Finite Group Databasebeta

← #174062

Subgroups — #174062

Order 512.

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

normal subgroups only

This group has 823 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#29141C212yesmaximal
3#28440C211yesmaximal, characteristic
4#7759C218yesmaximal
5#7571C211yesmaximal, characteristic
6#7571C212yesmaximal
7C2×#494—21no—
8C2×#462—22no—
9#2868C2211yescharacteristic
10#2811C2214yes—
11#2567—24no—
12#2567—22no—
13#2551—22no—
14#2551—21no—
15#2369—21no—
16#2369—22no—
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2212yes—
19#2024C2218yes—
20#2016C2214yes—
21#1956—21no—
22#1956—22no—
23#1834—22no—
24#1803—24no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2212yes—
27#1272C2218yes—
28#1272C2214yes—
29C22×Q16—24no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—24no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×C4⋊Q8D412yes—
36C2×C4⋊Q8C2312yes—
37C2×C4×Q8D414yes—
38C2×C4×Q8D412yes—
39#494—44no—
40#462—48no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—22no—
43C2×#103—24no—
44C2×#103—28no—
45C2×C4×C8—22no—
46C2×C4×C8—21no—
47#395—24no—
48#395—28no—
49#395—24no—
50#391—24no—
51#391—28no—
52#391C2312yes—
53#391C2314yes—
54#389—24no—
55#389—24no—
56#389—24no—
57#389—22no—
58#389—22no—
59#389—22no—
60#384—24no—
61#384—22no—
62#384—22no—
63#384D412yes—
64#384D411yescharacteristic
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#336C2312yes—
73#336C2314yes—
74C22×Q8—24no—
75C22×Q8—22no—
76C22×Q8C2×D412yes—
77C22×Q8C2×D414yes—
78C2×Q16—48no—
79C2×Q16—48no—
80C2×Q16—84no—
81C22×C8—22no—
82C22×C8—24no—
83C4⋊Q8—22no—
84C4⋊Q8—24no—
85C4⋊Q8—44no—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C4⋊Q8—82no—
89C4⋊Q8—84no—
90C4⋊Q8—88no—
91C4⋊Q8SD1614yes—
92C4⋊Q8Q1618yescomplemented
93C4×Q8—48no—
94C4×Q8—44no—
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—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—28no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4C2×D414yes—
110C2×C4⋊C4C2×D412yes—
111C2×C42—22no—
112C2×C42—24no—
113C2×C42—24no—
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D412yes—
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D412yes—
118C2×C42C2411yescharacteristic
119C4⋊C8—24no—
120C4⋊C8—44no—
121#103—48no—
122#103—88no—
123C4×C8—42no—
124C4×C8—44no—
125#95—28no—
126#95—24no—
127#95—24no—
128#95—24no—
129#95—24no—
130C2×Q8—22no—
131C2×Q8—24no—
132C2×Q8—44no—
133C2×Q8—42no—
134C2×Q8—44no—
135C2×Q8—42no—
136C2×Q8—44no—
137C2×Q8—44no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—84no—
141C2×Q8—88no—
142C2×Q8C2×SD1614yes—
143C2×Q8C2×Q1618yes—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—42no—
149C22×C4#12011yescharacteristic
150C22×C4#12012yes—
151C22×C4#12011yescharacteristic
152C22×C4C22×D411yescharacteristic
153C22×C4C22×D412yes—
154Q16—816no—
155C2×C8—24no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—44no—
159C2×C8—44no—
160C2×C8—44no—
161C4⋊C4—28no—
162C4⋊C4—28no—
163C4⋊C4—28no—
164C4⋊C4—42no—
165C4⋊C4—44no—
166C4⋊C4—48no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—82no—
171C4⋊C4—84no—
172C4⋊C4—82no—
173C4⋊C4—84no—
174C4⋊C4—88no—
175C4⋊C4—84no—
176C4⋊C4—82no—
177C42—22no—
178C42—24no—
179C42—41no—
180C42—44no—
181C42—48no—
182C42—84no—
183C42C2×SD1612yes—
184C42C2×Q1614yes—
185C42#13712yes—
186C42#13714yes—
187C23C2×#12011yescharacteristic
188Q8—44no—
189Q8—48no—
190Q8—82no—
191Q8—84no—
192Q8—84no—
193Q8—84no—
194Q8—168no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—28no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4#44814yes—
216C2×C4#45114yes—
217C2×C4#45114yes—
218C2×C4#45312yes—
219C2×C4#45514yes—
220C2×C4C22×SD1611yescharacteristic
221C2×C4C22×Q1612yes—
222C2×C4C2×#13711yescharacteristic
223C2×C4C2×#13712yes—
224C8—44no—
225C8—84no—
226C22C2×#44811yescharacteristic
227C22C2×#45112yes—
228C22C2×#45311yescharacteristic
229C22C2×#45512yes—
230C22#302711yescharacteristic
231C4—21no—
232C4—21no—
233C4—22no—
234C4—22no—
235C4—41no—
236C4—44no—
237C4—48no—
238C4—82no—
239C4—84no—
240C4—162no—
241C4#163314yes—
242C4#164518yes—
243C2#1838911yescharacteristic
244C2#1839012yes—
245C2C2×#163311yescharacteristic
246C2C2×#164512yes—
247C2#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

Complements

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

NH ≅ G/Ntype
92. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#599242
#291412
#284402
#77592
#75712