Finite Group Databasebeta

← #352639

Subgroups — #352639

Order 512.

698 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 218 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#33310C211yesmaximal, characteristic
2#33294C211yesmaximal, characteristic
3#33256C211yesmaximal, characteristic
4#33232C211yesmaximal, characteristic
5#30228C211yesmaximal, characteristic
6#30220C211yesmaximal, characteristic
7#9828C214yesmaximal
8#9819C214yesmaximal
9C8×#97C211yesmaximal, characteristic
10C2×#498C2212yes—
11C2×#498C2212yes—
12C2×#491—22no—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15#2859C2211yescharacteristic
16#2859C2211yescharacteristic
17#2099—24no—
18#2043—24no—
19#1992—22no—
20#1990C2214yes—
21#1990C2214yes—
22#1960—22no—
23#1956C2214yes—
24#1956C2214yes—
25#1941—24no—
26#1844—24no—
27#1844C2212yes—
28#1836—22no—
29#1834C2212yes—
30#1744C2211yescharacteristic
31C4×#97C2211yescharacteristic
32C42×C8C2211yescharacteristic
33C2×#322C2212yes—
34C2×C82C2212yes—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8C2312yes—
39C2×C4⋊Q8C2312yes—
40C2×#125—21no—
41C2×#125—21no—
42#498—28no—
43#498—28no—
44#491—48no—
45C2×#106—24no—
46C2×#106—24no—
47C2×#106—24no—
48C2×#103—28no—
49C2×#103—28no—
50C2×#97D412yes—
51C2×#97D412yes—
52C2×C4×C8D414yes—
53C2×C4×C8C2312yes—
54C2×C4×C8C2312yes—
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#389—22no—
58#389—22no—
59#389—22no—
60#389—22no—
61#384—21no—
62#384—22no—
63#384—22no—
64#384—22no—
65#384—21no—
66#384C2312yes—
67#384C2312yes—
68#382—21no—
69#382—21no—
70C43C2311yescharacteristic
71#340—24no—
72#340—28no—
73#340—28no—
74#336D414yes—
75#322—24no—
76C82—24no—
77C22×Q8—22no—
78C22×Q8—22no—
79C22×C8C2×D414yes—
80C22×C8C2×D412yes—
81C22×C8C2×D414yes—
82C22×C8C2×D412yes—
83C4⋊Q8—24no—
84C4⋊Q8—24no—
85C4⋊Q8—24no—
86C4⋊Q8—24no—
87C4⋊Q8—44no—
88C4⋊Q8—44no—
89C4⋊Q8—84no—
90#125—44no—
91#125—44no—
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—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C42C2×D412yes—
109C2×C42C2×D412yes—
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42C2411yescharacteristic
113#106—48no—
114#106—48no—
115#106—48no—
116#103—416no—
117#103—88no—
118#97—24no—
119#97SD1618yes—
120C4×C8—28no—
121C4×C8—22no—
122C4×C8—22no—
123C4×C8—22no—
124C4×C8—22no—
125C4×C8#4114yes—
126C4×C8#4114yes—
127#95—42no—
128#95—42no—
129C2×Q8—44no—
130C2×Q8—44no—
131C2×Q8—44no—
132C2×Q8—44no—
133C2×Q8—48no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4#12712yes—
139C22×C4#12712yes—
140C22×C4C22×D411yescharacteristic
141C22×C4C22×D411yescharacteristic
142C22×C4C22×D411yescharacteristic
143C2×C8—24no—
144C2×C8—24no—
145C2×C8—24no—
146C2×C8—24no—
147C2×C8—28no—
148C2×C8#12414yes—
149C2×C8#12414yes—
150C2×C8C2×SD1618yes—
151C2×C8#13718yes—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C4⋊C4—84no—
167C4⋊C4—84no—
168C4⋊C4—84no—
169C4⋊C4—84no—
170C42—24no—
171C42—24no—
172C42C2×SD1614yes—
173C42#13514yes—
174C42C2×#4112yes—
175C42C2×#4112yes—
176C23C2×#12711yescharacteristic
177Q8—88no—
178Q8—88no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—84no—
194C2×C4#49214yes—
195C2×C4#49514yes—
196C2×C4C2×#12411yescharacteristic
197C2×C4C2×#12411yescharacteristic
198C2×C4#53212yes—
199C2×C4#53212yes—
200C2×C4C22×SD1612yes—
201C2×C4C2×#13512yes—
202C2×C4C2×#13714yes—
203C2×C4#57714yes—
204C8—24no—
205C8—28no—
206C8—44no—
207C8#49118yescomplemented
208C22#285811yescharacteristic
209C22C2×#49211yescharacteristic
210C22C2×#49511yescharacteristic
211C22#315212yes—
212C22#315412yes—
213C4—24no—
214C4—24no—
215C4—162no—
216C4—162no—
217C4—162no—
218C4—162no—
219C4C2×#49112yes—
220C4#313912yes—
221C4#316714yes—
222C4#317414yes—
223C2#1962212yes—
224C2#1962612yes—
225C2#3323211yescharacteristic
226C2#3323911yescharacteristic
227C2#3324611yescharacteristic
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

Complements

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

NH ≅ G/Ntype
207. C8#491semidirect

Maximal subgroups

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

subgroupindex
#333102
#332942
#332562
#332322
#302282
#302202
#98282
#98192
C8×#972