Finite Group Databasebeta

← #317130

Subgroups — #317130

Order 512.

568 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 126 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#22261C211yesmaximal, characteristic
2#22258C211yesmaximal, characteristic, complemented
3#19105C212yesmaximal, complemented
4#10630C212yesmaximal, complemented
5#10015C212yesmaximal
6#9050C214yesmaximal, complemented
7#8782C211yesmaximal, characteristic, complemented
8#8660C212yesmaximal, complemented
9#3160C2211yescharacteristic
10#3101C2214yes—
11#2961C2212yes—
12C2×C8×D4C2212yes—
13#2397C2211yescharacteristic
14#1995C2212yes—
15#1955C2212yes—
16#1922—24no—
17#1869—24no—
18#1837C2211yescharacteristic
19#1832C2211yescharacteristic
20#1746C2211yescharacteristic
21#1675—22no—
22#1673—22no—
23#1671—24no—
24#1564C2212yes—
25#1542C2214yes—
26#1486C2214yes—
27#1301C2214yes—
28#1281C2212yes—
29#1274C2212yes—
30C23×C8—22no—
31#533C2312yes—
32C2×C4×D4—22no—
33C2×C4×D4C2312yes—
34#500D412yescomplemented
35#499—22no—
36#498D412yescomplemented
37#483—24no—
38#476—24no—
39#461—24no—
40#450—24no—
41#446—44no—
42C4×SD16—44no—
43C8×D4—22no—
44C8×D4—22no—
45C8×D4—22no—
46C8×D4—22no—
47#431C2314yes—
48#427—22no—
49#424—22no—
50#423C2312yes—
51C2×C4⋊C8C2312yes—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—24no—
54#405—22no—
55C2×#97C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#399—22no—
58#392—22no—
59#376C2311yescharacteristic
60#341—22no—
61#339—24no—
62#336—22no—
63#336D414yes—
64#335—22no—
65#334—22no—
66#333—24no—
67#332—24no—
68D4⋊C8—44no—
69C22×D4—22no—
70C22×D4#4112yes—
71C23×C4—22no—
72C2×SD16—44no—
73C2×M16—22no—
74C22×C8—22no—
75C22×C8—42no—
76C22×C8—42no—
77C22×C8—84no—
78C22×C8C2×D412yes—
79C22×C8C2×D412yes—
80C4⋊Q8C2×D414yes—
81#125—22no—
82C22⋊Q8—24no—
83C4×Q8—24no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—84no—
89#117—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4#4112yes—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—24no—
94C2×C42C2×D412yes—
95C2×C42C2411yescharacteristic
96#107—24no—
97#106—24no—
98#106—44no—
99C4⋊C8—22no—
100C4⋊C8—24no—
101C4⋊C8—22no—
102C4⋊C8—24no—
103C4⋊C8—44no—
104#103—44no—
105#102—44no—
106C22⋊C8—24no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109C22⋊C8—84no—
110#97—22no—
111#97—42no—
112C4×C8—24no—
113C4×C8—42no—
114C4×C8C2×D411yescharacteristic
115C4×C8C2×D411yescharacteristic
116C4×C8#4112yes—
117#95—22no—
118C24—22no—
119C2×Q8—24no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—44no—
123C2×D4—48no—
124C2×D4—82no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4C22×D411yescharacteristic
132C22×C4C2×#4112yes—
133SD16—88no—
134M16—44no—
135C2×C8—24no—
136C2×C8—22no—
137C2×C8—24no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—44no—
141C2×C8—44no—
142C2×C8—84no—
143C2×C8—84no—
144C2×C8#12112yes—
145C2×C8#12112yes—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—44no—
151C4⋊C4—82no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—84no—
156C42—24no—
157C42—42no—
158C42C22×D411yescharacteristic
159C42C2×#4111yescharacteristic
160C42H(5,𝔽2)12yes—
161C23—24no—
162C23—44no—
163C23—42no—
164C23—44no—
165C23—42no—
166C23#52511yescharacteristic
167Q8—84no—
168D4—24no—
169D4—24no—
170D4—42no—
171D4—42no—
172D4—88no—
173C2×C4—42no—
174C2×C4—44no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—84no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4C2×#12111yescharacteristic
183C2×C4#53511yescharacteristic
184C2×C4#54612yes—
185C2×C4#55312yes—
186C2×C4#57714yes—
187C8—22no—
188C8—22no—
189C8—42no—
190C8—84no—
191C8—82no—
192C8—82no—
193C22—22no—
194C22—48no—
195C22—44no—
196C22—84no—
197C22—82no—
198C22—82no—
199C22#253511yescharacteristic
200C22#312312yes—
201C4—22no—
202C4—22no—
203C4—82no—
204C4—82no—
205C4—84no—
206C4#304312yes—
207C4#322112yes—
208C2—22no—
209C2—84no—
210C2#2816611yescharacteristic
211C2#2818012yes—
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2535
prime radical#2397C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22258C2semidirect
3. #19105C2semidirect
4. #10630C2semidirect
6. #9050C2semidirect
7. #8782C2semidirect
8. #8660C2semidirect
34. #500D4semidirect
36. #498D4semidirect

Maximal subgroups

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

subgroupindex
#222612
#222582
#191052
#106302
#100152
#90502
#87822
#86602