Finite Group Databasebeta

← #323309

Subgroups — #323309

Order 512.

656 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 214 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#33311C211yesmaximal, characteristic
2#29193C212yesmaximal
3#29179C212yesmaximal
4C4×#498C211yesmaximal, characteristic
5#10679C212yesmaximal
6#10674C212yesmaximal
7#9819C212yesmaximal
8#8814C212yesmaximal
9C8×C4⋊C8C211yesmaximal, characteristic
10C2×#499—22no—
11C2×#498C2211yescharacteristic
12#2859C2211yescharacteristic
13#2859C2211yescharacteristic
14C4×C4⋊Q8C2212yes—
15#2087—24no—
16#2068—24no—
17#1990C2212yes—
18#1990C2212yes—
19#1990C2212yes—
20#1956C2212yes—
21#1956C2212yes—
22#1941—22no—
23#1921—24no—
24#1863—24no—
25#1846—22no—
26#1845C2212yes—
27#1845C2212yes—
28#1845C2212yes—
29#1844C2212yes—
30#1844C2211yescharacteristic
31#1834C2212yes—
32#1834C2212yes—
33C4×#106—22no—
34C4×C4⋊C8C2212yes—
35C42×C8C2211yescharacteristic
36C2×C82C2212yes—
37#1273C2211yescharacteristic
38#1273C2211yescharacteristic
39C2×C4⋊Q8C2312yes—
40C2×#125—21no—
41C2×#125—21no—
42C2×C4×Q8—22no—
43#499—48no—
44#498—22no—
45#498Q814yes—
46C2×#107—24no—
47C2×#107—24no—
48C2×#107—24no—
49C2×#106—22no—
50C2×#106—24no—
51C2×#106—22no—
52C2×C4⋊C8D414yes—
53C2×#103—24no—
54C2×#103—24no—
55C2×C4×C8D412yes—
56C2×C4×C8D412yes—
57C2×C4×C8C2312yes—
58C2×C4×C8C2312yes—
59C2×C4×C8C2311yescharacteristic
60C2×C4×C8C2311yescharacteristic
61#391—22no—
62#389—22no—
63#389—22no—
64#389—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68#384C2312yes—
69#384C2312yes—
70#384C2312yes—
71#382—21no—
72#382—21no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C43C2311yescharacteristic
76#340—24no—
77#340—24no—
78#340—24no—
79#340—24no—
80#340—24no—
81#336D414yes—
82C82—24no—
83C22×Q8—22no—
84C22×C8C2×D414yes—
85C22×C8C2×D412yes—
86C22×C8C2×D414yes—
87C22×C8C2×D412yes—
88C4⋊Q8—22no—
89C4⋊Q8—22no—
90C4⋊Q8—22no—
91C4⋊Q8C2×Q814yes—
92#125—44no—
93#125—44no—
94C4×Q8—84no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C42—42no—
110C2×C42C2×D412yes—
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D412yes—
113C2×C42C2×D411yescharacteristic
114C2×C42C2411yescharacteristic
115#107—48no—
116#107—48no—
117#107—84no—
118#106—28no—
119#106—44no—
120#106—48no—
121C4⋊C8—28no—
122#103—48no—
123#103—84no—
124C4×C8—24no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—22no—
130C4×C8Q1618yes—
131C4×C8C2×Q812yes—
132C4×C8#4112yes—
133#95—42no—
134#95—42no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95—41no—
139C2×Q8—24no—
140C2×Q8—42no—
141C2×Q8—44no—
142C2×Q8—44no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4#12712yes—
148C22×C4#12711yescharacteristic
149C22×C4#12711yescharacteristic
150C22×C4C22×D411yescharacteristic
151C22×C4C22×D412yes—
152C2×C8—22no—
153C2×C8—24no—
154C2×C8—22no—
155C2×C8—24no—
156C2×C8—24no—
157C2×C8—24no—
158C2×C8—24no—
159C2×C8C22⋊Q814yes—
160C2×C8C2×Q1614yes—
161C2×C8C2×Q1614yes—
162C2×C8#13518yes—
163C2×C8AGL(1,ℤ/8)14yes—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—84no—
178C4⋊C4—84no—
179C4⋊C4—84no—
180C42—24no—
181C42—22no—
182C42—24no—
183C42—84no—
184C42C2×Q1614yes—
185C42C22×Q811yescharacteristic
186C42C2×#4111yescharacteristic
187C422−1+412yes—
188C23C2×#12711yescharacteristic
189Q8—48no—
190Q8—84no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—44no—
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#49412yes—
209C2×C4#49412yes—
210C2×C4C2×C22⋊Q811yescharacteristic
211C2×C4#53611yescharacteristic
212C2×C4#54712yes—
213C2×C4D4×Q812yes—
214C2×C4C22×Q1612yes—
215C2×C4C2×#13514yes—
216C2×C4C2×AGL(1,ℤ/8)12yes—
217C2×C4#57714yes—
218C8—28no—
219C8—24no—
220C8—44no—
221C8#47718yes—
222C22#254211yescharacteristic
223C22C2×#49411yescharacteristic
224C22#314712yes—
225C22#315111yescharacteristic
226C22#315412yes—
227C4—22no—
228C4—22no—
229C4—42no—
230C4—84no—
231C4—162no—
232C4—162no—
233C4—162no—
234C4C2×#47712yes—
235C4#323012yes—
236C4#333714yes—
237C4#338114yes—
238C2#1960411yescharacteristic
239C2#1964112yes—
240C2#1966311yescharacteristic
241C2#2851111yescharacteristic
242C2#2853312yes—
subgroupG/H
center, socle, prime radicalC23C2×#127
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
#333112
#291932
#291792
C4×#4982
#106792
#106742
#98192
#88142
C8×C4⋊C82