Finite Group Databasebeta

← #318358

Subgroups — #318358

Order 512.

630 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 170 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
4C2×#1670C212yesmaximal
5#10487C214yesmaximal
6#8810C211yesmaximal, characteristic
7#8649C212yesmaximal
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
26#1670—28no—
27C2×#335C2212yes—
28C2×#335C2212yes—
29C2×#332C2214yes—
30#1296C2214yes—
31C22×Q16—22no—
32C2×C4⋊Q8D411yescharacteristic
33C2×C4⋊Q8C2312yes—
34C2×C4×Q8—22no—
35C2×C4×Q8C2312yes—
36#494—22no—
37#494D414yes—
38#475—28no—
39#475—44no—
40C8×Q8—24no—
41C8×Q8—24no—
42C2×#106—24no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8C2314yes—
47C2×C4⋊C8C2312yes—
48C2×#103—24no—
49C2×#103—24no—
50C2×#103—24no—
51C2×C4×C8—22no—
52C2×C4×C8C2311yescharacteristic
53C2×C4×C8C2311yescharacteristic
54#391—22no—
55#391D412yes—
56#389—22no—
57#389—22no—
58#389—22no—
59#384D411yescharacteristic
60#384C2312yes—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C43C2311yescharacteristic
65#340—22no—
66#340—24no—
67#336—24no—
68#336—22no—
69#336D414yes—
70#335—24no—
71#335—42no—
72#332—28no—
73C22×Q8—22no—
74C22×Q8C2×D412yes—
75C2×Q16—28no—
76C2×Q16—84no—
77C22×C8—22no—
78C22×C8—24no—
79C22×C8C2×D412yes—
80C22×C8C2×D412yes—
81C4⋊Q8—22no—
82C4⋊Q8—22no—
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85C4⋊Q8Q1614yescomplemented
86C4⋊Q8C2×D414yes—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—84no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4C2×D412yes—
102C2×C42—22no—
103C2×C42—42no—
104C2×C42C2×D412yes—
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42#4112yes—
108C2×C42C2411yescharacteristic
109#106—48no—
110C4⋊C8—22no—
111C4⋊C8—24no—
112C4⋊C8—22no—
113C4⋊C8—24no—
114C4⋊C8—44no—
115C4⋊C8—44no—
116C4⋊C8—82no—
117#103—48no—
118#103—84no—
119#103—84no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8—44no—
123C4×C8C2×D412yes—
124C4×C8#4112yes—
125#95—24no—
126#95—42no—
127#95—42no—
128C2×Q8—22no—
129C2×Q8—24no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—42no—
135C2×Q8—44no—
136C2×Q8—44no—
137C2×Q8C2×Q1614yes—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4#12112yes—
143C22×C4#12112yes—
144C22×C4C22×D411yescharacteristic
145C22×C4C22×D411yescharacteristic
146C22×C4C2×#4111yescharacteristic
147Q16—88no—
148C2×C8—24no—
149C2×C8—22no—
150C2×C8—24no—
151C2×C8—42no—
152C2×C8—44no—
153C2×C8—44no—
154C2×C8—44no—
155C2×C8—44no—
156C2×C8#12114yes—
157C4⋊C4—28no—
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—24no—
161C4⋊C4—22no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—44no—
168C4⋊C4—42no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C4⋊C4—84no—
172C42—21no—
173C42—24no—
174C42—21no—
175C42—42no—
176C42—42no—
177C42—42no—
178C42—84no—
179C42C2×Q1612yes—
180C42#13512yes—
181C42C22×D411yescharacteristic
182C42C2×#4111yescharacteristic
183C42H(5,𝔽2)11yescharacteristic
184C422−1+411yescharacteristic
185C23C2×#12111yescharacteristic
186Q8—24no—
187Q8—24no—
188Q8—42no—
189Q8—42no—
190Q8—48no—
191Q8—84no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4#46714yes—
210C2×C4C2×#12111yescharacteristic
211C2×C4#53711yescharacteristic
212C2×C4D4211yescharacteristic
213C2×C4#54711yescharacteristic
214C2×C4#54711yescharacteristic
215C2×C4#55511yescharacteristic
216C2×C4C22×Q1611yescharacteristic
217C2×C4C2×#13511yescharacteristic
218C2×C4#57611yescharacteristic
219C2×C4#57714yes—
220C2×C4#57811yescharacteristic
221C8—24no—
222C8—42no—
223C8—84no—
224C8—82no—
225C22#253711yescharacteristic
226C22#304312yes—
227C22C2×#46711yescharacteristic
228C22#306111yescharacteristic
229C22#306912yes—
230C4—21no—
231C4—21no—
232C4—24no—
233C4—44no—
234C4—42no—
235C4—82no—
236C4—84no—
237C4—162no—
238C4#304212yes—
239C4#326012yes—
240C4D4×Q1611yescharacteristic
241C4#328711yescharacteristic
242C4#333811yescharacteristic
243C4#334211yescharacteristic
244C2#1939912yes—
245C2#1952712yes—
246C2#2822811yescharacteristic
247C2#2824411yescharacteristic
248C2#2826011yescharacteristic
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
85. 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
C2×#16702
#104872
#88102
#86492
#85912