Finite Group Databasebeta

← #7231168

Subgroups — #7231168

Order 512.

1509 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#46147C211yesmaximal, characteristic, complemented
2#36185C211yesmaximal, characteristic
3#33066C212yesmaximal, complemented
4#32820C212yesmaximal, complemented
5#32694C212yesmaximal, complemented
6#32039C212yesmaximal, complemented
7#31506C214yesmaximal, complemented
8#30655C214yesmaximal, complemented
9#27058C214yesmaximal, complemented
10#26355C214yesmaximal, complemented
11#25661C212yesmaximal, complemented
12#24104C212yesmaximal
13#23353C211yesmaximal, characteristic, complemented
14C2×#563—21no—
15C2×#556C2211yescharacteristic
16C2×#555C2211yescharacteristic
17C2×#489C2212yes—
18C2×#481C2212yes—
19C2×#477C2212yes—
20C2×#428C2212yes—
21#2862C2211yescharacteristic
22#2859C2212yes—
23#2850—21no—
24#2822—21no—
25#2734—21no—
26#2724—21no—
27#2698—24no—
28#2567C2211yescharacteristic
29#2546C2212yes—
30#2542C2211yescharacteristic
31#2521—24no—
32#2500—21no—
33#2499—21no—
34#2496C2214yes—
35#2486C2212yes—
36#2440C2214yes—
37#2427C2214yes—
38#2362C2214yes—
39C4×#124C2211yescharacteristic
40#2285—21no—
41C42×D4C2211yescharacteristic
42#2096C2214yes—
43#2088C2214yes—
44#2072C2214yes—
45#2066C2214yes—
46#2066C2214yes—
47#2058C2214yes—
48#2058C2214yes—
49#2043C2214yes—
50#2042C2214yes—
51#2036C2214yes—
52#2036C2214yes—
53#2034C2214yes—
54#1985C2212yes—
55#1983C2212yes—
56#1947C2212yes—
57#1947C2212yes—
58#1943C2214yes—
59#1924C2214yes—
60#1921C2214yes—
61#1918C2214yes—
62#1918C2214yes—
63#1895C2212yes—
64#1894C2214yes—
65#1872C2214yes—
66#1870C2212yes—
67#1863C2214yes—
68#1862C2214yes—
69#1862C2214yes—
70#1854C2214yes—
71#1851C2214yes—
72#1831C2212yes—
73#1825C2212yes—
74#1802C2212yes—
75#1802C2212yes—
76#1763C2212yes—
77C22×M16C2312yes—
78#563—48no—
79#556—42no—
80#556—42no—
81#555—42no—
82#555—42no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#126—22no—
88C2×#126—24no—
89C2×#125—22no—
90C2×#124—22no—
91C2×#124D411yescharacteristic
92C2×#124C2312yes—
93C2×#123—22no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8C2314yes—
97C2×#121—21no—
98C2×#121C2312yes—
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2311yescharacteristic
101C2×C4×D4C2311yescharacteristic
102C2×C4×D4C2312yes—
103C2×C4×D4C2311yescharacteristic
104C2×#117—21no—
105C2×#117—21no—
106C2×#117C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C42—21no—
109#489—44no—
110#481—44no—
111#477—44no—
112#428—44no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2312yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2314yes—
128C2×#97C2312yes—
129#400—24no—
130#399—22no—
131#398—22no—
132#398—22no—
133#398—22no—
134#398—24no—
135#397—22no—
136#397—24no—
137#397—24no—
138#395—22no—
139#395—24no—
140#394—24no—
141#391—21no—
142#391—22no—
143#391—22no—
144#391D411yescharacteristic
145#391C2314yes—
146#390—21no—
147#390D411yescharacteristic
148#389—22no—
149#389—22no—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#389C2314yes—
154#388—24no—
155#388—24no—
156#388D414yes—
157#387—24no—
158#387—24no—
159#386—24no—
160#385—24no—
161#384—21no—
162#384—21no—
163#384C2312yes—
164#384C2312yes—
165#382—21no—
166#382—22no—
167#382—21no—
168#382D411yescharacteristic
169#381—22no—
170#380—24no—
171#380C2312yes—
172#380C2314yes—
173C4×C4⋊C4—24no—
174C4×C4⋊C4C2311yescharacteristic
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—22no—
179C4×C22⋊C4C2314yes—
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189#336C2314yes—
190C22×Q8—22no—
191C22×Q8—21no—
192C22×Q8C2×D411yescharacteristic
193C22×Q8C2412yes—
194C22×D4—22no—
195C22×D4C2×D411yescharacteristic
196C22×D4C2411yescharacteristic
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4C2412yes—
201C2×M16—44no—
202C2×M16—42no—
203C22×C8C2414yes—
204C22×C8C2414yes—
205C4⋊Q8—41no—
206C4⋊Q8—41no—
207C4⋊Q8—42no—
208C4⋊Q8—42no—
209C4⋊Q8—41no—
210C4⋊Q8—41no—
211C4⋊Q8—48no—
212#126—48no—
