Finite Group Databasebeta

← #318416

Subgroups — #318416

Order 512.

614 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 162 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#28283C212yesmaximal
2#22270C211yesmaximal, characteristic
3#22242C211yesmaximal, characteristic
4C2×#1690C212yesmaximal
5#10486C214yesmaximal
6#8809C211yesmaximal, characteristic
7#8657C212yesmaximal
8#8591C212yesmaximal
9C2×#494C2211yescharacteristic
10C2×#475C2214yes—
11C2×#446C2212yes—
12C4×C4⋊Q8C2212yes—
13C4×C4⋊Q8C2211yescharacteristic
14#2066—24no—
15#1984C2212yes—
16#1981—22no—
17#1956C2212yes—
18#1948—22no—
19#1917—24no—
20#1878—24no—
21#1842C2212yes—
22#1834C2211yescharacteristic
23#1831C2211yescharacteristic
24#1803C2214yes—
25#1744C2211yescharacteristic
26#1690—28no—
27C2×#334C2212yes—
28C2×#334C2212yes—
29C2×#332C2214yes—
30#1296C2214yes—
31C22×Q16—22no—
32C2×C4⋊Q8D411yescharacteristic
33C2×C4⋊Q8C2312yes—
34C2×C4×Q8—22no—
35C2×C4×Q8C2312yes—
36#494—22no—
37#494D414yes—
38#475—28no—
39#475—44no—
40#446—24no—
41#446—24no—
42C2×#106—24no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8C2314yes—
47C2×C4⋊C8C2312yes—
48C2×#103—24no—
49C2×#103—24no—
50C2×#103—24no—
51C2×#97—22no—
52C2×C4×C8C2311yescharacteristic
53C2×C4×C8C2311yescharacteristic
54#391—22no—
55#391D412yes—
56#389—22no—
57#389—22no—
58#389—22no—
59#384D411yescharacteristic
60#384C2312yes—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C43C2311yescharacteristic
65#340—22no—
66#340—24no—
67#336—24no—
68#336—22no—
69#336D414yes—
70#334—24no—
71#334—42no—
72#332—28no—
73C22×Q8—22no—
74C22×Q8C2×D412yes—
75C2×Q16—28no—
76C2×Q16—84no—
77C22×C8—22no—
78C22×C8—24no—
79C22×C8C2×D412yes—
80C22×C8C2×D412yes—
81C4⋊Q8—24no—
82C4⋊Q8—22no—
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85C4⋊Q8C2×D414yes—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—84no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4C2×D412yes—
101C2×C42—22no—
102C2×C42—42no—
103C2×C42C2×D412yes—
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42#4112yes—
107C2×C42C2411yescharacteristic
108#106—48no—
109C4⋊C8—22no—
110C4⋊C8—24no—
111C4⋊C8—22no—
112C4⋊C8—24no—
113C4⋊C8—44no—
114C4⋊C8—44no—
115C4⋊C8—82no—
116#103—48no—
117#103—84no—
118#103—84no—
119#97—44no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8C2×D412yes—
123C4×C8#4112yes—
124#95—24no—
125#95—42no—
126#95—42no—
127C2×Q8—24no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—24no—
131C2×Q8—24no—
132C2×Q8—42no—
133C2×Q8—44no—
134C2×Q8—44no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4#12112yes—
140C22×C4#12112yes—
141C22×C4C22×D411yescharacteristic
142C22×C4C22×D411yescharacteristic
143C22×C4C2×#4111yescharacteristic
144Q16—88no—
145C2×C8—24no—
146C2×C8—22no—
147C2×C8—24no—
148C2×C8—42no—
149C2×C8—44no—
150C2×C8—44no—
151C2×C8—44no—
152C2×C8—44no—
153C2×C8#12114yes—
154C4⋊C4—24no—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4—22no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—44no—
163C4⋊C4—42no—
164C4⋊C4—44no—
165C4⋊C4—42no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4—84no—
169C42—22no—
170C42—24no—
171C42—42no—
172C42—42no—
173C42—42no—
174C42—84no—
175C42#13714yes—
176C42C22×D411yescharacteristic
177C42C2×#4111yescharacteristic
178C42H(5,𝔽2)11yescharacteristic
179C422−1+411yescharacteristic
180C23C2×#12111yescharacteristic
181Q8—44no—
182Q8—44no—
183Q8—48no—
184Q8—84no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4#47014yes—
203C2×C4C2×#12111yescharacteristic
204C2×C4#53711yescharacteristic
205C2×C4D4211yescharacteristic
206C2×C4#54711yescharacteristic
207C2×C4#54711yescharacteristic
208C2×C4#55511yescharacteristic
209C2×C4C2×#13712yes—
210C2×C4#57712yes—
211C2×C4#57714yes—
212C8—24no—
213C8—42no—
214C8—84no—
215C8—82no—
216C22#253711yescharacteristic
217C22#304312yes—
218C22C2×#47011yescharacteristic
219C22#306011yescharacteristic
220C22#306912yes—
221C4—21no—
222C4—21no—
223C4—24no—
224C4—42no—
225C4—82no—
226C4—82no—
227C4—84no—
228C4—162no—
229C4#304212yes—
230C4#326012yes—
231C4#327812yes—
232C4#335612yes—
233C2#1943512yes—
234C2#1952412yes—
235C2#2823211yescharacteristic
236C2#2824411yescharacteristic
237C2#2826011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#282832
#222702
#222422
C2×#16902
#104862
#88092
#86572
#85912