Finite Group Databasebeta

← #325462

Subgroups — #325462

Order 512.

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

normal subgroups only

This group has 597 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#33289C211yesmaximal, characteristic, complemented
2#28726C212yesmaximal
3#21631C211yesmaximal, characteristic
4#10624C212yesmaximal, complemented
5#10622C212yesmaximal
6#9905C212yesmaximal, complemented
7#8657C212yesmaximal, complemented
8#8656C212yesmaximal
9#8273C211yesmaximal, characteristic, complemented
10C2×#489—22no—
11C2×#446C2211yescharacteristic
12C2×#436C2211yescharacteristic
13#2868C2212yes—
14C4×#124C2211yescharacteristic
15#2072—22no—
16#2071—22no—
17#1993—22no—
18#1990—22no—
19#1990—22no—
20#1984C2211yescharacteristic
21#1954C2211yescharacteristic
22#1923—22no—
23#1834C2212yes—
24#1834C2212yes—
25#1831C2211yescharacteristic
26#1803C2214yes—
27#1802C2214yes—
28#1767C2212yes—
29#1753—22no—
30C42×C8C2211yescharacteristic
31C2×#334C2212yes—
32#1296C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35C22×M16—21no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—22no—
39C2×#124D411yescharacteristic
40C2×C4×Q8C2311yescharacteristic
41C2×C4×D4C2311yescharacteristic
42#489—48no—
43#446—24no—
44#436—24no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8C2314yes—
47C2×C4⋊C8C2311yescharacteristic
48C2×C4⋊C8C2311yescharacteristic
49C2×#103—24no—
50C2×#102—24no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—22no—
53C2×#97—21no—
54C2×#97—22no—
55C2×C4×C8D412yes—
56C2×C4×C8D412yes—
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2311yescharacteristic
59#391D411yescharacteristic
60#390D411yescharacteristic
61#388—22no—
62#384—22no—
63#384—22no—
64#384—22no—
65#384C2314yes—
66#382D411yescharacteristic
67C4×C22⋊C4—22no—
68C43C2311yescharacteristic
69#340—24no—
70#340—22no—
71#340—22no—
72#340—24no—
73#340—24no—
74#336—22no—
75#336—22no—
76#336—22no—
77#334—24no—
78C22×Q8—42no—
79C22×Q8C2×D411yescharacteristic
80C22×D4C2×D411yescharacteristic
81C23×C4—21no—
82C2×M16—44no—
83C2×M16—42no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—24no—
87C22×C8C2×D412yes—
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C4⋊Q8—44no—
91C4⋊Q8—44no—
92C4⋊Q8—44no—
93C4⋊Q8—44no—
94C4⋊Q8—84no—
95#124—24no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×D4—22no—
99C4×D4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4C2×D411yescharacteristic
108C2×C4⋊C4C2×D411yescharacteristic
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—21no—
112C2×C42—21no—
113C2×C42C2×D412yes—
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42#4112yes—
117C2×C42C2411yescharacteristic
118C4⋊C8—24no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—24no—
122C4⋊C8—42no—
123#103—48no—
124#102—48no—
125C22⋊C8—44no—
126C22⋊C8—42no—
127#97—42no—
128#97—44no—
129C4×C8—22no—
130C4×C8—42no—
131C4×C8—42no—
132C4×C8#4114yes—
133#95—22no—
134#95—22no—
135C24—21no—
136C2×Q8—22no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8—84no—
140C2×Q8—84no—
141C2×Q8—84no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4#12112yes—
154C22×C4#12112yes—
155C22×C4C22×D411yescharacteristic
156C22×C4C22×D411yescharacteristic
157C22×C4C2×#4111yescharacteristic
158M16—84no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—44no—
167C2×C8—44no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—44no—
171C2×C8#12412yes—
172C2×C8#12412yes—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—84no—
181C4⋊C4—84no—
182C4⋊C4—84no—
183C4⋊C4—84no—
184C22⋊C4—42no—
185C22⋊C4—44no—
186C22⋊C4—44no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—24no—
191C42—41no—
192C42—41no—
193C42AGL(1,ℤ/8)14yes—
194C42C2×#4112yes—
195C422−1+412yes—
196C23—42no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—42no—
201C23C2×#12111yescharacteristic
202Q8—44no—
203Q8—168no—
204D4—44no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—82no—
220C2×C4—84no—
221C2×C4—84no—
222C2×C4—84no—
223C2×C4#46914yes—
224C2×C4#47114yes—
225C2×C4C2×#12411yescharacteristic
226C2×C4#53611yescharacteristic
227C2×C4#54712yes—
228C2×C4#55612yes—
229C2×C4C2×AGL(1,ℤ/8)12yes—
230C2×C4#57512yes—
231C2×C4#57512yes—
232C2×C4#57812yes—
233C8—22no—
234C8—22no—
235C8—42no—
236C8—84no—
237C8—81no—
238C8—81no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—82no—
243C22—82no—
244C22#254711yescharacteristic
245C22#304012yes—
246C22C2×#46911yescharacteristic
247C22#305911yescharacteristic
248C22C2×#47112yes—
249C4—22no—
250C4—24no—
251C4—24no—
252C4—81no—
253C4—81no—
254C4—81no—
255C4—164no—
256C4#313511yescharacteristic
257C4#314011yescharacteristic
258C4#324012yes—
259C4#335314yes—
260C2—161no—
261C2#1930112yes—
262C2#1936512yes—
263C2#2871211yescharacteristic
264C2#2872711yescharacteristic
265C2#2873211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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
1. #33289C2semidirect
4. #10624C2semidirect
6. #9905C2semidirect
7. #8657C2semidirect
9. #8273C2semidirect

Maximal subgroups

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

subgroupindex
#332892
#287262
#216312
#106242
#106222
#99052
#86572
#86562
#82732