Finite Group Databasebeta

← #508978

Subgroups — #508978

Order 512.

486 conjugacy classes of subgroups.

The subgroup lattice has 269 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33312C211yesmaximal, characteristic
2#33257C211yesmaximal, characteristic
3#13506C212yesmaximal
4#13505C212yesmaximal
5#13452C212yesmaximal
6#13451C212yesmaximal
7#12908C212yesmaximal
8#12890C212yesmaximal
9#12591C211yesmaximal, characteristic
10C2×#500—21no—
11C2×#500C2212yes—
12#2868—21no—
13#2864—21no—
14C2×#367C2214yes—
15C2×#366C2214yes—
16C2×#346C2212yes—
17#1942C2211yescharacteristic
18#1942C2211yescharacteristic
19#1942C2211yescharacteristic
20#1942C2211yescharacteristic
21#1941—21no—
22#1941—21no—
23#1846C2211yescharacteristic
24#1846C2211yescharacteristic
25#1845—21no—
26#1845—21no—
27#1845C2212yes—
28#1845C2212yes—
29#1835—21no—
30C42×C8C2211yescharacteristic
31#1380C2214yes—
32#1380C2214yes—
33#1374C2212yes—
34#1374C2212yes—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—41no—
37C2×C4⋊Q8—41no—
38C22×C16C2314yes—
39#500—44no—
40#500—44no—
41C2×#107—22no—
42C2×#107—22no—
43C2×#107—22no—
44C2×#107C2314yes—
45C2×#107C2311yescharacteristic
46C2×#107C2311yescharacteristic
47C2×#107C2311yescharacteristic
48C2×#107C2311yescharacteristic
49C2×#106—21no—
50C2×#106—21no—
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#389—21no—
58#389—21no—
59#389—21no—
60#389—21no—
61#387—41no—
62#387—41no—
63#387—41no—
64#387—41no—
65#384—22no—
66#384—22no—
67#384—41no—
68#384—41no—
69#382—21no—
70#382—21no—
71C43D411yescharacteristic
72#367—44no—
73#366—44no—
74#346Q818yescomplemented
75#340—21no—
76#340—21no—
77#340—21no—
78#340—21no—
79C22×Q8—42no—
80C22×C8C2×D411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2411yescharacteristic
84C4⋊Q8—44no—
85C4⋊Q8—44no—
86C4⋊Q8—84no—
87C4⋊Q8—84no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C42D811yescharacteristic
103C2×C42D811yescharacteristic
104C2×C42D811yescharacteristic
105C2×C42D811yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C16—24no—
110C2×C16—24no—
111C2×C16C2×Q818yes—
112#107—22no—
113#107—24no—
114#107—22no—
115#107—24no—
116#107—22no—
117#107—22no—
118#107—44no—
119#107—44no—
120#107—44no—
121#106—42no—
122#106—42no—
123C4×C8—22no—
124C4×C8—22no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8Q1614yes—
128C4×C8C2×Q814yes—
129#95—41no—
130#95—41no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—41no—
136#95—41no—
137C2×Q8—84no—
138C2×Q8—84no—
139C2×Q8—84no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—44no—
145C22×C4C2×D811yescharacteristic
146C22×C4C2×D811yescharacteristic
147C22×C4C2×D811yescharacteristic
148C22×C4C2×D811yescharacteristic
149C22×C4C2×D811yescharacteristic
150C22×C4C2×D811yescharacteristic
151C22×C4C22×D411yescharacteristic
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8C4⋊Q812yes—
157C2×C8C2×Q1612yes—
158C2×C8C2×Q1612yes—
159C2×C8AGL(1,ℤ/8)12yes—
160C2×C8#13712yes—
161C2×C8C22×Q812yes—
162C2×C8H(5,𝔽2)12yes—
163C2×C82−1+412yes—
164C4⋊C4—44no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—44no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—84no—
175C4⋊C4—84no—
176C4⋊C4—84no—
177C4⋊C4—84no—
178C4⋊C4—82no—
179C4⋊C4—82no—
180C4⋊C4—84no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—22no—
191C42C4⋊Q812yes—
192C42C2×Q1612yes—
193C16—28no—
194C23C22×D811yescharacteristic
195Q8—168no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—84no—
209C2×C4—84no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—84no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4#50012yes—
224C2×C4#50012yes—
225C2×C4#50912yes—
226C2×C4#50912yes—
227C2×C4#51012yes—
228C2×C4#51012yes—
229C2×C4C2×C4⋊Q811yescharacteristic
230C2×C4#53511yescharacteristic
231C2×C4#53611yescharacteristic
232C2×C4C22×Q1611yescharacteristic
233C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
234C2×C4C2×#13711yescharacteristic
235C8—22no—
236C8—22no—
237C8#50114yes—
238C8#54414yes—
239C22C2×#50011yescharacteristic
240C22#320511yescharacteristic
241C22#322611yescharacteristic
242C22C2×#50911yescharacteristic
243C22C2×#50911yescharacteristic
244C22C2×#51011yescharacteristic
245C22C2×#51011yescharacteristic
246C4—22no—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—22no—
251C4—22no—
252C4—161no—
253C4—161no—
254C4—161no—
255C4—164no—
256C4—161no—
257C4#266011yescharacteristic
258C4C2×#50111yescharacteristic
259C4#320911yescharacteristic
260C4#323011yescharacteristic
261C2#3030411yescharacteristic
262C2#3373111yescharacteristic
263C2#3373111yescharacteristic
264C2#3378311yescharacteristic
265C2#3378411yescharacteristic
266C2#3380411yescharacteristic
267C2#3380511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
74. #346Q8semidirect

Maximal subgroups

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

subgroupindex
#333122
#332572
#135062
#135052
#134522
#134512
#129082
#128902
#125912