213#126—416no—
214#125—48no—
215#124—24no—
216#124—24no—
217#124—42no—
218#124—48no—
219#123—48no—
220C22⋊Q8—28no—
221C22⋊Q8—416no—
222C22⋊Q8—44no—
223C22⋊Q8—44no—
224#121—24no—
225#121—42no—
226#121—44no—
227C4×Q8—24no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8—42no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—24no—
234C4×D4—42no—
235C4×D4—41no—
236C4×D4—41no—
237#117—24no—
238#117—42no—
239#117—44no—
240#117—44no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2411yescharacteristic
265C2×C4⋊C4C2412yes—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D414yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—44no—
284C2×C42C2×D412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2412yes—
289C2×C42C2412yes—
290#107—28no—
291#107—28no—
292#106—44no—
293#106—44no—
294C4⋊C8—42no—
295C4⋊C8—42no—
296C4⋊C8—42no—
297#103—44no—
298#103—44no—
299#103—44no—
300#103—44no—
301#102—28no—
302#102—44no—
303C22⋊C8—44no—
304C22⋊C8—42no—
305#97—42no—
306#95—22no—
307#95—21no—
308#95—24no—
309#95—22no—
310#95—24no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95C2×D414yes—
318#95C2×D414yes—
319C24—21no—
320C24C22×D412yes—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—44no—
329C2×Q8—42no—
330C2×Q8—82no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—44no—
340C2×D4—84no—
341C2×D4H(5,𝔽2)12yes—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4C22×D412yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C2511yescharacteristic
369M16—48no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—48no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—48no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—44no—
398C4⋊C4—82no—
399C4⋊C4—84no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4H(5,𝔽2)12yes—
403C4⋊C42−1+414yes—
404C22⋊C4—24no—
405C22⋊C4—28no—
406C22⋊C4—44no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—48no—
413C22⋊C4—44no—
414C22⋊C4—48no—
415C22⋊C4—44no—
416C42—22no—
417C42—22no—
418C42—21no—
419C42—21no—
420C42—22no—
421C42—22no—
422C42—41no—
423C42—42no—
424C42—44no—
425C42—44no—
426C42—44no—
427C42—44no—
428C42—42no—
429C42AGL(1,ℤ/8)14yes—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—22no—
436C23—41no—
437C23—42no—
438C23—42no—
439C23—41no—
440C23C23×D411yescharacteristic
441Q8—42no—
442Q8—42no—
443Q8—44no—
444Q8—82no—
445Q8—164no—
446Q8—162no—
447D4—24no—
448D4—41no—
449D4—41no—
450D4—82no—
451D4—164no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—48no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—88no—
489C2×C4—164no—
490C2×C4C2×AGL(1,ℤ/8)14yes—
491C2×C4C2×AGL(1,ℤ/8)12yes—
492C2×C4C2×H(5,𝔽2)11yescharacteristic
493C2×C4C2×H(5,𝔽2)11yescharacteristic
494C2×C4C2×2−1+412yes—
495C2×C4#58512yes—
496C8—44no—
497C8—44no—
498C22—44no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—44no—
504C22—42no—
505C22—42no—
506C22—82no—
507C22—82no—
508C22#348411yescharacteristic
509C22#348912yes—
510C22C22×AGL(1,ℤ/8)11yescharacteristic
511C22#358611yescharacteristic
512C22#358612yes—
513C4—22no—
514C4—24no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—44no—
520C4—42no—
521C4—41no—
522C4—84no—
523C4—82no—
524C4—84no—
525C4—82no—
526C4#352911yescharacteristic
527C4#355011yescharacteristic
528C4#355912yes—
529C2—82no—
530C2—82no—
531C2#4622811yescharacteristic
532C2#6158511yescharacteristic
533C2#6167411yescharacteristic
534C2#6210712yes—
535C2#6224012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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. #33066C2semidirect
4. #32820C2semidirect
5. #32694C2semidirect
6. #32039C2semidirect
7. #31506C2semidirect
8. #30655C2semidirect
9. #27058C2semidirect
10. #26355C2semidirect
11. #25661C2semidirect
13. #23353C2semidirect

Maximal subgroups

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

subgroupindex
#461472
#361852
#330662
#328202
#326942
#320392
#315062
#306552
#270582
#263552
#256612
#241042
#233532