Finite Group Databasebeta

← #511558

Subgroups — #511558

Order 512.

450 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28676C211yesmaximal, characteristic
2#28675C211yesmaximal, characteristic
3#13516C212yesmaximal
4#13452C212yesmaximal
5#13452C212yesmaximal
6#13263C211yesmaximal, characteristic
7#13166C212yesmaximal
8#13111C212yesmaximal
9#13110C212yesmaximal
10C2×#500C2211yescharacteristic
11C2×#500C2211yescharacteristic
12C2×C8×Q8C2211yescharacteristic
13#2567—21no—
14#2546—21no—
15C2×#367C2212yes—
16C2×#367C2212yes—
17C2×#367C2212yes—
18C2×C4⋊C16C2211yescharacteristic
19C2×C4×C16C2211yescharacteristic
20#2087—21no—
21#2087—21no—
22#2086—21no—
23#2086—21no—
24#1988—21no—
25#1987—21no—
26#1942C2212yes—
27#1942C2212yes—
28#1942C2212yes—
29#1942C2212yes—
30C4×#107C2211yescharacteristic
31C4×#107C2211yescharacteristic
32#1380C2212yes—
33#1380C2212yes—
34#1380C2212yes—
35#1380C2212yes—
36#1380C2212yes—
37#1374C2212yes—
38#1374C2212yes—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—41no—
42C2×C4×Q8D411yescharacteristic
43C22×C16C2312yes—
44C22×C16C2312yes—
45#500—41no—
46#500—41no—
47#500—41no—
48#500—41no—
49C8×Q8—41no—
50C8×Q8—41no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107C2312yes—
54C2×#107C2312yes—
55C2×#107C2312yes—
56C2×#107C2312yes—
57C2×C4⋊C8D411yescharacteristic
58C2×C4⋊C8D411yescharacteristic
59C2×C4⋊C8D411yescharacteristic
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63#398—41no—
64#398—41no—
65#395—41no—
66#395—41no—
67#391—41no—
68#391—41no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#384—41no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—21no—
76#367—42no—
77#367—42no—
78#367Q818yescomplemented
79C4⋊C16—41no—
80C4×C16—41no—
81#340—21no—
82#340—22no—
83#340—22no—
84#340—21no—
85#340—22no—
86#340—22no—
87C22×Q8—41no—
88C22×Q8—41no—
89C22×Q8D811yescharacteristic
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D411yescharacteristic
92C22×C8C2×D411yescharacteristic
93C22×C8C2411yescharacteristic
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C4⋊Q8—42no—
97C4⋊Q8—42no—
98C4⋊Q8—84no—
99C4×Q8—41no—
100C4×Q8—41no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4D811yescharacteristic
112C2×C4⋊C4D811yescharacteristic
113C2×C4⋊C4D811yescharacteristic
114C2×C42—41no—
115C2×C42—41no—
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C16—22no—
120C2×C16—22no—
121C2×C16—22no—
122C2×C16—24no—
123C2×C16C2×Q814yes—
124#107—22no—
125#107—22no—
126#107—22no—
127#107—22no—
128#107—22no—
129#107—22no—
130#107—42no—
131#107—42no—
132#107C2×Q814yes—
133#107C2×Q814yes—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8—21no—
143#95—41no—
144#95—41no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—42no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—82no—
152C2×Q8—82no—
153C2×Q8—82no—
154C2×Q8—82no—
155C2×Q8—162no—
156C2×Q8SD3214yes—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4C2×D811yescharacteristic
162C22×C4C2×D811yescharacteristic
163C22×C4C2×D811yescharacteristic
164C22×C4C2×D811yescharacteristic
165C22×C4C2×D811yescharacteristic
166C22×C4C2×D811yescharacteristic
167C22×C4C22×D411yescharacteristic
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8C22×Q812yes—
175C2×C8H(5,𝔽2)11yescharacteristic
176C2×C8H(5,𝔽2)11yescharacteristic
177C2×C82−1+412yes—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—82no—
190C4⋊C4—82no—
191C4⋊C4—84no—
192C4⋊C4—84no—
193C4⋊C4—162no—
194C4⋊C4—162no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—82no—
202C42—82no—
203C16—24no—
204C16—42no—
205C23C22×D811yescharacteristic
206Q8—44no—
207Q8—164no—
208Q8—164no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4C2×SD3212yes—
234C2×C4C2×SD3212yes—
235C2×C4C2×SD3212yes—
236C2×C4#53411yescharacteristic
237C2×C4D4×Q812yes—
238C2×C4#57511yescharacteristic
239C2×C4#57812yes—
240C8—41no—
241C8—41no—
242C8—41no—
243C8#54412yes—
244C8Q8211yescharacteristic
245C8#56311yescharacteristic
246C22#318611yescharacteristic
247C22Q8×D812yes—
248C22C22×SD3211yescharacteristic
249C22#341411yescharacteristic
250C22#341612yes—
251C4—22no—
252C4—22no—
253C4—22no—
254C4—41no—
255C4—41no—
256C4—41no—
257C4—84no—
258C4—84no—
259C4—162no—
260C4—162no—
261C4#273411yescharacteristic
262C4#320211yescharacteristic
263C4#339012yes—
264C2#3268211yescharacteristic
265C2#3375311yescharacteristic
266C2#3376611yescharacteristic
267C2Q8×SD3212yes—
268C2#3395512yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
78. #367Q8semidirect

Maximal subgroups

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

subgroupindex
#286762
#286752
#135162
#134522
#132632
#131662
#131112
#131102