Finite Group Databasebeta

← #281022

Subgroups — #281022

Order 512.

538 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28265C211yesmaximal, characteristic
2#25061C212yesmaximal
3#22808C211yesmaximal, characteristic
4#10620C212yesmaximal
5#10438C212yesmaximal
6#9656C211yesmaximal, characteristic
7#9614C211yesmaximal, characteristic
8#8895C212yesmaximal
9#8814C211yesmaximal, characteristic
10#8645C211yesmaximal, characteristic
11C8×#103C211yesmaximal, characteristic
12C2×C8×Q8C2211yescharacteristic
13C2×C4×Q16C2211yescharacteristic
14#2582C2212yes—
15#2355C2212yes—
16#2097—22no—
17#2085—22no—
18#2068—22no—
19#1984C2212yes—
20#1969—21no—
21#1969—21no—
22#1947C2212yes—
23#1942—21no—
24#1928—22no—
25#1927—22no—
26#1919—22no—
27#1845C2211yescharacteristic
28#1845C2211yescharacteristic
29#1844C2212yes—
30#1834C2212yes—
31#1803C2212yes—
32#1803C2212yes—
33#1803C2212yes—
34C4×#107—21no—
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×Q8⋊C8C2211yescharacteristic
39C2×Q8⋊C8C2212yes—
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1277C2211yescharacteristic
45C22×Q16D411yescharacteristic
46C2×C4×Q8C2311yescharacteristic
47C2×C4×Q8C2312yes—
48C8×Q8—22no—
49C8×Q8—22no—
50C4×Q16—22no—
51C4×Q16—22no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—22no—
57C2×#107D411yescharacteristic
58C2×#106—22no—
59C2×#106—22no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2312yes—
64C2×#103—22no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103D412yes—
68C2×#103D411yescharacteristic
69C2×#103D411yescharacteristic
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#398—42no—
76#391—22no—
77#391—22no—
78#391—22no—
79#389—22no—
80#389—22no—
81#387—42no—
82#384C2312yes—
83#384C2312yes—
84#382—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2311yescharacteristic
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—42no—
93#340D411yescharacteristic
94#340D411yescharacteristic
95#336—22no—
96#336—22no—
97#336—21no—
98#335—22no—
99Q8⋊C8—22no—
100Q8⋊C8—24no—
101C82—22no—
102C22×Q8C2×D412yes—
103C22×Q8C2×D411yescharacteristic
104C2×Q16—22no—
105C2×Q16—22no—
106C2×Q16—22no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C4×Q8—22no—
116C4×Q8—21no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C2×D411yescharacteristic
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C42—22no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—42no—
140C2×C42C2411yescharacteristic
141#107—22no—
142#107—42no—
143#107—42no—
144#107—82no—
145#107—81no—
146#107—81no—
147#106—44no—
148#106—82no—
149C4⋊C8—22no—
150C4⋊C8—21no—
151C4⋊C8—22no—
152C4⋊C8—21no—
153C4⋊C8—21no—
154C4⋊C8—21no—
155C4⋊C8—42no—
156#103—24no—
157#103—22no—
158#103—82no—
159#103—82no—
160#103—82no—
161#103Q1614yescomplemented
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—42no—
166C4×C8—42no—
167C4×C8#4112yes—
168C4×C8#4112yes—
169C4×C8#4112yes—
170#95—21no—
171#95—42no—
172#95—42no—
173#95—42no—
174#95—42no—
175#95—42no—
176#95—42no—
177C2×Q8—21no—
178C2×Q8—22no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—21no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8C2×Q1612yes—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4C22×D411yescharacteristic
198C22×C4C22×D411yescharacteristic
199C22×C4H(5,𝔽2)11yescharacteristic
200Q16—44no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8#12312yes—
220C2×C8#12312yes—
221C2×C8C2×Q1612yes—
222C2×C8#13512yes—
223C2×C8#13512yes—
224C2×C8#13712yes—
225C4⋊C4—21no—
226C4⋊C4—22no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—162no—
247C4⋊C4C2×Q1612yes—
248C42—41no—
249C42—42no—
250C42—41no—
251C42—42no—
252C42—41no—
253C42—41no—
254C42—162no—
255C42C2×#4111yescharacteristic
256C42C2×#4111yescharacteristic
257C42C2×#4111yescharacteristic
258C422−1+411yescharacteristic
259C23D4211yescharacteristic
260Q8—22no—
261Q8—22no—
262Q8—41no—
263Q8—41no—
264Q8—42no—
265Q8—42no—
266Q8—42no—
267Q8—42no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4C2×#12311yescharacteristic
295C2×C4#53211yescharacteristic
296C2×C4#54211yescharacteristic
297C2×C4#54711yescharacteristic
298C2×C4#54711yescharacteristic
299C2×C4#55311yescharacteristic
300C2×C4C22×Q1611yescharacteristic
301C2×C4C2×#13511yescharacteristic
302C2×C4C2×#13511yescharacteristic
303C2×C4C2×#13711yescharacteristic
304C2×C4#57611yescharacteristic
305C2×C4#57711yescharacteristic
306C2×C4#57711yescharacteristic
307C2×C4#57811yescharacteristic
308C8—22no—
309C8—22no—
310C8—22no—
311C8—42no—
312C8—81no—
313C8—81no—
314C8—82no—
315C8#48514yes—
316C22#242211yescharacteristic
317C22#327811yescharacteristic
318C22#328411yescharacteristic
319C22D4×Q1611yescharacteristic
320C22#328711yescharacteristic
321C22#329011yescharacteristic
322C22#329011yescharacteristic
323C4—21no—
324C4—21no—
325C4—42no—
326C4—42no—
327C4—81no—
328C4—82no—
329C4—82no—
330C4—81no—
331C4—162no—
332C4—162no—
333C4C2×#48511yescharacteristic
334C4#309911yescharacteristic
335C4#317011yescharacteristic
336C4#317511yescharacteristic
337C4#333611yescharacteristic
338C4#333811yescharacteristic
339C4#334011yescharacteristic
340C4#335611yescharacteristic
341C2#1993211yescharacteristic
342C2#1996611yescharacteristic
343C2#2000711yescharacteristic
344C2#2001811yescharacteristic
345C2#2501511yescharacteristic
346C2#2503511yescharacteristic
347C2#2505211yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
161. #103Q16semidirect

Maximal subgroups

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

subgroupindex
#282652
#250612
#228082
#106202
#104382
#96562
#96142
#88952
#88142
#86452
C8×#1032