Finite Group Databasebeta

← #200843

Subgroups — #200843

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60663C211yesmaximal, characteristic
2#34348C211yesmaximal, characteristic
3#17739C211yesmaximal, characteristic
4#17562C212yesmaximal
5#9427C218yesmaximal, complemented
6#7973C212yesmaximal
7C2×#533C2211yescharacteristic
8C2×#533C2211yescharacteristic
9#3166—22no—
10#3166—22no—
11#3165C414yescomplemented
12#3165C2214yes—
13#3162—22no—
14#3160C414yescomplemented
15#2920C2212yes—
16#2911—22no—
17C2×C4×M16C2211yescharacteristic
18#2382—22no—
19#2381C2214yes—
20#2378—24no—
21#2298—22no—
22#2295—24no—
23#2283C2212yes—
24#1561—22no—
25#1556C2214yes—
26#1544—22no—
27#1515C2218yes—
28#1511—24no—
29#1503—24no—
30#1288C2218yes—
31C22×M16—21no—
32#533—21no—
33#533—21no—
34#533—22no—
35#533—22no—
36#533—22no—
37#533—22no—
38#533C2×C412yes—
39#533C2×C412yes—
40#533C2×C412yes—
41#533D412yes—
42#533D412yes—
43#533C2312yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2312yes—
46C2×#125—21no—
47C2×#125D412yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×#117—21no—
53C2×#117D412yes—
54C22×C42C2311yescharacteristic
55#501—28no—
56#501—28no—
57#500C2×C418yes—
58#499—28no—
59#498C2×C418yes—
60#432—22no—
61#432C2314yes—
62#431—22no—
63#431—22no—
64#428—28no—
65#428—28no—
66#427—28no—
67#423—22no—
68#407—22no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—22no—
72C4×M16C2×C412yes—
73C4×M16C2312yes—
74C2×#97—21no—
75C2×C4×C8C2×C412yes—
76#391—28no—
77#389—28no—
78#388—28no—
79#387—28no—
80#387—28no—
81#387—28no—
82#383C2314yes—
83#382—22no—
84#381D414yes—
85#380—22no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4D414yes—
88C4×C22⋊C4—22no—
89#376—22no—
90#376—22no—
91C43—22no—
92#332—28no—
93#328—216no—
94C22×Q8—22no—
95C22×Q8—22no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D412yes—
98C2×M16—28no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—42no—
102C22×C8—22no—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8C22×C414yes—
109C4⋊Q8C22×C414yes—
110C4⋊Q8C2×D414yes—
111#125—22no—
112#125—24no—
113#125—22no—
114#125—42no—
115#125C22⋊C414yes—
116#125C22⋊C414yes—
117C22⋊Q8—24no—
118C22⋊Q8—24no—
119C22⋊Q8—24no—
120C22⋊Q8—24no—
121C22⋊Q8—24no—
122C22⋊Q8—24no—
123C22⋊Q8—42no—
124C22⋊Q8—42no—
125C4×Q8—24no—
126C4×Q8—24no—
127C4×Q8—24no—
128C4×Q8—24no—
129C4×Q8—24no—
130C4×Q8—24no—
131C4×Q8—42no—
132C4×Q8—42no—
133#117—22no—
134#117—22no—
135#117—22no—
136#117—22no—
137#117—41no—
138#117—41no—
139#117C22⋊C414yes—
140#117C22⋊C414yes—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4C2×D414yes—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—41no—
152C2×C42—22no—
153C2×C42—24no—
154C2×C42—28no—
155C2×C42—41no—
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D412yes—
162C2×C42#4112yes—
163C2×C42#4112yes—
164C2×C42C2411yescharacteristic
165#107—216no—
166#106—216no—
167C4⋊C8—24no—
168C4⋊C8—42no—
169C22⋊C8—24no—
170C22⋊C8—42no—
171#97—22no—
172#97—22no—
173#97—24no—
174C4×C8—22no—
175C4×C8—42no—
176C4×C8C22×C414yes—
177#95—28no—
178#95—28no—
179#95—24no—
180#95—24no—
181#95—24no—
182#95C2×D418yes—
183C24#12011yescharacteristic
184C2×Q8—24no—
185C2×Q8—24no—
186C2×Q8—24no—
187C2×Q8—42no—
188C2×Q8—84no—
189C2×Q8—84no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—28no—
193C22×C4—42no—
194C22×C4—82no—
195C22×C4C2×C22⋊C411yescharacteristic
196C22×C4C2×C22⋊C411yescharacteristic
197C22×C4C4×D412yes—
198C22×C4C4×D412yes—
199C22×C4#12012yes—
200C22×C4#12011yescharacteristic
201C22×C4#12112yes—
202C22×C4#12112yes—
203C22×C4#12311yescharacteristic
204C22×C4#12311yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D412yes—
207M16—28no—
208M16—28no—
209M16—84no—
210C2×C8—28no—
211C2×C8—24no—
212C2×C8—44no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—28no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—44no—
224C4⋊C4—84no—
225C4⋊C4—88no—
226C4⋊C4C2×C22⋊C418yes—
227C22⋊C4—24no—
228C22⋊C4—24no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—84no—
232C42—24no—
233C42—24no—
234C42—42no—
235C42—42no—
236C42—48no—
237C42—88no—
238C42C2×C22⋊C411yescharacteristic
239C42C2×C22⋊C411yescharacteristic
240C42C4×D412yes—
241C42C4×D412yes—
242C42#12112yes—
243C42#12112yes—
244C42#12312yes—
245C42C23×C411yescharacteristic
246C42C22×D411yescharacteristic
247C42C2×#4112yes—
248C23—22no—
249C23—42no—
250C23#38611yescharacteristic
251C23#38611yescharacteristic
252C23C2×#12011yescharacteristic
253Q8—84no—
254Q8—84no—
255Q8—84no—
256Q8—84no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—28no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—48no—
270C2×C4—48no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—44no—
274C2×C4—84no—
275C2×C4#38612yes—
276C2×C4#38612yes—
277C2×C4#38611yescharacteristic
278C2×C4#38611yescharacteristic
279C2×C4#45614yes—
280C2×C4C22×C22⋊C411yescharacteristic
281C2×C4C2×C4×D412yes—
282C2×C4C2×#12112yes—
283C2×C4C2×#12311yescharacteristic
284C8—28no—
285C8—84no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—21no—
290C22—21no—
291C22—82no—
292C22C2×#38611yescharacteristic
293C22C2×#45612yes—
294C4—22no—
295C4—22no—
296C4—22no—
297C4—22no—
298C4—21no—
299C4—21no—
300C4—48no—
301C4—88no—
302C4—84no—
303C4#188612yes—
304C4#188612yes—
305C2—22no—
306C2—41no—
307C2—41no—
308C2C2×#188611yescharacteristic
309C2#2099812yes—
subgroupG/H
centerC2×C4#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9427C2semidirect
11. #3165C4semidirect
14. #3160C4semidirect

Maximal subgroups

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

subgroupindex
#606632
#343482
#177392
#175622
#94272
#79732