Finite Group Databasebeta

← #347647

Subgroups — #347647

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2#33207C211yesmaximal, characteristic, complemented
3#32956C211yesmaximal, characteristic
4#32905C211yesmaximal, characteristic
5C2×#1671C212yesmaximal, complemented
6#9217C212yesmaximal, complemented
7#9215C212yesmaximal, complemented
8#9129C211yesmaximal, characteristic, complemented
9#9127C211yesmaximal, characteristic, complemented
10#8642C212yesmaximal, complemented
11#8595C211yesmaximal, characteristic, complemented
12C24×C8—21no—
13C22×C4×D4C2211yescharacteristic
14C2×#498C2211yescharacteristic, complemented
15C2×#476C2212yes—
16C2×#461C2212yes—
17C2×#450—21no—
18C2×C8×D4—28no—
19C2×C8×D4—21no—
20C2×C8×D4—21no—
21C2×C8×D4C2212yes—
22C2×C8×D4C2212yes—
23C22×C4⋊C8C2211yescharacteristic
24C22×C22⋊C8—21no—
25C22×C4×C8C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#1990C2211yescharacteristic, complemented
29#1990C2211yescharacteristic, complemented
30#1941—21no—
31#1941—21no—
32#1877—22no—
33#1872—21no—
34#1864—22no—
35#1853—21no—
36#1844C2211yescharacteristic, complemented
37#1807—21no—
38#1807—21no—
39#1802C2212yes—
40#1802C2212yes—
41#1797C2211yescharacteristic
42#1797C2211yescharacteristic
43#1786—21no—
44#1671—28no—
45C2×#334C2212yescomplemented
46C2×#333C2212yescomplemented
47C2×#332C2212yescomplemented
48#1296C2212yescomplemented
49#1296C2212yescomplemented
50#1277C2212yescomplemented
51C23×D4D411yescharacteristic
52C24×C4—21no—
53C22×SD16—22no—
54C23×C8—28no—
55C23×C8—21no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—44no—
59C23×C8—44no—
60C23×C8D411yescharacteristic
61C23×C8D411yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4D412yes—
71C2×C4×D4C2312yes—
72C22×C4⋊C4D411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C42C2311yescharacteristic
75#498—24no—
76#476—24no—
77#476—24no—
78#461—24no—
79#461—24no—
80#450—44no—
81C8×D4—28no—
82C8×D4—28no—
83C8×D4—24no—
84C8×D4—24no—
85C8×D4—24no—
86C8×D4D418yes—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—22no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97C2×#103—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×C22⋊C8—28no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C4×C8—24no—
107C2×C4×C8D411yescharacteristic
108C2×C4×C8D411yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#391—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—22no—
128#340—22no—
129#336—21no—
130#336—21no—
131#334—24no—
132#333—24no—
133#332—24no—
134C25—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4SD1614yes—
142C22×D4C2×D412yes—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4#4111yescharacteristic
155C2×SD16—44no—
156C2×SD16—44no—
157C2×SD16—44no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—24no—
161C22×C8—28no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—28no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—41no—
169C22×C8—44no—
170C22×C8—432no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—44no—
174C22×C8—44no—
175C22×C8—44no—
176C22×C8SD1614yescomplemented
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C22⋊Q8—82no—
188C4×D4—24no—
189C4×D4—28no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4C2×D414yes—
196C4×D4#4114yes—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4C2×D412yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42C2411yescharacteristic
223#106—42no—
224#106—42no—
225#106—42no—
226#106—42no—
227#106—44no—
228C4⋊C8—24no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8C2×D414yes—
235#103—44no—
236#102—44no—
237#102—44no—
238#102—44no—
239C22⋊C8—28no—
240C22⋊C8—28no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243C4×C8—22no—
244C4×C8—22no—
245C4×C8—22no—
246C4×C8C2×D414yes—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24—24no—
252C24—24no—
253C24—22no—
254C24—24no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24#12111yescharacteristic
