Finite Group Databasebeta

← #318418

Subgroups — #318418

Order 512.

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

normal subgroups only

This group has 604 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#28205C212yesmaximal
2#22270C211yesmaximal, characteristic
3#22242C211yesmaximal, characteristic
4C2×#1690C212yesmaximal
5#10487C214yesmaximal
6#8811C211yesmaximal, characteristic
7#8657C212yesmaximal
8#8593C212yesmaximal
9C2×#494C2211yescharacteristic
10C2×#475C2214yes—
11C2×#446C2212yes—
12C4×C4⋊Q8C2212yes—
13C4×C4⋊Q8C2211yescharacteristic
14#2084—22no—
15#2043—22no—
16#1984C2212yes—
17#1956C2212yes—
18#1927—24no—
19#1917—24no—
20#1878—24no—
21#1834C2211yescharacteristic
22#1831C2212yes—
23#1831C2211yescharacteristic
24#1803C2214yes—
25#1744C2211yescharacteristic
26#1690—28no—
27C2×#335C2212yes—
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⋊C8C2314yes—
45C2×C4⋊C8C2312yes—
46C2×#103—24no—
47C2×#103—24no—
48C2×#103—24no—
49C2×#97—22no—
50C2×C4×C8C2311yescharacteristic
51C2×C4×C8C2311yescharacteristic
52#391—22no—
53#391D412yes—
54#389—22no—
55#389—22no—
56#389—22no—
57#384D411yescharacteristic
58#384C2312yes—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C43C2311yescharacteristic
63#340—22no—
64#340—24no—
65#336—24no—
66#336—22no—
67#336—22no—
68#336—22no—
69#336D414yes—
70#335—42no—
71#334—24no—
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—22no—
82C4⋊Q8—22no—
83C4⋊Q8—22no—
84C4⋊Q8—42no—
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—
114#103—48no—
115#103—48no—
116#103—84no—
117#97—44no—
118C4×C8—21no—
119C4×C8—21no—
120C4×C8C2×D412yes—
121C4×C8#4112yes—
122#95—24no—
123#95—42no—
124#95—42no—
125C2×Q8—24no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—24no—
129C2×Q8—24no—
130C2×Q8—42no—
131C2×Q8—44no—
132C2×Q8—44no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4#12112yes—
138C22×C4#12112yes—
139C22×C4C22×D411yescharacteristic
140C22×C4C22×D411yescharacteristic
141C22×C4C2×#4111yescharacteristic
142Q16—88no—
143C2×C8—24no—
144C2×C8—22no—
145C2×C8—24no—
146C2×C8—42no—
147C2×C8—44no—
148C2×C8—44no—
149C2×C8—44no—
150C2×C8—44no—
151C2×C8#12114yes—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—24no—
155C4⋊C4—22no—
156C4⋊C4—44no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—82no—
164C4⋊C4—82no—
165C4⋊C4—84no—
166C42—22no—
167C42—24no—
168C42—22no—
169C42—42no—
170C42—42no—
171C42—42no—
172C42—84no—
173C42C22×D411yescharacteristic
174C42C2×#4111yescharacteristic
175C42H(5,𝔽2)11yescharacteristic
176C422−1+411yescharacteristic
177C23C2×#12111yescharacteristic
178Q8—44no—
179Q8—44no—
180Q8—48no—
181Q8—84no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4C2×#12111yescharacteristic
201C2×C4#53711yescharacteristic
202C2×C4D4211yescharacteristic
203C2×C4#54711yescharacteristic
204C2×C4#54711yescharacteristic
205C2×C4#55511yescharacteristic
206C2×C4#57512yes—
207C2×C4#57714yes—
208C2×C4#57812yes—
209C8—24no—
210C8—42no—
211C8—84no—
212C8—82no—
213C22#253711yescharacteristic
214C22#304312yes—
215C22#305411yescharacteristic
216C22#305911yescharacteristic
217C22#306912yes—
218C4—21no—
219C4—21no—
220C4—42no—
221C4—42no—
222C4—82no—
223C4—82no—
224C4—84no—
225C4—162no—
226C4#304212yes—
227C4#326012yes—
228C4#327712yes—
229C4#335212yes—
230C2#1943612yes—
231C2#1952312yes—
232C2#2823411yescharacteristic
233C2#2824411yescharacteristic
234C2#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
#282052
#222702
#222422
C2×#16902
#104872
#88112
#86572
#85932