C212yesmaximal, complemented
9#28850C214yesmaximal, complemented
10#26071C214yesmaximal, complemented
11#25851C212yesmaximal, complemented
12#24088C212yesmaximal
13#23273C211yesmaximal, characteristic, complemented
14C2×#563—21no—
15C2×#554C2211yescharacteristic
16C2×#547C2211yescharacteristic
17C2×#501C2212yes—
18C2×#480C2212yes—
19C2×#476C2212yes—
20C22×#107C2212yes—
21#2862C2211yescharacteristic
22#2859C2212yes—
23#2822—21no—
24#2734—21no—
25#2724—21no—
26#2698—24no—
27#2687—24no—
28#2621—21no—
29#2563C2211yescharacteristic
30#2546C2211yescharacteristic
31#2546C2212yes—
32#2503—21no—
33#2496C2214yes—
34#2486C2212yes—
35#2442C2214yes—
36#2440C2214yes—
37#2432—21no—
38#2420C2214yes—
39#2311—21no—
40C4×#124C2211yescharacteristic
41#2293C2211yescharacteristic
42#2100C2212yes—
43#2094C2212yes—
44#2087C2214yes—
45#2085C2214yes—
46#2085C2214yes—
47#2083C2212yes—
48#2077C2214yes—
49#2077C2214yes—
50#2071C2212yes—
51#2059C2214yes—
52#2058C2214yes—
53#2057C2214yes—
54#2057C2214yes—
55#2033C2214yes—
56#1989C2212yes—
57#1985C2212yes—
58#1944C2214yes—
59#1943C2214yes—
60#1936C2214yes—
61#1924C2214yes—
62#1919C2214yes—
63#1919C2214yes—
64#1893C2212yes—
65#1869C2212yes—
66#1864C2214yes—
67#1862C2214yes—
68#1853C2214yes—
69#1852C2214yes—
70#1847C2214yes—
71#1836C2214yes—
72#1823C2212yes—
73#1807C2214yes—
74#1762C2212yes—
75#1752C2214yes—
76#1745C2212yes—
77C22×#41—21no—
78C23×C8C2312yes—
79#563—48no—
80#554—42no—
81#554—42no—
82#547—42no—
83#547—42no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×#126—24no—
90C2×#125—22no—
91C2×#125C2312yes—
92C2×#124D411yescharacteristic
93C2×#123—22no—
94C2×#123—22no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—24no—
99C2×#121C2312yes—
100C2×C4×Q8C2311yescharacteristic
101C2×C4×D4C2311yescharacteristic
102C2×C4×D4C2312yes—
103C2×C4×D4C2311yescharacteristic
104C2×#117—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108#501—44no—
109#480—44no—
110#476—44no—
111C2×#107—44no—
112C2×#107C2314yes—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2312yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×C22⋊C8C2314yes—
122C2×C22⋊C8C2312yes—
123C2×#97C2314yes—
124C2×#97C2312yes—
125#400—24no—
126#400—24no—
127#398—22no—
128#398—22no—
129#398—22no—
130#398—22no—
131#398—24no—
132#398—24no—
133#397—22no—
134#397—24no—
135#397—24no—
136#395—24no—
137#394—24no—
138#393—22no—
139#391—21no—
140#391—22no—
141#391—21no—
142#391—22no—
143#391—21no—
144#391D411yescharacteristic
145#390—21no—
146#390D411yescharacteristic
147#390C2312yes—
148#389—22no—
149#389—22no—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#389C2312yes—
154#389C2314yes—
155#388—22no—
156#388D414yes—
157#387—24no—
158#386—24no—
159#385—24no—
160#385—24no—
161#384C2312yes—
162#384C2312yes—
163#382—21no—
164#382—22no—
165#382D411yescharacteristic
166#382C2314yes—
167#381—22no—
168#381—24no—
169#380—24no—
170#380C2314yes—
171#380C2314yes—
172#380C2312yes—
173C4×C4⋊C4C2311yescharacteristic
174C4×C4⋊C4C2312yes—
175C4×C22⋊C4—21no—
176C4×C22⋊C4C2314yes—
177C43C2311yescharacteristic
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2312yes—
189#336C2312yes—
190C2×#41—44no—
191C2×#41—44no—
192C2×#41—82no—
193C22×Q8—21no—
194C22×Q8—22no—
195C22×Q8—21no—
196C22×Q8C2×D411yescharacteristic
197C22×D4—22no—
198C22×D4C2×D411yescharacteristic
199C22×D4C2411yescharacteristic
200C23×C4—21no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4C2412yes—
205C22×C8—28no—
206C22×C8—42no—
207C22×C8C2414yes—
208C22×C8C2414yes—
209C4⋊Q8—41no—
210C4⋊Q8—41no—
211C4⋊Q8—41no—
212C4⋊Q8—41no—
213C4⋊Q8—42no—
