Finite Group Databasebeta

← #331149

Subgroups — #331149

Order 512.

718 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33207C212yesmaximal
2#33049C212yesmaximal
3#29876C211yesmaximal, characteristic
4#29808C211yesmaximal, characteristic
5#10681C212yesmaximal, complemented
6#10681C212yesmaximal, complemented
7#10318C212yesmaximal, complemented
8#10314C212yesmaximal, complemented
9#7848C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#498C2212yes—
12C2×#494C2211yescharacteristic
13C2×#487C2212yes—
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2857C2212yes—
18#2857C2212yes—
19#2086—22no—
20#2075—22no—
21#2056—22no—
22#2001—22no—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2212yes—
27#1956C2212yes—
28#1956C2212yes—
29#1941—24no—
30#1935—24no—
31#1894—24no—
32#1844C2212yes—
33#1834C2212yes—
34#1834C2211yescharacteristic
35#1834C2211yescharacteristic
36#1807—24no—
37#1273C2212yes—
38#1273C2212yes—
39C22×Q16—22no—
40C22×M16D412yes—
41C23×C8D412yes—
42C2×C4⋊Q8C2312yes—
43C2×C4⋊Q8C2312yes—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C22×C42C2311yescharacteristic
47#500—41no—
48#500—41no—
49#498—42no—
50#498—42no—
51#494—41no—
52#494—41no—
53#487—42no—
54#487—42no—
55#431—41no—
56#431—41no—
57C2×#107—22no—
58C2×#106—24no—
59C2×#106—24no—
60C2×C4⋊C8D412yes—
61C2×#103—24no—
62C2×#103—24no—
63C2×#103—24no—
64C2×C22⋊C8D412yes—
65C4×M16—41no—
66C4×M16—41no—
67C2×#97D412yes—
68C2×C4×C8—41no—
69C2×C4×C8—41no—
70C2×C4×C8D412yes—
71C2×C4×C8C2312yes—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2312yes—
75#391—22no—
76#391—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#384C2312yes—
82#384C2312yes—
83#380—22no—
84#380—22no—
85#380—22no—
86#380—22no—
87#340—22no—
88#340—24no—
89#340—24no—
90#340—22no—
91#340—24no—
92#340—24no—
93#340—24no—
94C22×Q8—22no—
95C22×Q8—22no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C2×Q16—44no—
100C2×Q16—84no—
101C2×M16—24no—
102C2×M16—42no—
103C2×M16—42no—
104C22×C8—28no—
105C22×C8—42no—
106C22×C8C2×D414yes—
107C22×C8C2×D412yes—
108C22×C8C2×D412yes—
109C22×C8C2×D414yes—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C22⋊Q8—48no—
119C22⋊Q8—84no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2411yescharacteristic
142#107—44no—
143#106—48no—
144#106—48no—
145C4⋊C8—42no—
146#103—48no—
147#103—48no—
148#103—84no—
149C22⋊C8—42no—
150#97—42no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—21no—
159C4×C8—42no—
160#95—42no—
161#95—42no—
162#95—42no—
163#95—42no—
164C24#12711yescharacteristic
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4#12711yescharacteristic
187C22×C4#12711yescharacteristic
188C22×C4#12711yescharacteristic
189C22×C4#13514yes—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193Q16—88no—
194M16—48no—
195C2×C8—24no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—24no—
201C2×C8—24no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—48no—
205C2×C8#13718yes—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—84no—
221C4⋊C4—84no—
222C22⋊C4—84no—
223C22⋊C4—84no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42H(5,𝔽2)11yescharacteristic
231C42H(5,𝔽2)11yescharacteristic
232C42H(5,𝔽2)11yescharacteristic
233C422−1+411yescharacteristic
234C23—21no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23#49512yes—
239C23C2×#12711yescharacteristic
240Q8—84no—
241Q8—84no—
242Q8—84no—
243Q8—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4#49512yes—
269C2×C4#53411yescharacteristic
270C2×C4#53411yescharacteristic
271C2×C4#53411yescharacteristic
272C2×C4#53711yescharacteristic
273C2×C4#53711yescharacteristic
274C2×C4#53711yescharacteristic
275C2×C4C2×#13512yes—
276C2×C4C2×#13714yes—
277C2×C4#57512yes—
278C2×C4#57714yes—
279C8—28no—
280C8—44no—
281C8—42no—
282C8—42no—
283C22—22no—
284C22—24no—
285C22—41no—
286C22—41no—
287C22—42no—
288C22#265611yescharacteristic
289C22C2×#49511yescharacteristic
290C22#315011yescharacteristic
291C22#315212yes—
292C22#315412yes—
293C4—22no—
294C4—41no—
295C4—41no—
296C4—162no—
297C4—162no—
298C4—162no—
299C4—162no—
300C4#319112yes—
301C4#319512yes—
302C4#320012yes—
303C4#325012yes—
304C4#325512yes—
305C4#326012yes—
306C2—42no—
307C2#1962612yes—
308C2#1969312yes—
309C2#3024111yescharacteristic
310C2#3025011yescharacteristic
311C2#3025411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
5. #10681C2semidirect
6. #10681C2semidirect
7. #10318C2semidirect
8. #10314C2semidirect

Maximal subgroups

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

subgroupindex
#332072
#330492
#298762
#298082
#106812
#103182
#103142
#78482