Finite Group Databasebeta

← #174064

Subgroups — #174064

Order 512.

801 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 192 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#28717C212yesmaximal
3#28512C211yesmaximal, characteristic
4#7812C218yesmaximal
5#7571C212yesmaximal
6#7571C211yesmaximal, characteristic
7C2×#487—22no—
8C2×#475—22no—
9#2868C2211yescharacteristic
10#2811C2214yes—
11#2567—22no—
12#2567—24no—
13#2551—21no—
14#2551—22no—
15#2369—22no—
16#2369—21no—
17C4×C4⋊Q8C2212yes—
18C4×C4⋊Q8C2211yescharacteristic
19#2096C2214yes—
20#2072C2218yes—
21#1990—22no—
22#1956—21no—
23#1844—21no—
24#1803—24no—
25C4×C4⋊C8C2212yes—
26C4×C4⋊C8C2211yescharacteristic
27#1272C2214yes—
28#1272C2218yes—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—24no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8D412yes—
34C2×C4⋊Q8D411yescharacteristic
35C2×C4×Q8D412yes—
36C2×C4×Q8D414yes—
37#487—48no—
38#475—48no—
39C2×#106—24no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—22no—
42C2×#103—24no—
43C2×#103—28no—
44C2×C4×C8—21no—
45C2×C4×C8—22no—
46#395—24no—
47#395—28no—
48#395—24no—
49#391—28no—
50#391—24no—
51#391C2314yes—
52#391C2312yes—
53#389—24no—
54#389—24no—
55#389—24no—
56#389—22no—
57#389—22no—
58#389—22no—
59#384—24no—
60#384—22no—
61#384—21no—
62#384D411yescharacteristic
63#384D412yes—
64#384C2312yes—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—24no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C43C2311yescharacteristic
71#340—28no—
72#336C2314yes—
73#336C2312yes—
74C22×Q8—24no—
75C22×Q8—41no—
76C22×Q8C2×D414yes—
77C22×Q8C2×D412yes—
78C22×C8—22no—
79C22×C8—24no—
80C4⋊Q8—24no—
81C4⋊Q8—22no—
82C4⋊Q8—44no—
83C4⋊Q8—44no—
84C4⋊Q8—44no—
85C4⋊Q8—84no—
86C4⋊Q8—88no—
87C4⋊Q8SD1618yes—
88C4⋊Q8Q1614yes—
89C4×Q8—48no—
90C4×Q8—44no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—28no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4C2×D414yes—
106C2×C4⋊C4C2×D412yes—
107C2×C42—22no—
108C2×C42—24no—
109C2×C42—24no—
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D412yes—
112C2×C42C2×D412yes—
113C2×C42C2×D411yescharacteristic
114C2×C42C2411yescharacteristic
115#106—48no—
116C4⋊C8—24no—
117C4⋊C8—44no—
118#103—48no—
119#103—416no—
120C4×C8—44no—
121C4×C8—42no—
122#95—24no—
123#95—28no—
124#95—24no—
125#95—24no—
126#95—24no—
127C2×Q8—24no—
128C2×Q8—22no—
129C2×Q8—42no—
130C2×Q8—44no—
131C2×Q8—44no—
132C2×Q8—42no—
133C2×Q8—84no—
134C2×Q8—88no—
135C2×Q8—84no—
136C2×Q8—82no—
137C2×Q8C2×SD1618yes—
138C2×Q8C2×Q1614yes—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—42no—
144C22×C4#12011yescharacteristic
145C22×C4#12011yescharacteristic
146C22×C4#12012yes—
147C22×C4C22×D412yes—
148C22×C4C22×D411yescharacteristic
149C2×C8—24no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—44no—
153C2×C8—44no—
154C2×C8—44no—
155C4⋊C4—28no—
156C4⋊C4—28no—
157C4⋊C4—28no—
158C4⋊C4—42no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—82no—
167C4⋊C4—84no—
168C4⋊C4—82no—
169C4⋊C4—84no—
170C4⋊C4—88no—
171C4⋊C4—84no—
172C42—22no—
173C42—24no—
174C42—41no—
175C42—44no—
176C42—48no—
177C42—84no—
178C42C2×SD1614yes—
179C42C2×Q1612yes—
180C42#13714yes—
181C42#13712yes—
182C23C2×#12011yescharacteristic
183Q8—48no—
184Q8—44no—
185Q8—84no—
186Q8—82no—
187Q8—168no—
188Q8—164no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—28no—
194C2×C4—24no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—84no—
207C2×C4—82no—
208C2×C4#44814yes—
209C2×C4#44814yes—
210C2×C4#45114yes—
211C2×C4#45312yes—
212C2×C4#45514yes—
213C2×C4C22×SD1612yes—
214C2×C4C22×Q1611yescharacteristic
215C2×C4C2×#13712yes—
216C2×C4C2×#13711yescharacteristic
217C8—44no—
218C8—84no—
219C22C2×#44812yes—
220C22C2×#45111yescharacteristic
221C22C2×#45311yescharacteristic
222C22C2×#45512yes—
223C22#302811yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—21no—
228C4—41no—
229C4—48no—
230C4—44no—
231C4—82no—
232C4—84no—
233C4—162no—
234C4#162314yes—
235C4#164518yes—
236C2#1839612yes—
237C2#1839711yescharacteristic
238C2C2×#162311yescharacteristic
239C2C2×#164512yes—
240C2#1907411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#599242
#287172
#285122
#78122
#75712