Finite Group Databasebeta

← #513186

Subgroups — #513186

Order 512.

430 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C2×#2264C212yesmaximal, complemented
2#33851C212yesmaximal, complemented
3#13728C211yesmaximal, characteristic, complemented
4#13699C212yesmaximal, complemented
5#13690C212yesmaximal, complemented
6#13665C212yesmaximal, complemented
7#13663C212yesmaximal, complemented
8C4×Q64C211yesmaximal, characteristic, complemented
9#13646C211yesmaximal, characteristic, complemented
10C2×#508C2212yes—
11C22×Q32C2212yes—
12#3285—22no—
13#2264—28no—
14C2×Q64C2212yes—
15C2×Q64C2211yescharacteristic
16C2×Q64C2211yescharacteristic
17C2×SD64C2212yes—
18C2×SD64C2212yes—
19C2×D32C2211yescharacteristic
20C2×M64C2212yes—
21#2249C2212yes—
22#2216C2212yes—
23#2215C2212yes—
24#2211—22no—
25#2189—22no—
26#2185—22no—
27C4×Q32C2212yes—
28D4×C16C2211yescharacteristic
29#1423C2211yescharacteristic
30C4⋊C32C2211yescharacteristic
31#1416C2212yes—
32#1416C2212yes—
33#1415C2212yes—
34C22⋊C32C2212yes—
35C4×C32C2211yescharacteristic
36C2×#135—22no—
37C22×Q16—22no—
38#548—42no—
39#508—24no—
40#508—42no—
41C2×Q32—24no—
42C2×Q32—22no—
43C2×Q32—42no—
44C2×Q32C2312yes—
45C2×Q32C2312yes—
46C2×SD32—22no—
47C2×SD32—22no—
48C2×D16C2312yes—
49C22×C16C2312yes—
50#495—22no—
51#467—22no—
52#466—22no—
53#464—42no—
54#452—42no—
55#448—42no—
56C4×Q16—22no—
57C8×D4D411yescharacteristic
58Q64—24no—
59Q64—41no—
60Q64D414yescomplemented
61SD64—24no—
62SD64—24no—
63D32—41no—
64M64—24no—
65C2×C32C2311yescharacteristic
66C2×C32C2312yes—
67C2×C32C2311yescharacteristic
68#366C2312yes—
69C4⋊C16D411yescharacteristic
70#358—22no—
71#358—22no—
72#357—22no—
73C22⋊C16D412yes—
74C4×C16C2311yescharacteristic
75C2×#41—42no—
76C22×Q8—42no—
77#135—44no—
78#135—82no—
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—44no—
82C2×Q16—42no—
83C2×Q16—82no—
84C2×SD16—42no—
85C2×SD16—42no—
86C2×D8—22no—
87C22×C8C2×D412yes—
88#124—42no—
89#124—82no—
90C22⋊Q8—42no—
91C22⋊Q8—82no—
92#121—42no—
93#121—82no—
94C4×Q8—42no—
95C4×D4—82no—
96C4×D4D811yescharacteristic
97Q32—22no—
98Q32—22no—
99Q32—22no—
100Q32—44no—
101Q32C2×D414yes—
102SD32—44no—
103SD32—82no—
104D16—22no—
105D16—22no—
106C2×C16—22no—
107C2×C16—22no—
108C2×C16C2×D411yescharacteristic
109C2×C16C2×D412yes—
110C2×C16C2411yescharacteristic
111#107—22no—
112C4⋊C8D811yescharacteristic
113#103—42no—
114#103—42no—
115#102—42no—
116C22⋊C8D812yes—
117C4×C8C2×D411yescharacteristic
118C32—22no—
119C32—21no—
120C32C2×D412yes—
121#41—84no—
122#41—162no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—84no—
126C2×Q8—82no—
127C2×Q8—162no—
128C2×D4—42no—
129C2×D4—82no—
130C2×D4—82no—
131C2×D4D1611yescharacteristic
132C22×C4—82no—
133C22×C4—82no—
134C22×C4C2×D812yes—
135Q16—24no—
136Q16—42no—
137Q16—42no—
138Q16—42no—
139Q16—84no—
140SD16—84no—
141SD16—162no—
142D8—42no—
143D8—42no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8C2×D811yescharacteristic
147C2×C8C2×D812yes—
148C2×C8C22×D411yescharacteristic
149C4⋊C4—42no—
150C4⋊C4—82no—
151C4⋊C4—82no—
152C4⋊C4D1611yescharacteristic
153C22⋊C4—82no—
154C22⋊C4—82no—
155C22⋊C4—82no—
156C22⋊C4—82no—
157C22⋊C4D1612yes—
158C42—82no—
159C42C2×D811yescharacteristic
160C16—22no—
161C16—21no—
162C16C22×D411yescharacteristic
163C16H(5,𝔽2)11yescharacteristic
164C23—82no—
165C23C2×D1612yes—
166Q8—44no—
167Q8—82no—
168Q8—82no—
169Q8—82no—
170Q8—164no—
171D4—41no—
172D4—82no—
173D4—82no—
174D4—164no—
175D4—164no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—164no—
182C2×C4—164no—
183C2×C4C2×D1611yescharacteristic
184C2×C4C2×D1611yescharacteristic
185C2×C4C2×D1612yes—
186C2×C4C22×D811yescharacteristic
187C8—21no—
188C8—22no—
189C8D4211yescharacteristic
190C8#57611yescharacteristic
191C22—22no—
192C22—22no—
193C22—22no—
194C22—162no—
195C22—162no—
196C22—162no—
197C22C22×D1611yescharacteristic
198C4—22no—
199C4—21no—
200C4—21no—
201C4—84no—
202C4—162no—
203C4—162no—
204C4D4×D811yescharacteristic
205C4#341511yescharacteristic
206C2—42no—
207C2—322no—
208C2D4×D1611yescharacteristic
209C2#3398111yescharacteristic
210C2#3398411yescharacteristic
subgroupG/H
center, socleC22C22×D16
commutator, Frattini, cosocleC2×C16C24
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2264C2semidirect
2. #33851C2semidirect
3. #13728C2semidirect
4. #13699C2semidirect
5. #13690C2semidirect
6. #13665C2semidirect
7. #13663C2semidirect
8. C4×Q64C2semidirect
9. #13646C2semidirect
60. Q64D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22642
#338512
#137282
#136992
#136902
#136652
#136632
C4×Q642
#136462