Finite Group Databasebeta

← #318410

Subgroups — #318410

Order 512.

664 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 162 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#28280C212yesmaximal
2#22264C211yesmaximal, characteristic
3#22248C211yesmaximal, characteristic, complemented
4C2×#1695C212yesmaximal, complemented
5#10480C214yesmaximal, complemented
6#8809C211yesmaximal, characteristic, complemented
7#8656C212yesmaximal, complemented
8#8590C212yesmaximal, complemented
9C2×#500C2211yescharacteristic
10C2×#476C2214yes—
11C2×#436C2212yes—
12C4×C4⋊Q8C2212yes—
13C4×#127C2211yescharacteristic
14#2065—24no—
15#1990C2212yes—
16#1982—22no—
17#1954C2212yes—
18#1948—22no—
19#1921—24no—
20#1870—24no—
21#1842C2212yes—
22#1834C2211yescharacteristic
23#1831C2211yescharacteristic
24#1802C2214yes—
25#1744C2211yescharacteristic
26#1695—28no—
27C2×#334C2212yes—
28C2×#334C2212yes—
29C2×#333C2214yes—
30#1296C2214yes—
31C22×M16—22no—
32C2×C4⋊Q8C2312yes—
33C2×#127D411yescharacteristic
34C2×C4×Q8—22no—
35C2×C4×D4—22no—
36C2×C4×D4C2312yes—
37#500—22no—
38#500D414yescomplemented
39#476—28no—
40#476—44no—
41#436—24no—
42#436—24no—
43C2×#107—22no—
44C2×#106—24no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8C2312yes—
48C2×C4⋊C8C2314yes—
49C2×#102—24no—
50C2×#102—24no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—24no—
53C2×C4×C8C2311yescharacteristic
54C2×C4×C8C2311yescharacteristic
55#391—22no—
56#390—22no—
57#390D412yes—
58#389—22no—
59#389—22no—
60#384D411yescharacteristic
61#384C2312yes—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C43C2311yescharacteristic
65#340—22no—
66#340—24no—
67#340—24no—
68#336—22no—
69#336D414yes—
70#334—24no—
71#334—42no—
72#333—28no—
73C22×Q8—22no—
74C22×D4—22no—
75C22×D4C2×D412yes—
76C23×C4—22no—
77C2×M16—44no—
78C2×M16—84no—
79C22×C8—22no—
80C22×C8—24no—
81C22×C8C2×D412yes—
82C22×C8C2×D412yes—
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85C4⋊Q8—22no—
86C4⋊Q8C2×D414yes—
87#127—24no—
88C4×Q8—84no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—84no—
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×C22⋊C4—22no—
103C2×C22⋊C4—24no—
104C2×C42—42no—
105C2×C42C2×D412yes—
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42#4112yes—
109C2×C42C2411yescharacteristic
110#107—28no—
111#106—48no—
112C4⋊C8—22no—
113C4⋊C8—24no—
114C4⋊C8—22no—
115C4⋊C8—24no—
116C4⋊C8—44no—
117C4⋊C8—82no—
118#102—48no—
119#102—84no—
120C22⋊C8—44no—
121C22⋊C8—84no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8C2×D412yes—
125C4×C8#4112yes—
126#95—42no—
127#95—42no—
128C24—22no—
129C2×Q8—24no—
130C2×Q8—42no—
131C2×Q8—82no—
132C2×Q8—82no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—44no—
138C2×D4—82no—
139C2×D4—82no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4#12112yes—
148C22×C4#12112yes—
149C22×C4C22×D411yescharacteristic
150C22×C4C22×D411yescharacteristic
151C22×C4C2×#4111yescharacteristic
152M16—88no—
153C2×C8—24no—
154C2×C8—22no—
155C2×C8—24no—
156C2×C8—42no—
157C2×C8—44no—
158C2×C8—44no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8#12114yes—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—44no—
170C4⋊C4—42no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—84no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—84no—
177C42—22no—
178C42—24no—
179C42—42no—
180C42—84no—
181C42#13714yes—
182C42C22×D411yescharacteristic
183C42C2×#4111yescharacteristic
184C42H(5,𝔽2)11yescharacteristic
185C422−1+411yescharacteristic
186C23—44no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—44no—
191C23C2×#12111yescharacteristic
192Q8—84no—
193Q8—84no—
194D4—44no—
195D4—44no—
196D4—88no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—84no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4#47014yes—
215C2×C4C2×#12111yescharacteristic
216C2×C4#53711yescharacteristic
217C2×C4D4211yescharacteristic
218C2×C4#54711yescharacteristic
219C2×C4#54711yescharacteristic
220C2×C4#55511yescharacteristic
221C2×C4C2×#13712yes—
222C2×C4#57712yes—
223C2×C4#57714yes—
224C8—24no—
225C8—42no—
226C8—84no—
227C8—82no—
228C22—84no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22—84no—
233C22#253711yescharacteristic
234C22#304212yes—
235C22C2×#47011yescharacteristic
236C22#306011yescharacteristic
237C22#306812yes—
238C4—21no—
239C4—21no—
240C4—24no—
241C4—42no—
242C4—84no—
243C4—82no—
244C4—162no—
245C4#304312yes—
246C4#326012yes—
247C4#327812yes—
248C4#335612yes—
249C2—162no—
250C2#1942912yes—
251C2#1951612yes—
252C2#2823211yescharacteristic
253C2#2824311yescharacteristic
254C2#2826011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
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
3. #22248C2semidirect
4. C2×#1695C2semidirect
5. #10480C2semidirect
6. #8809C2semidirect
7. #8656C2semidirect
8. #8590C2semidirect
38. #500D4semidirect

Maximal subgroups

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

subgroupindex
#282802
#222642
#222482
C2×#16952
#104802
#88092
#86562
#85902