Finite Group Databasebeta

← #173938

Subgroups — #173938

Order 512.

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

normal subgroups only

This group has 907 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#29208C212yesmaximal
3#28440C211yesmaximal, characteristic
4#7757C218yesmaximal
5#7571C211yesmaximal, characteristic
6#7570C212yesmaximal
7C2×Q82—21no—
8C2×#494—21no—
9C2×#462—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#2057C2218yes—
20#2016C2214yes—
21#1990—22no—
22#1956—21no—
23#1947—24no—
24#1845—22no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2212yes—
27#1272C2214yes—
28#1271C2218yes—
29C22×Q16—24no—
30Q82—44no—
31Q82—44no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8—24no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8D411yescharacteristic
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8C2312yes—
40C2×C4×Q8—24no—
41C2×C4×Q8—22no—
42C2×C4×Q8D412yes—
43#494—44no—
44#462—48no—
45C2×#107—28no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8—21no—
48C2×#103—24no—
49C2×C4×C8—22no—
50C2×C4×C8—21no—
51#398—24no—
52#398—24no—
53#395—28no—
54#391—24no—
55#391—24no—
56#391—22no—
57#391C2312yes—
58#389—24no—
59#389—24no—
60#389—22no—
61#389—22no—
62#389C2314yes—
63#387—28no—
64#384—22no—
65#384—22no—
66#384D411yescharacteristic
67#384D412yes—
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—22no—
79C22×Q8C2×D412yes—
80C2×Q16—48no—
81C2×Q16—48no—
82C2×Q16—84no—
83C22×C8—22no—
84C22×C8—24no—
85C4⋊Q8—22no—
86C4⋊Q8—24no—
87C4⋊Q8—44no—
88C4⋊Q8—416no—
89C4⋊Q8—48no—
90C4⋊Q8—48no—
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93C4⋊Q8—82no—
94C4⋊Q8—84no—
95C4⋊Q8D814yes—
96C4⋊Q8Q1618yescomplemented
97C4×Q8—44no—
98C4×Q8—416no—
99C4×Q8—44no—
100C4×Q8—44no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—28no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4C2×D414yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D414yes—
117C2×C42—24no—
118C2×C42—24no—
119C2×C42—22no—
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D412yes—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D412yes—
124C2×C42C2411yescharacteristic
125#107—88no—
126C4⋊C8—28no—
127C4⋊C8—24no—
128#103—48no—
129C4×C8—42no—
130C4×C8—44no—
131#95—24no—
132#95—28no—
133#95—24no—
134#95—24no—
135#95—24no—
136C2×Q8—22no—
137C2×Q8—28no—
138C2×Q8—48no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—44no—
142C2×Q8—44no—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—84no—
148C2×Q8—84no—
149C2×Q8C2×D814yes—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—42no—
155C22×C4#12011yescharacteristic
156C22×C4#12012yes—
157C22×C4#12011yescharacteristic
158C22×C4C22×D411yescharacteristic
159C22×C4C22×D412yes—
160Q16—816no—
161C2×C8—24no—
162C2×C8—28no—
163C2×C8—44no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—44no—
167C4⋊C4—24no—
168C4⋊C4—28no—
169C4⋊C4—28no—
170C4⋊C4—42no—
171C4⋊C4—44no—
172C4⋊C4—416no—
173C4⋊C4—48no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4—84no—
182C4⋊C4—84no—
183C4⋊C4—82no—
184C4⋊C4C2×Q1618yes—
185C42—22no—
186C42—24no—
187C42—41no—
188C42—48no—
189C42—48no—
190C42—44no—
191C42C2×D812yes—
192C42C2×Q1614yes—
193C42AGL(1,ℤ/8)12yes—
194C42#13714yes—
195C23C2×#12011yescharacteristic
196Q8—44no—
197Q8—416no—
198Q8—82no—
199Q8—84no—
200Q8—84no—
201Q8—168no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—28no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4#44714yes—
223C2×C4#45114yes—
224C2×C4#45114yes—
225C2×C4#45312yes—
226C2×C4#45614yes—
227C2×C4C22×D811yescharacteristic
228C2×C4C22×Q1612yes—
229C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
230C2×C4C2×#13712yes—
231C8—48no—
232C8—44no—
233C22C2×#44711yescharacteristic
234C22C2×#45112yes—
235C22C2×#45311yescharacteristic
236C22C2×#45612yes—
237C22#302711yescharacteristic
238C4—21no—
239C4—21no—
240C4—22no—
241C4—22no—
242C4—48no—
243C4—41no—
244C4—44no—
245C4—82no—
246C4—84no—
247C4—162no—
248C4#163314yes—
249C4#164918yes—
250C2#1838711yescharacteristic
251C2#1840412yes—
252C2C2×#164912yes—
253C2C2×#163311yescharacteristic
254C2#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

Complements

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

NH ≅ G/Ntype
96. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#598322
#292082
#284402
#77572
#75712
#75702