Finite Group Databasebeta

← #6484488

Subgroups — #6484488

Order 512.

1428 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C212yesmaximal
2#33211C214yesmaximal
3#30297C214yesmaximal
4#29321C218yesmaximal, complemented
5#29210C214yesmaximal, complemented
6#28691C214yesmaximal, complemented
7#21964C212yesmaximal
8#21964C212yesmaximal
9C4×#423C211yesmaximal, characteristic
10C2×#499C2214yes—
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2212yes—
14#2864—24no—
15#2859C2214yes—
16#2859C2212yes—
17#2859C2212yes—
18#2851—24no—
19#2851C2214yes—
20#2851C2214yes—
21C4×#125—24no—
22C4×#125C2214yes—
23C4×#125C2214yes—
24#2296—24no—
25#2285C2214yes—
26C2×C43C2211yescharacteristic
27#2085C22116yes—
28#2059C22116yes—
29#1991C2218yes—
30#1948C2218yes—
31#1948C2218yes—
32#1947C2218yes—
33#1947C2218yes—
34#1944C2218yes—
35#1943C2218yes—
36#1846C2214yes—
37#1809C2218yes—
38#1809C2218yes—
39#1767C2214yes—
40C4×C4⋊C8C2214yes—
41#1728C2212yes—
42C22×M16C2312yes—
43C22×M16C2312yes—
44C2×#125—24no—
45C2×#125C2314yes—
46C2×#125C2314yes—
47C2×#117—24no—
48C22×C42D414yes—
49C22×C42C2311yescharacteristic
50C22×C42C2311yescharacteristic
51C22×C42C2311yescharacteristic
52#499—48no—
53C2×#107C2318yes—
54C2×#107C2318yes—
55C2×#106C2318yes—
56C2×#106C2318yes—
57#423Q818yes—
58#423Q818yes—
59C2×C4⋊C8C2314yes—
60C2×C4⋊C8C2314yes—
61C4×M16—44no—
62C2×#97C2314yes—
63C2×C4×C8C2314yes—
64#389—24no—
65#389—24no—
66#389—24no—
67#389—24no—
68#389—24no—
69#389—24no—
70#389C2318yes—
71#389C2318yes—
72#387—28no—
73#387—28no—
74#387—28no—
75#387—28no—
76#385—28no—
77#385—28no—
78#384C2314yes—
79#384C2314yes—
80#382—24no—
81#382—24no—
82#382C2314yes—
83#382C2314yes—
84#380—24no—
85#380C2318yes—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—28no—
88C4×C4⋊C4—28no—
89C4×C4⋊C4C2318yes—
90C43D412yes—
91C43D412yes—
92C43C2312yes—
93C43C2312yes—
94#340C23116yes—
95#340C23116yes—
96#336C2318yes—
97C23×C4C2×D412yes—
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4C2411yescharacteristic
101C2×M16—42no—
102C2×M16—42no—
103C2×M16—42no—
104C2×M16—42no—
105C2×M16C2×Q818yes—
106C2×M16C2×Q818yes—
107C22×C8C2414yes—
108C22×C8C2414yes—
109#125—28no—
110#125—28no—
111#125—44no—
112#125—44no—
113#125—416no—
114#117—416no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—28no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4C2418yes—
124C2×C4⋊C4C2418yes—
125C2×C22⋊C4—28no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—28no—
129C2×C42—48no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42C2×D414yes—
133C2×C42C2×D412yes—
134C2×C42C2×D414yes—
135C2×C42C2×D412yes—
136C2×C42C2×D414yes—
137C2×C42C2×D414yes—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×Q814yes—
141C2×C42C2×Q814yes—
142C2×C42C2412yes—
143C2×C42C2411yescharacteristic
144C2×C42C2412yes—
145C2×C42C2411yescharacteristic
146#107—48no—
147#107—48no—
148#106—48no—
149#106—48no—
150C4⋊C8C2×Q8116yes—
151C4⋊C8C2×Q8116yes—
152#97—44no—
153C4×C8—28no—
154#95—28no—
155#95—28no—
156#95—28no—
157#95—28no—
158#95—28no—
159#95—28no—
160C24C22×D411yescharacteristic
161C22×C4—28no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—24no—
167C22×C4—28no—
168C22×C4—28no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4C4⋊Q818yes—
173C22×C4C22×D412yes—
174C22×C4C22×D414yes—
175C22×C4C22×D411yescharacteristic
176C22×C4C22×D412yes—
177C22×C4C22×D412yes—
178C22×C4C22×D412yes—
179C22×C4C22×Q812yes—
180C22×C4C22×Q812yes—
181C22×C4C2511yescharacteristic
182M16—48no—
183M16—48no—
184C2×C8—24no—
185C2×C8—24no—
186C2×C8—24no—
187C2×C8—24no—
188C2×C8C22×Q818yes—
189C2×C8C22×Q818yes—
190C4⋊C4—28no—
191C4⋊C4—28no—
192C4⋊C4—28no—
193C4⋊C4—28no—
194C4⋊C4—48no—
195C4⋊C4—48no—
196C4⋊C4—416no—
197C4⋊C4—416no—
198C4⋊C4—416no—
199C4⋊C4—416no—
200C4⋊C4—48no—
201C4⋊C4—48no—
202C22⋊C4—416no—
203C42—24no—
204C42—28no—
205C42—28no—
206C42—21no—
207C42—21no—
208C42—22no—
209C42—22no—
210C42—42no—
211C42—44no—
212C42—44no—
213C42—416no—
214C42C4⋊Q814yes—
215C42C4⋊Q814yes—
216C42C22×Q814yes—
217C42C22×Q814yes—
218C42H(5,𝔽2)12yes—
219C422−1+412yes—
220C23—21no—
221C23—21no—
222C23—24no—
223C23C2×C4⋊Q812yes—
224C23C23×D411yescharacteristic
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—216no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—48no—
240C2×C4—48no—
241C2×C4—48no—
242C2×C4—48no—
243C2×C4—48no—
244C2×C4—48no—
245C2×C4—48no—
246C2×C4—48no—
247C2×C4C2×C4⋊Q812yes—
248C2×C4C2×C4⋊Q814yes—
249C2×C4C2×C4⋊Q814yes—
250C2×C4#53714yes—
251C2×C4#57518yes—
252C2×C4C23×Q811yescharacteristic
253C2×C4C23×Q811yescharacteristic
254C2×C4C2×H(5,𝔽2)11yescharacteristic
255C2×C4C2×2−1+411yescharacteristic
256C2×C4#58511yescharacteristic
257C2×C4#58511yescharacteristic
258C8—28no—
259C8—28no—
260C22—22no—
261C22—24no—
262C22—42no—
263C22—42no—
264C22C22×C4⋊Q811yescharacteristic
265C22C2×#53711yescharacteristic
266C22#344911yescharacteristic
267C22C2×#57514yes—
268C4—22no—
269C4—24no—
270C4—24no—
271C4—44no—
272C4—42no—
273C4—88no—
274C4—88no—
275C4#316618yes—
276C4#350111yescharacteristic
277C4#350111yescharacteristic
278C4#350211yescharacteristic
279C4#350211yescharacteristic
280C2—42no—
281C2#3517311yescharacteristic
282C2C2×#316612yes—
283C2#6090712yes—
284C2#6096712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #29321C2semidirect
5. #29210C2semidirect
6. #28691C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#332112
#302972
#293212
#292102
#286912
#219642
C4×#4232