Finite Group Databasebeta

← #325370

Subgroups — #325370

Order 512.

548 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 170 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#33256C211yesmaximal, characteristic
2#29150C211yesmaximal, characteristic
3#28725C211yesmaximal, characteristic
4C8×C4⋊Q8C211yesmaximal, characteristic
5#10626C214yesmaximal
6#9904C211yesmaximal, characteristic
7#9902C211yesmaximal, characteristic
8#8643C214yesmaximal
9#8273C211yesmaximal, characteristic
10C2×#491—21no—
11C2×C8×Q8C2212yes—
12#2859C2211yescharacteristic
13#2859C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2074—22no—
16#2026—22no—
17#1992—21no—
18#1984C2212yes—
19#1960—21no—
20#1941—22no—
21#1941—22no—
22#1939—22no—
23#1844C2212yes—
24#1842C2211yescharacteristic
25#1836—21no—
26#1834C2212yes—
27#1803C2214yes—
28#1803C2214yes—
29C8×C4⋊C4—22no—
30#1767C2211yescharacteristic
31#1767C2211yescharacteristic
32C42×C8C2211yescharacteristic
33C2×#334C2212yes—
34#1296C2214yes—
35#1296C2214yes—
36#1277C2212yes—
37C2×C4⋊Q8D411yescharacteristic
38C2×#125—21no—
39C2×#125—21no—
40C2×C4×Q8C2312yes—
41#491—44no—
42C8×Q8—28no—
43C2×#106—22no—
44C2×#106—22no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8C2312yes—
48C2×C4⋊C8C2312yes—
49C2×C4⋊C8C2312yes—
50C2×#103—24no—
51C2×#103—24no—
52C2×#97—21no—
53C2×#97—21no—
54C2×C4×C8—22no—
55C2×C4×C8D412yes—
56C2×C4×C8D412yes—
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2311yescharacteristic
59#391D412yes—
60#389—22no—
61#389—22no—
62#389—22no—
63#389—22no—
64#389—22no—
65#384D411yescharacteristic
66#384C2312yes—
67#384C2312yes—
68#382—21no—
69#382—21no—
70C4×C4⋊C4—22no—
71C43C2311yescharacteristic
72#340—24no—
73#340—24no—
74#340—24no—
75#336—24no—
76#336—21no—
77#336—21no—
78#334—24no—
79C22×Q8C2×D412yes—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8C2×D412yes—
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C4⋊Q8—22no—
87C4⋊Q8SD1614yes—
88#125—42no—
89#125—42no—
90#125—82no—
91C4×Q8—24no—
92C4×Q8—24no—
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⋊C4C2×D412yes—
105C2×C42—22no—
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D412yes—
108C2×C42C2×D411yescharacteristic
109C2×C42#4112yes—
110C2×C42C2411yescharacteristic
111#106—44no—
112#106—44no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—24no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—44no—
119C4⋊C8—44no—
120#103—48no—
121#103—84no—
122#97—42no—
123#97—42no—
124C4×C8—24no—
125C4×C8—22no—
126C4×C8—42no—
127C4×C8—44no—
128C4×C8#4114yes—
129#95—24no—
130#95—42no—
131#95—42no—
132C2×Q8—22no—
133C2×Q8—24no—
134C2×Q8—24no—
135C2×Q8C2×SD1614yes—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4#12112yes—
141C22×C4#12112yes—
142C22×C4C22×D411yescharacteristic
143C22×C4C22×D411yescharacteristic
144C22×C4C2×#4111yescharacteristic
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—44no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8#12412yes—
158C2×C8#12412yes—
159C2×C8#13714yes—
160C4⋊C4—28no—
161C4⋊C4—24no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—44no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—84no—
174C4⋊C4—84no—
175C42—22no—
176C42—22no—
177C42—22no—
178C42—42no—
179C42—42no—
180C42—42no—
181C42C2×SD1612yes—
182C42#13512yes—
183C42C2×#4112yes—
184C422−1+412yes—
185C23C2×#12111yescharacteristic
186Q8—28no—
187Q8—44no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4#46514yes—
206C2×C4C2×#12411yescharacteristic
207C2×C4#53611yescharacteristic
208C2×C4#54712yes—
209C2×C4#55612yes—
210C2×C4C22×SD1611yescharacteristic
211C2×C4C2×#13511yescharacteristic
212C2×C4C2×#13712yes—
213C2×C4#57512yes—
214C2×C4#57712yes—
215C8—22no—
216C8—42no—
217C8—82no—
218C8—82no—
219C8—82no—
220C8#49114yescomplemented
221C22#254711yescharacteristic
222C22#303712yes—
223C22C2×#46511yescharacteristic
224C22#305711yescharacteristic
225C22#306712yes—
226C4—24no—
227C4—22no—
228C4—44no—
229C4—42no—
230C4—82no—
231C4—162no—
232C4—162no—
233C4C2×#49111yescharacteristic
234C4#314211yescharacteristic
235C4#323712yes—
236C4#333512yes—
237C4#334012yes—
238C2#1926012yes—
239C2#1934012yes—
240C2#2870611yescharacteristic
241C2#2872511yescharacteristic
242C2#2873311yescharacteristic
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

Complements

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

NH ≅ G/Ntype
220. C8#491semidirect

Maximal subgroups

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

subgroupindex
#332562
#291502
#287252
C8×C4⋊Q82
#106262
#99042
#99022
#86432
#82732