214C4⋊Q8—42no—
215C4⋊Q8—48no—
216C4⋊Q8—82no—
217#126—416no—
218#125—24no—
219#125—42no—
220#125—48no—
221#124—24no—
222#123—48no—
223#123—48no—
224C22⋊Q8—48no—
225C22⋊Q8—416no—
226C22⋊Q8—44no—
227C22⋊Q8—44no—
228#121—24no—
229#121—42no—
230C4×Q8—41no—
231C4×Q8—41no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—24no—
235C4×D4—42no—
236C4×D4—41no—
237C4×D4—41no—
238#117—82no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2411yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D414yes—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D412yes—
285C2×C42C2411yescharacteristic
286C2×C42C2412yes—
287C2×C42C2412yes—
288#107—28no—
289#107—44no—
290#107—44no—
291#106—28no—
292#106—44no—
293#106—44no—
294C4⋊C8—42no—
295#102—28no—
296#102—28no—
297C22⋊C8—28no—
298C22⋊C8—42no—
299#97—28no—
300#97—42no—
301#95—22no—
302#95—21no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95C2×D414yes—
314#95C2×D414yes—
315C24—21no—
316C24C22×D412yes—
317#41—816no—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—44no—
325C2×Q8—42no—
326C2×Q8—82no—
327C2×Q8—82no—
328C2×Q8—82no—
329C2×D4—21no—
330C2×D4—24no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—82no—
336C2×D4AGL(1,ℤ/8)14yes—
337C2×D4H(5,𝔽2)12yes—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—84no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4C22×D412yes—
360C22×C4C22×D412yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C2511yescharacteristic
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—48no—
373C2×C82−1+418yes—
374C4⋊C4—21no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—84no—
403C4⋊C4H(5,𝔽2)12yes—
404C22⋊C4—24no—
405C22⋊C4—28no—
406C22⋊C4—44no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—48no—
411C22⋊C4—44no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—84no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—21no—
419C42—21no—
420C42—24no—
421C42—42no—
422C42—41no—
423C42—42no—
424C42—44no—
425C23—22no—
426C23—24no—
427C23—22no—
428C23—24no—
429C23—22no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23C2×AGL(1,ℤ/8)14yes—
435C23C23×D411yescharacteristic
436Q8—82no—
437Q8—84no—
438Q8—164no—
439Q8—162no—
440D4—24no—
441D4—44no—
442D4—42no—
443D4—88no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—48no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—88no—
483C2×C4—164no—
484C2×C4—164no—
485C2×C4C2×AGL(1,ℤ/8)12yes—
486C2×C4C2×H(5,𝔽2)11yescharacteristic
487C2×C4C2×H(5,𝔽2)11yescharacteristic
488C2×C4C2×2−1+414yes—
489C8—28no—
490C8—44no—
491C22—28no—
492C22—24no—
493C22—44no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—82no—
500C22—82no—
501C22#348411yescharacteristic
502C22#349912yes—
503C22C22×AGL(1,ℤ/8)11yescharacteristic
504C22#358511yescharacteristic
505C22#358612yes—
506C4—22no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—41no—
514C4—82no—
515C4—84no—
516C4—82no—
517C4—84no—
518C4—82no—
519C4#353812yes—
520C4#354412yes—
521C2—44no—
522C2—82no—
523C2#5154211yescharacteristic
524C2#6156911yescharacteristic
525C2#6167411yescharacteristic
526C2#6250012yes—
527C2#6250312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #46147C2semidirect
3. #32693C2semidirect
4. #32259C2semidirect
5. #31714C2semidirect
6. #31104C2semidirect
7. #30310C2semidirect
8. #29580C2semidirect
9. #28850C2semidirect
10. #26071C2semidirect
11. #25851C2semidirect
13. #23273C2semidirect

Maximal subgroups

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

subgroupindex
#461472
#389552
#326932
#322592
#317142
#311042
#303102
#295802
#288502
#260712
#258512
#240882
#232732