Finite Group Databasebeta

← #316833

Subgroups — #316833

Order 512.

600 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 158 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#28283C212yesmaximal
2#22270C211yesmaximal, characteristic
3#22238C211yesmaximal, characteristic
4#10624C212yesmaximal
5#10276C212yesmaximal
6#9220C214yesmaximal
7#8787C211yesmaximal, characteristic
8#8591C212yesmaximal
9C2×#494C2211yescharacteristic
10C2×#475C2214yes—
11C2×C8×Q8C2212yes—
12C4×C4⋊Q8C2212yes—
13C4×C4⋊Q8C2211yescharacteristic
14#2066—24no—
15#1984C2212yes—
16#1981—22no—
17#1956C2212yes—
18#1948—22no—
19#1916—24no—
20#1878—24no—
21#1842C2212yes—
22#1834C2211yescharacteristic
23#1831C2211yescharacteristic
24#1803C2214yes—
25#1744C2211yescharacteristic
26C2×#332C2214yes—
27#1296C2214yes—
28#1277C2212yes—
29#1277C2212yes—
30C22×Q16—22no—
31C2×C4⋊Q8D411yescharacteristic
32C2×C4⋊Q8C2312yes—
33C2×C4×Q8—22no—
34C2×C4×Q8C2312yes—
35#494—24no—
36#475—28no—
37#475—44no—
38C8×Q8—44no—
39C2×#106—24no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8C2314yes—
44C2×C4⋊C8C2312yes—
45C2×#103—24no—
46C2×#103—24no—
47C2×#103—24no—
48C2×C4×C8—22no—
49C2×C4×C8C2311yescharacteristic
50C2×C4×C8C2311yescharacteristic
51#391—22no—
52#391D412yes—
53#389—22no—
54#389—22no—
55#389—22no—
56#384D411yescharacteristic
57#384C2312yes—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C43C2311yescharacteristic
62#340—22no—
63#340—24no—
64#336—24no—
65#336—22no—
66#336D414yes—
67#332—44no—
68C22×Q8—22no—
69C22×Q8C2×D412yes—
70C2×Q16—28no—
71C2×Q16—84no—
72C22×C8—22no—
73C22×C8—24no—
74C22×C8C2×D412yes—
75C22×C8C2×D412yes—
76C4⋊Q8—22no—
77C4⋊Q8—22no—
78C4⋊Q8—22no—
79C4⋊Q8—22no—
80C4⋊Q8Q1614yescomplemented
81C4⋊Q8#4114yes—
82C4×Q8—24no—
83C4×Q8—24no—
84C4×Q8—84no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4C2×D412yes—
95C2×C42—22no—
96C2×C42—42no—
97C2×C42C2×D412yes—
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C42#4112yes—
101C2×C42C2411yescharacteristic
102#106—48no—
103C4⋊C8—24no—
104C4⋊C8—24no—
105C4⋊C8—24no—
106C4⋊C8—44no—
107C4⋊C8—44no—
108C4⋊C8—82no—
109#103—48no—
110#103—84no—
111#103—84no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8—44no—
115#95—24no—
116#95—42no—
117#95—42no—
118C2×Q8—22no—
119C2×Q8—24no—
120C2×Q8—24no—
121C2×Q8—24no—
122C2×Q8—42no—
123C2×Q8—44no—
124C2×Q8—44no—
125C2×Q8C2×Q1614yes—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4#12112yes—
131C22×C4#12112yes—
132C22×C4C22×D411yescharacteristic
133C22×C4C22×D411yescharacteristic
134C22×C4C2×#4111yescharacteristic
135Q16—88no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—24no—
139C2×C8—24no—
140C2×C8—42no—
141C2×C8—44no—
142C2×C8—44no—
143C2×C8—44no—
144C2×C8—44no—
145C4⋊C4—28no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—42no—
150C4⋊C4—44no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4—82no—
156C4⋊C4—82no—
157C4⋊C4—84no—
158C42—22no—
159C42—24no—
160C42—42no—
161C42—42no—
162C42—42no—
163C42—84no—
164C42C2×Q1612yes—
165C42#13512yes—
166C42C2×#4112yes—
167C42H(5,𝔽2)12yes—
168C23C2×#12111yescharacteristic
169Q8—28no—
170Q8—44no—
171Q8—48no—
172Q8—84no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4#46714yes—
191C2×C4#52511yescharacteristic
192C2×C4#53511yescharacteristic
193C2×C4#54612yes—
194C2×C4#55312yes—
195C2×C4C22×Q1611yescharacteristic
196C2×C4C2×#13511yescharacteristic
197C2×C4#57714yes—
198C2×C4#57712yes—
199C8—42no—
200C8—42no—
201C8—82no—
202C8—84no—
203C22#253511yescharacteristic
204C22#304312yes—
205C22C2×#46711yescharacteristic
206C22#306011yescharacteristic
207C22#306912yes—
208C4—22no—
209C4—24no—
210C4—42no—
211C4—44no—
212C4—84no—
213C4—82no—
214C4—162no—
215C4#312312yes—
216C4#322112yes—
217C4#329912yes—
218C4#330412yes—
219C2#1939912yes—
220C2#1952412yes—
221C2#2814211yescharacteristic
222C2#2816611yescharacteristic
223C2#2817911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
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
80. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#282832
#222702
#222382
#106242
#102762
#92202
#87872
#85912