Finite Group Databasebeta

← #508981

Subgroups — #508981

Order 512.

460 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33311C211yesmaximal, characteristic
2#33258C211yesmaximal, characteristic
3#13506C212yesmaximal
4#13505C212yesmaximal
5#13453C214yesmaximal
6#12908C212yesmaximal
7#12890C212yesmaximal
8#12591C211yesmaximal, characteristic
9C2×#499—21no—
10C2×#499C2212yes—
11#2859—21no—
12#2859—21no—
13C2×#367C2214yes—
14C2×#366C2214yes—
15C2×#346C2212yes—
16#1942—21no—
17#1942—21no—
18#1942C2212yes—
19#1942C2212yes—
20#1846—21no—
21#1846C2212yes—
22#1845C2211yescharacteristic
23#1845C2211yescharacteristic
24#1845C2211yescharacteristic
25#1845C2211yescharacteristic
26#1844—21no—
27#1834—21no—
28C42×C8C2211yescharacteristic
29#1380C2214yes—
30#1380C2214yes—
31#1374C2212yes—
32#1374C2212yes—
33C2×#125—22no—
34C2×#125—41no—
35C22×C16C2314yes—
36#499—44no—
37#499—44no—
38C2×#107—21no—
39C2×#107—21no—
40C2×#107—21no—
41C2×#107—21no—
42C2×#107C2312yes—
43C2×#107C2312yes—
44C2×#107C2312yes—
45C2×#107C2312yes—
46C2×#106—22no—
47C2×#106—22no—
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8C2311yescharacteristic
52C2×C4×C8C2311yescharacteristic
53C2×C4×C8C2311yescharacteristic
54#389—22no—
55#389—22no—
56#389—41no—
57#389—41no—
58#389—41no—
59#389—41no—
60#389—41no—
61#389—41no—
62#384—21no—
63#384—21no—
64#384—21no—
65#384—21no—
66#382—22no—
67#382—41no—
68C43D411yescharacteristic
69#367—44no—
70#366—44no—
71#346Q818yes—
72#340—22no—
73#340—22no—
74C22×C8C2×D411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C22×C8C2411yescharacteristic
78#125—44no—
79#125—44no—
80#125—84no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C42D811yescharacteristic
98C2×C42D811yescharacteristic
99C2×C42D811yescharacteristic
100C2×C42D811yescharacteristic
101C2×C42C2×D411yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C16—24no—
105C2×C16—24no—
106C2×C16C2×Q818yes—
107#107—22no—
108#107—24no—
109#107—22no—
110#107—24no—
111#107—24no—
112#107—42no—
113#107—42no—
114#107—42no—
115#107—42no—
116#106—44no—
117#106—44no—
118C4×C8—22no—
119C4×C8—22no—
120C4×C8—22no—
121C4×C8—22no—
122C4×C8Q1614yes—
123C4×C8C2×Q814yes—
124#95—42no—
125#95—42no—
126#95—42no—
127#95—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4C2×D811yescharacteristic
133C22×C4C2×D811yescharacteristic
134C22×C4C2×D811yescharacteristic
135C22×C4C2×D811yescharacteristic
136C22×C4C2×D811yescharacteristic
137C22×C4C2×D811yescharacteristic
138C22×C4C22×D411yescharacteristic
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8C4⋊Q812yes—
144C2×C8C2×Q1612yes—
145C2×C8C2×Q1612yes—
146C2×C8AGL(1,ℤ/8)12yes—
147C2×C8#13712yes—
148C2×C8C22×Q812yes—
149C2×C8H(5,𝔽2)12yes—
150C2×C82−1+412yes—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C4⋊C4—84no—
168C4⋊C4—82no—
169C4⋊C4—84no—
170C4⋊C4—82no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—22no—
176C42—22no—
177C42—41no—
178C42—41no—
179C42C4⋊Q812yes—
180C42C2×Q1612yes—
181C16—28no—
182C23C22×D811yescharacteristic
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4#50012yes—
210C2×C4#50012yes—
211C2×C4#50912yes—
212C2×C4#51012yes—
213C2×C4C2×C4⋊Q811yescharacteristic
214C2×C4#53511yescharacteristic
215C2×C4#53611yescharacteristic
216C2×C4C22×Q1611yescharacteristic
217C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
218C2×C4C2×#13711yescharacteristic
219C8—22no—
220C8—22no—
221C8#50114yes—
222C8#54414yes—
223C22C2×#50011yescharacteristic
224C22#320511yescharacteristic
225C22#322611yescharacteristic
226C22C2×#50911yescharacteristic
227C22C2×#51011yescharacteristic
228C22#341411yescharacteristic
229C22#341411yescharacteristic
230C4—22no—
231C4—22no—
232C4—22no—
233C4—22no—
234C4—41no—
235C4—41no—
236C4—162no—
237C4—162no—
238C4—162no—
239C4—162no—
240C4#266011yescharacteristic
241C4C2×#50111yescharacteristic
242C4#320911yescharacteristic
243C4#323011yescharacteristic
244C2#3030411yescharacteristic
245C2#3373111yescharacteristic
246C2#3373211yescharacteristic
247C2#3378011yescharacteristic
248C2#3378211yescharacteristic
249C2#3380011yescharacteristic
250C2#3380211yescharacteristic
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

Maximal subgroups

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

subgroupindex
#333112
#332582
#135062
#135052
#134532
#129082
#128902
#125912