Finite Group Databasebeta

← #329444

Subgroups — #329444

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33043C211yesmaximal, characteristic, complemented
2#32953C211yesmaximal, characteristic
3#32905C211yesmaximal, characteristic
4C2×#1954C211yesmaximal, characteristic
5#10625C212yesmaximal, complemented
6#10484C212yesmaximal, complemented
7#10477C212yesmaximal, complemented
8#10270C212yesmaximal, complemented
9#9131C211yesmaximal, characteristic, complemented
10#9129C211yesmaximal, characteristic, complemented
11#8595C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#487C2211yescharacteristic, complemented
14C2×#474C2212yes—
15C2×#461C2212yes—
16C2×#450—21no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8—21no—
19C22×C22⋊C8—21no—
20C22×C4×C8C2211yescharacteristic
21#2857C2211yescharacteristic
22#2857C2211yescharacteristic
23#1990C2211yescharacteristic, complemented
24#1956C2211yescharacteristic, complemented
25#1954—21no—
26#1954—21no—
27#1954C2212yes—
28#1954C2212yes—
29#1933—21no—
30#1933—21no—
31#1892—21no—
32#1892—21no—
33#1877—22no—
34#1872—21no—
35#1862—22no—
36#1852—21no—
37#1834C2211yescharacteristic, complemented
38#1802C2212yes—
39#1802C2212yes—
40#1797C2211yescharacteristic
41#1797C2211yescharacteristic
42#1786—21no—
43C2×#334C2212yescomplemented
44C2×#332C2212yescomplemented
45C2×#332C2212yescomplemented
46#1296C2212yescomplemented
47#1296C2212yescomplemented
48#1277C2212yescomplemented
49C23×D4D411yescharacteristic
50C24×C4—21no—
51C22×SD16—22no—
52C23×C8—21no—
53C23×C8D411yescharacteristic
54C23×C8D411yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2311yescharacteristic
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4—42no—
62C2×C4×D4—42no—
63C2×C4×D4D412yes—
64C2×C4×D4C2312yes—
65C22×C4⋊C4D411yescharacteristic
66C22×C22⋊C4—21no—
67C22×C42C2311yescharacteristic
68#487—42no—
69#474—42no—
70#474—42no—
71#461—42no—
72#461—42no—
73#450—44no—
74C2×#107—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—42no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×#103—21no—
81C2×#103—21no—
82C2×#103—21no—
83C2×#103—21no—
84C2×#103—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—44no—
95C2×C22⋊C8—44no—
96C2×C4×C8—42no—
97C2×C4×C8D411yescharacteristic
98C2×C4×C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#391—21no—
102#391—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#340—21no—
114#340—21no—
115#340—21no—
116#340—21no—
117#340—22no—
118#340—22no—
119#336—21no—
120#336—21no—
121#334—42no—
122#332—42no—
123#332—42no—
124C25—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—42no—
130C22×D4—42no—
131C22×D4SD1614yes—
132C22×D4C2×D412yes—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4#4111yescharacteristic
145C2×SD16—44no—
146C2×SD16—44no—
147C2×SD16—44no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—41no—
155C22×C8—41no—
156C22×C8—41no—
157C22×C8—42no—
158C22×C8—42no—
159C22×C8—42no—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—44no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42C2411yescharacteristic
207#107—44no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—42no—
215#103—44no—
216#103—42no—
217#103—42no—
218#103—42no—
219#103—42no—
220#102—44no—
221#102—44no—
222#102—44no—
223C22⋊C8—44no—
224C22⋊C8—44no—
225C22⋊C8—44no—
226C22⋊C8—42no—
227C22⋊C8—42no—
228C22⋊C8—44no—
229C4×C8—22no—
230C4×C8—22no—
231C4×C8—41no—
232C4×C8—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237C24—22no—
238C24—24no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24#12111yescharacteristic
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—48no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—48no—
266C2×D4C2×SD1614yes—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—84no—
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4C2×SD1612yes—
301C22×C4#13512yes—
302C22×C4#13512yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307SD16—88no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—84no—
331C2×C8AGL(1,ℤ/8)12yes—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—81no—
348C4⋊C4—81no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C22⋊C4—44no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C42—21no—
357C42—22no—
358C42—21no—
359C42—22no—
360C42H(5,𝔽2)12yes—
361C42H(5,𝔽2)12yes—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—24no—
367C23—48no—
368C23—48no—
369C23—42no—
370C23—48no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—84no—
377C23—84no—
378C23—84no—
379C23—84no—
380C23#46312yes—
381C23#46512yes—
382C23#47312yes—
383C23C2×#12111yescharacteristic
384Q8—82no—
385Q8—81no—
386Q8—82no—
387Q8—81no—
388Q8—81no—
389Q8—81no—
390D4—24no—
391D4—24no—
392D4—48no—
393D4—42no—
394D4—42no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—88no—
415C2×C4—84no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4#46312yes—
433C2×C4#46512yes—
434C2×C4#53412yes—
435C2×C4#53411yescharacteristic
436C2×C4#53511yescharacteristic
437C2×C4#53812yes—
438C2×C4C22×SD1611yescharacteristic
439C2×C4C2×#13511yescharacteristic
440C2×C4C2×#13511yescharacteristic
441C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
442C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
443C2×C4#57512yes—
444C2×C4#57511yescharacteristic
445C8—22no—
446C8—41no—
447C8—41no—
448C8—41no—
449C8—82no—
450C8—82no—
451C8—82no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—44no—
459C22—44no—
460C22—44no—
461C22—816no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22#167914yes—
467C22#261711yescharacteristic
468C22C2×#46311yescharacteristic
469C22#303611yescharacteristic
470C22C2×#46511yescharacteristic
471C22#305411yescharacteristic
472C22C2×#47311yescharacteristic
473C22#306711yescharacteristic
474C4—22no—
475C4—22no—
476C4—22no—
477C4—41no—
478C4—82no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4#318712yes—
484C4#318811yescharacteristic
485C4#319112yes—
486C4#319411yescharacteristic
487C4#321111yescharacteristic
488C4#321711yescharacteristic
489C2—24no—
490C2—84no—
491C2C2×#167911yescharacteristic
492C2#1914211yescharacteristic
493C2#1925411yescharacteristic
494C2#1928411yescharacteristic
495C2#2971811yescharacteristic
496C2#2973411yescharacteristic
497C2#2974911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33043C2semidirect
5. #10625C2semidirect
6. #10484C2semidirect
7. #10477C2semidirect
8. #10270C2semidirect
9. #9131C2semidirect
10. #9129C2semidirect
11. #8595C2semidirect
13. C2×#487C22semidirect
23. #1990C22semidirect
24. #1956C22semidirect
37. #1834C22semidirect
43. C2×#334C22semidirect
44. C2×#332C22semidirect
45. C2×#332C22semidirect
46. #1296C22semidirect
47. #1296C22semidirect
48. #1277C22semidirect

Maximal subgroups

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

subgroupindex
#330432
#329532
#329052
C2×#19542
#106252
#104842
#104772
#102702
#91312
#91292
#85952