Finite Group Databasebeta

← #318352

Subgroups — #318352

Order 512.

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

normal subgroups only

This group has 686 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#28280C212yesmaximal
2#22260C211yesmaximal, characteristic
3#22248C211yesmaximal, characteristic, complemented
4C2×#1675C212yesmaximal, complemented
5#10481C214yesmaximal, complemented
6#8810C211yesmaximal, characteristic, complemented
7#8648C212yesmaximal, complemented
8#8590C212yesmaximal, complemented
9C2×#500C2211yescharacteristic
10C2×#476C2214yes—
11C2×C8×D4C2212yes—
12C4×C4⋊Q8C2212yes—
13C4×#127C2211yescharacteristic
14#2065—24no—
15#1990C2212yes—
16#1982—22no—
17#1954C2212yes—
18#1948—22no—
19#1921—24no—
20#1869—24no—
21#1842C2212yes—
22#1834C2211yescharacteristic
23#1831C2211yescharacteristic
24#1802C2214yes—
25#1744C2211yescharacteristic
26#1675—28no—
27C2×#335C2212yes—
28C2×#335C2212yes—
29C2×#333C2214yes—
30#1296C2214yes—
31C23×C8—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—
41C8×D4—24no—
42C8×D4—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#335—24no—
71#335—42no—
72#333—28no—
73C22×Q8—22no—
74C22×D4—22no—
75C22×D4C2×D412yes—
76C23×C4—22no—
77C22×C8—22no—
78C22×C8—24no—
79C22×C8—44no—
80C22×C8—84no—
81C22×C8C2×D412yes—
82C22×C8C2×D412yes—
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85C4⋊Q8—22no—
86C4⋊Q8C2×D414yes—
87#127—22no—
88#127Q1614yes—
89C4×Q8—84no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—84no—
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—42no—
102C2×C4⋊C4C2×D412yes—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—24no—
105C2×C42—42no—
106C2×C42C2×D412yes—
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42#4112yes—
110C2×C42C2411yescharacteristic
111#107—28no—
112#106—48no—
113C4⋊C8—22no—
114C4⋊C8—24no—
115C4⋊C8—22no—
116C4⋊C8—24no—
117C4⋊C8—44no—
118C4⋊C8—82no—
119#102—48no—
120#102—84no—
121C22⋊C8—44no—
122C22⋊C8—84no—
123C4×C8—21no—
124C4×C8—21no—
125C4×C8C2×D412yes—
126C4×C8#4112yes—
127#95—42no—
128#95—42no—
129C24—22no—
130C2×Q8—24no—
131C2×Q8—42no—
132C2×Q8—82no—
133C2×Q8—82no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—28no—
140C2×D4—82no—
141C2×D4—82no—
142C2×D4C2×Q1614yes—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4#12112yes—
151C22×C4#12112yes—
152C22×C4C22×D411yescharacteristic
153C22×C4C22×D411yescharacteristic
154C22×C4C2×#4111yescharacteristic
155C2×C8—24no—
156C2×C8—22no—
157C2×C8—24no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—44no—
162C2×C8—44no—
163C2×C8—88no—
164C2×C8#12114yes—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—44no—
173C4⋊C4—42no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C4⋊C4—84no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C22⋊C4—84no—
180C42—21no—
181C42—24no—
182C42—21no—
183C42—42no—
184C42—84no—
185C42C2×Q1612yes—
186C42#13512yes—
187C42C22×D411yescharacteristic
188C42C2×#4111yescharacteristic
189C42H(5,𝔽2)11yescharacteristic
190C422−1+411yescharacteristic
191C23—24no—
192C23—44no—
193C23—42no—
194C23—42no—
195C23—44no—
196C23C2×#12111yescharacteristic
197Q8—84no—
198Q8—84no—
199D4—24no—
200D4—24no—
201D4—42no—
202D4—42no—
203D4—88no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—84no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4#46714yes—
222C2×C4C2×#12111yescharacteristic
223C2×C4#53711yescharacteristic
224C2×C4D4211yescharacteristic
225C2×C4#54711yescharacteristic
226C2×C4#54711yescharacteristic
227C2×C4#55511yescharacteristic
228C2×C4C22×Q1611yescharacteristic
229C2×C4C2×#13511yescharacteristic
230C2×C4#57611yescharacteristic
231C2×C4#57714yes—
232C2×C4#57811yescharacteristic
233C8—24no—
234C8—42no—
235C8—84no—
236C8—82no—
237C22—44no—
238C22—48no—
239C22—84no—
240C22—82no—
241C22—82no—
242C22#253711yescharacteristic
243C22#304212yes—
244C22C2×#46711yescharacteristic
245C22#306111yescharacteristic
246C22#306812yes—
247C4—21no—
248C4—21no—
249C4—24no—
250C4—42no—
251C4—84no—
252C4—82no—
253C4—162no—
254C4#304312yes—
255C4#326012yes—
256C4D4×Q1611yescharacteristic
257C4#328711yescharacteristic
258C4#333811yescharacteristic
259C4#334211yescharacteristic
260C2—84no—
261C2#1939312yes—
262C2#1951912yes—
263C2#2822811yescharacteristic
264C2#2824311yescharacteristic
265C2#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×#1675C2semidirect
5. #10481C2semidirect
6. #8810C2semidirect
7. #8648C2semidirect
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
#222602
#222482
C2×#16752
#104812
#88102
#86482
#85902