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—42no—
265C2×Q8—81no—
266C2×Q8—81no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—216no—
270C2×D4—216no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4#12114yes—
276C2×D4C2×SD1614yes—
277C2×D4#13514yes—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—28no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—41no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—48no—
295C22×C4—48no—
296C22×C4—48no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4C2×SD1612yes—
312C22×C4C2×SD1612yes—
313C22×C4#13512yes—
314C22×C4#13712yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318SD16—88no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—28no—
324C2×C8—21no—
325C2×C8—24no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—44no—
329C2×C8—416no—
330C2×C8—48no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—416no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8#12012yes—
341C2×C8#12012yes—
342C2×C8C2×SD1612yes—
343C2×C8C2×SD1612yes—
344C2×C8AGL(1,ℤ/8)12yes—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4#12114yes—
365C22⋊C4—28no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C42—21no—
371C42—22no—
372C42—21no—
373C42—22no—
374C42C22×D412yes—
375C42C2×#4112yes—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—216no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—48no—
385C23—48no—
386C23—48no—
387C23—48no—
388C23—48no—
389C23—48no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23#46112yes—
395C23#46512yes—
396C23#47112yes—
397C23C2×#12111yescharacteristic
398Q8—82no—
399Q8—82no—
400Q8—162no—
401D4—216no—
402D4—24no—
403D4—24no—
404D4—24no—
405D4#46518yes—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—44no—
423C2×C4—416no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4#46112yes—
444C2×C4#46512yes—
445C2×C4C2×#12011yescharacteristic
446C2×C4C2×#12112yes—
447C2×C4#52512yes—
448C2×C4#52511yescharacteristic
449C2×C4C22×SD1611yescharacteristic
450C2×C4C22×SD1611yescharacteristic
451C2×C4C2×#13512yes—
452C2×C4C2×#13511yescharacteristic
453C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
454C2×C4C2×#13711yescharacteristic
455C2×C4#57511yescharacteristic
456C8—22no—
457C8—22no—
458C8—44no—
459C8—41no—
460C8—82no—
461C8—82no—
462C8#45014yescomplemented
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—28no—
470C22—432no—
471C22—44no—
472C22—44no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22#167114yescomplemented
478C22#284711yescharacteristic
479C22C2×#46111yescharacteristic
480C22#303611yescharacteristic
481C22C2×#46511yescharacteristic
482C22#305011yescharacteristic
483C22C2×#47111yescharacteristic
484C22#306711yescharacteristic
485C4—22no—
486C4—22no—
487C4—22no—
488C4—44no—
489C4—41no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4C2×#45011yescharacteristic
495C4#300711yescharacteristic
496C4C2×#46512yes—
497C4#305012yes—
498C4#310111yescharacteristic
499C4#311011yescharacteristic
500C2—24no—
501C2—48no—
502C2C2×#167111yescharacteristic
503C2#1910511yescharacteristic
504C2#1925411yescharacteristic
505C2#1925711yescharacteristic
506C2#3290511yescharacteristic
507C2#3292911yescharacteristic
508C2#3294111yescharacteristic
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
2. #33207C2semidirect
5. C2×#1671C2semidirect
6. #9217C2semidirect
7. #9215C2semidirect
8. #9129C2semidirect
9. #9127C2semidirect
10. #8642C2semidirect
11. #8595C2semidirect
14. C2×#498C22semidirect
28. #1990C22semidirect
29. #1990C22semidirect
36. #1844C22semidirect
45. C2×#334C22semidirect
46. C2×#333C22semidirect
47. C2×#332C22semidirect
48. #1296C22semidirect
49. #1296C22semidirect
50. #1277C22semidirect
176. C22×C8SD16semidirect
462. C8#450semidirect
477. C22#1671semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
#332072
#329562
#329052
C2×#16712
#92172
#92152
#91292
#91272
#86422
#85952