Finite Group Databasebeta

← #323451

Subgroups — #323451

Order 512.

724 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 226 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#29219C212yesmaximal
3#29208C212yesmaximal
4C4×#494C211yesmaximal, characteristic
5#10675C214yesmaximal
6#9839C212yesmaximal
7#8820C212yesmaximal
8#8260C211yesmaximal, characteristic
9C2×#498—22no—
10C2×#494C2211yescharacteristic
11C2×C4×Q16—22no—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14C4×C4⋊Q8C2212yes—
15#2099—24no—
16#2057—24no—
17#1990C2214yes—
18#1990C2214yes—
19#1988—24no—
20#1969—22no—
21#1956C2212yes—
22#1947—24no—
23#1845C2214yes—
24#1845C2211yescharacteristic
25#1844—22no—
26#1844C2212yes—
27#1844C2212yes—
28#1834C2214yes—
29C4×C4⋊C8C2212yes—
30C42×C8C2211yescharacteristic
31C2×#322C2212yes—
32#1273C2212yes—
33C22×Q16—22no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8C2312yes—
39C2×C4×Q8—22no—
40#498—48no—
41#494—22no—
42#494Q814yescomplemented
43C4×Q16—48no—
44C2×#107—22no—
45C2×#107—28no—
46C2×#106—24no—
47C2×#106—24no—
48C2×C4⋊C8D414yes—
49C2×#103—24no—
50C2×C4×C8D412yes—
51C2×C4×C8D412yes—
52C2×C4×C8C2314yes—
53C2×C4×C8C2311yescharacteristic
54C2×C4×C8C2311yescharacteristic
55#391—22no—
56#389—22no—
57#389—22no—
58#384—22no—
59#384—21no—
60#384—21no—
61#384—22no—
62#384C2312yes—
63#384C2312yes—
64#384C2312yes—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C43C2311yescharacteristic
68#340—28no—
69#340—28no—
70#340—28no—
71#336D414yes—
72#322—24no—
73C22×Q8—22no—
74C22×Q8—41no—
75C22×Q8—41no—
76C2×Q16—28no—
77C2×Q16—44no—
78C2×Q16—44no—
79C22×C8C2×D418yes—
80C22×C8C2×D412yes—
81C22×C8C2×D412yes—
82C4⋊Q8—22no—
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85C4⋊Q8—44no—
86C4⋊Q8—44no—
87C4⋊Q8—48no—
88C4⋊Q8—84no—
89C4⋊Q8C2×Q814yes—
90C4×Q8—48no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C42—42no—
106C2×C42C2×D412yes—
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D412yes—
110C2×C42C2411yescharacteristic
111#107—44no—
112#107—88no—
113#106—48no—
114#106—48no—
115C4⋊C8—28no—
116#103—48no—
117C4×C8—24no—
118C4×C8—24no—
119C4×C8—22no—
120C4×C8—24no—
121C4×C8D818yes—
122C4×C8C2×Q812yes—
123C4×C8#4112yes—
124#95—42no—
125#95—42no—
126C2×Q8—24no—
127C2×Q8—42no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—84no—
131C2×Q8—82no—
132C2×Q8—82no—
133C2×Q8—84no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4#12711yescharacteristic
139C22×C4#12712yes—
140C22×C4#12711yescharacteristic
141C22×C4C22×D411yescharacteristic
142C22×C4C22×D412yes—
143Q16—416no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—28no—
147C2×C8—28no—
148C2×C8—28no—
149C2×C8C22⋊Q814yes—
150C2×C8C2×D814yes—
151C2×C8C2×D814yes—
152C2×C8#13714yes—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C4⋊C4—48no—
161C4⋊C4—44no—
162C4⋊C4—48no—
163C4⋊C4—44no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C4⋊C4—82no—
167C4⋊C4—82no—
168C4⋊C4—84no—
169C42—22no—
170C42—24no—
171C42—84no—
172C42C2×D814yes—
173C42#13718yes—
174C42C22×Q811yescharacteristic
175C42C2×#4111yescharacteristic
176C422−1+412yes—
177C23C2×#12711yescharacteristic
178Q8—48no—
179Q8—84no—
180Q8—164no—
181Q8—164no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—44no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—84no—
192C2×C4—84no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4#49312yes—
196C2×C4#49312yes—
197C2×C4#49714yes—
198C2×C4#49714yes—
199C2×C4C2×C22⋊Q811yescharacteristic
200C2×C4#53611yescharacteristic
201C2×C4#54712yes—
202C2×C4D4×Q812yes—
203C2×C4C22×D812yes—
204C2×C4C2×#13714yes—
205C2×C4C2×#13712yes—
206C2×C4#57714yes—
207C8—24no—
208C8—48no—
209C8#47418yes—
210C22#254211yescharacteristic
211C22C2×#49311yescharacteristic
212C22C2×#49712yes—
213C22#315211yescharacteristic
214C22#315412yes—
215C4—22no—
216C4—24no—
217C4—84no—
218C4—162no—
219C4—162no—
220C4—162no—
221C4#172214yes—
222C4C2×#47412yes—
223C4#322612yes—
224C4#335614yes—
225C4#338714yes—
226C2C2×#172211yescharacteristic
227C2#1966011yescharacteristic
228C2#1968012yes—
229C2#2850811yescharacteristic
230C2#2854212yes—
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
42. #494Q8semidirect

Maximal subgroups

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

subgroupindex
#333102
#292192
#292082
C4×#4942
#106752
#98392
#88202
#82602