Finite Group Databasebeta

← #303034

Subgroups — #303034

Order 512.

739 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 739 conjugacy classes of subgroups — too many to draw. Only its 180 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#59924C211yesmaximal, characteristic
2#29319C212yesmaximal
3#29144C211yesmaximal, characteristic
4#10575C214yesmaximal
5#9869C212yesmaximal
6#8927C214yesmaximal
7#8232C211yesmaximal, characteristic
8#2859C2211yescharacteristic
9#2811—22no—
10#2567—24no—
11#2567—22no—
12#2551—22no—
13#2551C2212yes—
14#2369—22no—
15#2369C2212yes—
16C4×C4⋊Q8C2212yes—
17C4×C4⋊Q8C2211yescharacteristic
18#2084—22no—
19#2057—22no—
20#2026—22no—
21#1956C2212yes—
22#1948C2214yes—
23#1947C2214yes—
24#1943—24no—
25#1834C2212yes—
26#1805—22no—
27#1767C2212yes—
28C4×C4⋊C8C2211yescharacteristic
29C2×#333C2212yes—
30#1296C2212yes—
31#1279C2218yes—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—24no—
34C2×C4⋊Q8D412yes—
35C2×C4⋊Q8C2311yescharacteristic
36C2×#125—21no—
37C2×C4×Q8—22no—
38C2×C4×Q8D412yes—
39C2×#107—24no—
40C2×#106—24no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8C2314yes—
43C2×#103—24no—
44C2×#97—22no—
45C2×C4×C8C2312yes—
46#395—24no—
47#395—28no—
48#395—24no—
49#391—28no—
50#391—24no—
51#391—22no—
52#391D412yes—
53#389—24no—
54#389—24no—
55#389—24no—
56#389—22no—
57#389—22no—
58#389—22no—
59#389—22no—
60#389D414yes—
61#384—24no—
62#384D412yes—
63#384C2312yes—
64#384C2311yescharacteristic
65#382—21no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2314yes—
71C43C2311yescharacteristic
72#340—24no—
73#340—24no—
74#340—24no—
75#336—22no—
76#336—21no—
77#333Q818yescomplemented
78C22×Q8—24no—
79C22×Q8—22no—
80C22×Q8C2×D412yes—
81C22×C8—24no—
82C22×C8—22no—
83C4⋊Q8—28no—
84C4⋊Q8—22no—
85C4⋊Q8—84no—
86C4⋊Q8—84no—
87C4⋊Q8—84no—
88C4⋊Q8C2×Q814yes—
89#125—82no—
90C4×Q8—44no—
91C4×Q8—82no—
92C4×Q8—82no—
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—24no—
101C2×C4⋊C4—28no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4C2×D414yes—
107C2×C4⋊C4C2×D414yes—
108C2×C4⋊C4C2×D412yes—
109C2×C42—22no—
110C2×C42—24no—
111C2×C42—24no—
112C2×C42C2×D412yes—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2411yescharacteristic
117#107—48no—
118#106—84no—
119C4⋊C8—24no—
120C4⋊C8—42no—
121C4⋊C8C2×Q818yes—
122#103—84no—
123#97—82no—
124C4×C8—24no—
125#95—28no—
126#95—24no—
127#95—24no—
128#95—24no—
129#95—24no—
130#95—41no—
131#95—41no—
132C2×Q8—28no—
133C2×Q8—22no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—82no—
137C2×Q8—82no—
138C2×Q8—88no—
139C2×Q8—84no—
140C2×Q8C4⋊Q814yes—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—42no—
146C22×C4#12012yes—
147C22×C4#12011yescharacteristic
148C22×C4#12011yescharacteristic
149C22×C4C22×D411yescharacteristic
150C22×C4C22×D412yes—
151C2×C8—28no—
152C2×C8—44no—
153C2×C8—44no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C4⋊C4—28no—
158C4⋊C4—28no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—42no—
163C4⋊C4—44no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—82no—
171C4⋊C4—84no—
172C4⋊C4—82no—
173C4⋊C4—84no—
174C4⋊C4—88no—
175C4⋊C4—84no—
176C4⋊C4#13718yes—
177C42—24no—
178C42—22no—
179C42—22no—
180C42—22no—
181C42—41no—
182C42—44no—
183C42—44no—
184C42—44no—
185C42—84no—
186C42C22×Q812yes—
187C422−1+412yes—
188C23C2×#12011yescharacteristic
189Q8—44no—
190Q8—82no—
191Q8—88no—
192Q8—168no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—28no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4#45512yes—
216C2×C4C2×C4⋊Q811yescharacteristic
217C2×C4#53611yescharacteristic
218C2×C4D4×Q814yes—
219C2×C4C2×#13714yes—
220C2×C4#57612yes—
221C2×C4#57712yes—
222C8—48no—
223C8—82no—
224C22#243411yescharacteristic
225C22#300412yes—
226C22#301212yes—
227C22C2×#45511yescharacteristic
228C22#302111yescharacteristic
229C4—22no—
230C4—21no—
231C4—21no—
232C4—41no—
233C4—42no—
234C4—48no—
235C4—44no—
236C4—84no—
237C4—82no—
238C4—162no—
239C4#338514yes—
240C4#338714yes—
241C2#1855111yescharacteristic
242C2#1869111yescharacteristic
243C2#1883912yes—
244C2#2582311yescharacteristic
245C2#2584712yes—
subgroupG/H
center, socle, prime radicalC23C2×#120
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
77. #333Q8semidirect

Maximal subgroups

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

subgroupindex
#599242
#293192
#291442
#105752
#98692
#89272
#82322