Finite Group Databasebeta

← #173980

Subgroups — #173980

Order 512.

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

normal subgroups only

This group has 885 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#59832C211yesmaximal, characteristic
2#28687C212yesmaximal
3#28512C211yesmaximal, characteristic
4#7817C218yesmaximal
5#7571C211yesmaximal, characteristic
6#7570C212yesmaximal
7C2×Q82—21no—
8C2×#498—22no—
9C2×#475—22no—
10#2868C2211yescharacteristic
11#2734C2214yes—
12#2582—21no—
13#2551—22no—
14#2546—22no—
15#2451—24no—
16#2365—22no—
17C4×C4⋊Q8C2212yes—
18C4×C4⋊Q8C2211yescharacteristic
19#2096C2214yes—
20#2084C2218yes—
21#1990—22no—
22#1956—21no—
23#1948—24no—
24#1844—21no—
25C4×C4⋊C8C2212yes—
26C4×C4⋊C8C2211yescharacteristic
27#1272C2214yes—
28#1271C2218yes—
29Q82—44no—
30Q82—44no—
31C2×C4⋊Q8—24no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8D411yescharacteristic
37C2×C4⋊Q8D412yes—
38C2×C4×Q8—24no—
39C2×C4×Q8—22no—
40C2×C4×Q8D412yes—
41#498—48no—
42#475—48no—
43C2×#106—24no—
44C2×#106—28no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—22no—
47C2×#103—24no—
48C2×C4×C8—21no—
49C2×C4×C8—22no—
50#398—24no—
51#398—24no—
52#395—28no—
53#391—24no—
54#391—24no—
55#391—22no—
56#391C2312yes—
57#389—24no—
58#389—24no—
59#389—22no—
60#389—22no—
61#389C2314yes—
62#387—28no—
63#384—21no—
64#384—22no—
65#384D411yescharacteristic
66#384D412yes—
67#384C2312yes—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4D414yes—
72C43C2311yescharacteristic
73#340—28no—
74#336C2314yes—
75#336C2312yes—
76C22×Q8—24no—
77C22×Q8—22no—
78C22×Q8—41no—
79C22×Q8C2×D412yes—
80C22×C8—24no—
81C22×C8—22no—
82C4⋊Q8—22no—
83C4⋊Q8—24no—
84C4⋊Q8—44no—
85C4⋊Q8—416no—
86C4⋊Q8—48no—
87C4⋊Q8—48no—
88C4⋊Q8—44no—
89C4⋊Q8—44no—
90C4⋊Q8—84no—
91C4⋊Q8SD1614yes—
92C4⋊Q8SD1618yes—
93C4×Q8—44no—
94C4×Q8—416no—
95C4×Q8—44no—
96C4×Q8—44no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—28no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4C2×D414yes—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D414yes—
113C2×C42—24no—
114C2×C42—24no—
115C2×C42—22no—
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D412yes—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D412yes—
120C2×C42C2411yescharacteristic
121#106—48no—
122#106—416no—
123C4⋊C8—28no—
124C4⋊C8—24no—
125#103—48no—
126C4×C8—44no—
127C4×C8—42no—
128#95—24no—
129#95—28no—
130#95—24no—
131#95—24no—
132#95—24no—
133C2×Q8—22no—
134C2×Q8—28no—
135C2×Q8—48no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—44no—
139C2×Q8—44no—
140C2×Q8—84no—
141C2×Q8—84no—
142C2×Q8—84no—
143C2×Q8—82no—
144C2×Q8C2×SD1614yes—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—42no—
150C22×C4#12011yescharacteristic
151C22×C4#12011yescharacteristic
152C22×C4#12012yes—
153C22×C4C22×D411yescharacteristic
154C22×C4C22×D412yes—
155C2×C8—24no—
156C2×C8—28no—
157C2×C8—44no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—42no—
161C4⋊C4—24no—
162C4⋊C4—28no—
163C4⋊C4—28no—
164C4⋊C4—42no—
165C4⋊C4—44no—
166C4⋊C4—416no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—82no—
176C4⋊C4—82no—
177C4⋊C4—84no—
178C4⋊C4—84no—
179C4⋊C4C2×SD1618yes—
180C42—22no—
181C42—24no—
182C42—41no—
183C42—48no—
184C42—48no—
185C42—44no—
186C42C2×SD1614yes—
187C42C2×SD1612yes—
188C42AGL(1,ℤ/8)12yes—
189C42#13714yes—
190C23C2×#12011yescharacteristic
191Q8—44no—
192Q8—416no—
193Q8—82no—
194Q8—168no—
195Q8—164no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—24no—
201C2×C4—28no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—84no—
214C2×C4—82no—
215C2×C4#44814yes—
216C2×C4#44814yes—
217C2×C4#45014yes—
218C2×C4#45312yes—
219C2×C4#45614yes—
220C2×C4C22×SD1611yescharacteristic
221C2×C4C22×SD1612yes—
222C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
223C2×C4C2×#13712yes—
224C8—44no—
225C8—48no—
226C22C2×#45011yescharacteristic
227C22C2×#44812yes—
228C22C2×#45311yescharacteristic
229C22C2×#45612yes—
230C22#302811yescharacteristic
231C4—21no—
232C4—21no—
233C4—22no—
234C4—22no—
235C4—48no—
236C4—41no—
237C4—44no—
238C4—82no—
239C4—84no—
240C4—162no—
241C4#162314yes—
242C4#165118yes—
243C2#1839511yescharacteristic
244C2#1841012yes—
245C2C2×#165112yes—
246C2C2×#162311yescharacteristic
247C2#1907811yescharacteristic
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
#598322
#286872
#285122
#78172
#75712
#75702