Finite Group Databasebeta

← #319254

Subgroups — #319254

Order 512.

1106 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#27972C211yesmaximal, characteristic, complemented
3#27948C211yesmaximal, characteristic, complemented
4#25855C211yesmaximal, characteristic
5#25717C211yesmaximal, characteristic
6C2×#1957C211yesmaximal, characteristic
7#10672C212yesmaximal, complemented
8#10308C212yesmaximal, complemented
9#9829C212yesmaximal, complemented
10#9827C212yesmaximal, complemented
11#8144C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C2×#500C2211yescharacteristic, complemented
14C2×#492C2212yes—
15C2×#486C2212yes—
16C2×#450—22no—
17C22×#102—22no—
18C22×C4×C8C2211yescharacteristic
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2084—22no—
23#2072—22no—
24#2041—22no—
25#2014—22no—
26#1990C2212yescomplemented
27#1990C2211yescharacteristic, complemented
28#1990C2211yescharacteristic, complemented
29#1957—22no—
30#1957C2212yes—
31#1957C2212yes—
32#1957C2212yes—
33#1957C2212yes—
34#1956C2212yescomplemented
35#1942—22no—
36#1872—22no—
37#1853—22no—
38#1845C2211yescharacteristic, complemented
39#1844C2212yescomplemented
40#1834C2212yescomplemented
41#1808—22no—
42#1786—22no—
43C4×C22⋊C8C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45C2×#322C2212yescomplemented
46#1273C2212yescomplemented
47C23×D4—21no—
48C23×D4—21no—
49C23×D4—21no—
50C22×SD16—24no—
51C23×C8D412yes—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×#127—24no—
55C2×#127—21no—
56C2×#127—21no—
57C2×#127—21no—
58C2×#127—42no—
59C2×#127C2312yes—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C42C2311yescharacteristic
66#500—42no—
67#492—24no—
68#492—42no—
69#486—24no—
70#486—42no—
71#450—48no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#106—24no—
75C2×#103—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—48no—
82C2×C22⋊C8D412yes—
83C2×C22⋊C8D412yes—
84C2×C4×C8—42no—
85C2×C4×C8D412yes—
86C2×C4×C8C2312yes—
87C2×C4×C8C2312yes—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#391—21no—
91#391—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384C2311yescharacteristic
100#384C2311yescharacteristic
101#384C2312yes—
102#380—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106#340—22no—
107#340—22no—
108#340—24no—
109#340—24no—
110#336D412yes—
111#336D412yes—
112#322—42no—
113C25—41no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×D4—24no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—44no—
121C22×D4—44no—
122C22×D4—44no—
123C22×D4—42no—
124C22×D4—41no—
125C22×D4—41no—
126C22×D4—41no—
127C22×D4—44no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4—82no—
131C22×D4—82no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×SD16—48no—
137C2×SD16—48no—
138C2×SD16—84no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8SD1618yescomplemented
142C22×C8C2×D414yes—
143C22×C8C2×D414yes—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150#127—24no—
151#127—22no—
152#127—22no—
153#127—22no—
154#127—44no—
155#127—44no—
156#127—42no—
157#127#4114yes—
158C22⋊Q8—44no—
159C22⋊Q8—44no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2411yescharacteristic
189#107—44no—
190#107—44no—
191#106—84no—
192#103—84no—
193#102—48no—
194#102—48no—
195#102—48no—
196#102—48no—
197#102—84no—
198C22⋊C8—24no—
199C22⋊C8—24no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—42no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211C24—44no—
212C24—42no—
213C24—82no—
214C24—82no—
215C24—82no—
216C24—82no—
217C24—82no—
218C24—82no—
219C24#12711yescharacteristic
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—416no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—816no—
243C2×D4—816no—
244C2×D4—82no—
245C2×D4—88no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4—82no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4#12711yescharacteristic
267C22×C4#12711yescharacteristic
268C22×C4#12711yescharacteristic
269C22×C4C2×SD1614yes—
270C22×C4AGL(1,ℤ/8)14yes—
271C22×C4#13714yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275SD16—816no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—48no—
287C2×C8C2×SD1614yes—
288C2×C8C2×SD1614yes—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42C2×#4112yes—
316C42H(5,𝔽2)12yes—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—44no—
322C23—88no—
323C23—88no—
324C23—88no—
325C23—82no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23—82no—
330C23—82no—
331C23—164no—
332C23—164no—
333C23—164no—
334C23—164no—
335C23#49212yes—
336C23#49612yes—
337C23#49712yes—
338C23C2×#12711yescharacteristic
339Q8—84no—
340Q8—84no—
341D4—48no—
342D4—48no—
343D4—48no—
344D4—88no—
345D4—88no—
346D4—88no—
347D4—84no—
348D4—84no—
349D4—84no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—164no—
376C2×C4#49212yes—
377C2×C4#49612yes—
378C2×C4#49712yes—
379C2×C4#52511yescharacteristic
380C2×C4#53411yescharacteristic
381C2×C4#54612yes—
382C2×C4#54612yes—
383C2×C4C22×SD1612yes—
384C2×C4C2×AGL(1,ℤ/8)12yes—
385C2×C4C2×#13712yes—
386C2×C4#57512yes—
387C2×C4#57612yes—
388C2×C4#57612yes—
389C8—24no—
390C8—24no—
391C8—44no—
392C8—44no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—84no—
400C22—84no—
401C22—84no—
402C22—1616no—
403C22—162no—
404C22—162no—
405C22—162no—
406C22—162no—
407C22#171414yes—
408C22#253911yescharacteristic
409C22C2×#49211yescharacteristic
410C22C2×#49611yescharacteristic
411C22C2×#49711yescharacteristic
412C22#315311yescharacteristic
413C22#315511yescharacteristic
414C22#315511yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—161no—
419C4—161no—
420C4—162no—
421C4—161no—
422C4—161no—
423C4#310112yes—
424C4#318712yes—
425C4#331414yes—
426C4#331814yes—
427C2—24no—
428C2—164no—
429C2C2×#171411yescharacteristic
430C2#1965211yescharacteristic
431C2#1967311yescharacteristic
432C2#1968211yescharacteristic
433C2#2830211yescharacteristic
434C2#2834111yescharacteristic
435C2#2834211yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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. #33209C2semidirect
2. #27972C2semidirect
3. #27948C2semidirect
7. #10672C2semidirect
8. #10308C2semidirect
9. #9829C2semidirect
10. #9827C2semidirect
11. #8144C2semidirect
13. C2×#500C22semidirect
26. #1990C22semidirect
27. #1990C22semidirect
28. #1990C22semidirect
34. #1956C22semidirect
38. #1845C22semidirect
39. #1844C22semidirect
40. #1834C22semidirect
45. C2×#322C22semidirect
46. #1273C22semidirect
141. C22×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#332092
#279722
#279482
#258552
#257172
C2×#19572
#106722
#103082
#98292
#98272
#81442