Finite Group Databasebeta

← #482989

Subgroups — #482989

Order 512.

708 conjugacy classes of subgroups; only the 276 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 276 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#62360C211yesmaximal, characteristic
2#33496C211yesmaximal, characteristic
3#22740C211yesmaximal, characteristic
4#11857C216yesmaximal
5#11743C216yesmaximal
6#3502—21no—
7#3395C414yescomplemented
8Q8×Q16C414yescomplemented
9#3363—24no—
10#3249—23no—
11#3170C2216yes—
12C2×C8×Q8C2211yescharacteristic
13Q8×C4⋊C4—21no—
14Q8×C16—24no—
15#2162C2216yes—
16#1947—23no—
17#1834C2213yes—
18C4×#107C2213yes—
19#1348C22112yes—
20#1348C2214yes—
21Q82—24no—
22#557—24no—
23#557—44no—
24#544—43no—
25#533—26no—
26#519—41no—
27C2×C4×Q8—41no—
28C2×C4×Q8D411yescharacteristic
29#500C2×C416yes—
30#494C2×C416yes—
31#487—26no—
32#477—212no—
33#467—26no—
34#467C2314yes—
35#462—212no—
36C8×Q8C2×C414yes—
37C8×Q8D414yes—
38C4×Q16—26no—
39C4×Q16C2×C4112yes—
40C2×#107—23no—
41C2×C4⋊C8D413yes—
42C2×C4×C8C2313yes—
43#389—43no—
44#384—23no—
45C4×C4⋊C4—23no—
46C4⋊C16—26no—
47C22⋊C16C2314yes—
48C4×C16—26no—
49#340—23no—
50#340C2313yes—
51#340C2311yescharacteristic
52C22×Q8D811yescharacteristic
53C2×Q16—26no—
54C2×Q16C22×C414yes—
55C22×C8C2×D413yes—
56C22×C8C2411yescharacteristic
57C4⋊Q8—26no—
58C4⋊Q8—26no—
59C4⋊Q8—412no—
60#125—412no—
61#125—46no—
62C22⋊Q8—24no—
63C22⋊Q8—46no—
64C4×Q8—212no—
65C4×Q8—44no—
66C4×Q8—44no—
67C4×Q8—46no—
68C4×Q8—84no—
69C4×Q8C22⋊C414yes—
70C4×Q8SD1614yes—
71#117—46no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—23no—
74C2×C4⋊C4—43no—
75C2×C4⋊C4—43no—
76C2×C4⋊C4D813yes—
77C2×C42—43no—
78C2×C42C2×D413yes—
79C2×C16—24no—
80#107—26no—
81#107—46no—
82#107C22×C414yes—
83C4⋊C8C22⋊C4112yes—
84#103—212no—
85#103—24no—
86C4×C8C22×C416yes—
87C4×C8C2×D416yes—
88#95—43no—
89C2×Q8—24no—
90C2×Q8—46no—
91C2×Q8#10214yes—
92C2×Q8SD3214yes—
93C2×Q8Q3214yes—
94C22×C4—41no—
95C22×C4—43no—
96C22×C4C2×D813yes—
97C22×C4C2×D813yes—
98C22×C4C22×D411yescharacteristic
99Q16—44no—
100Q16—86no—
101C2×C8—43no—
102C2×C8—41no—
103C2×C8C2×C22⋊C416yes—
104C2×C8C23×C411yescharacteristic
105C2×C8C22×D411yescharacteristic
106C4⋊C4—22no—
107C4⋊C4—24no—
108C4⋊C4—41no—
109C4⋊C4—43no—
110C4⋊C4—43no—
111C4⋊C4—44no—
112C4⋊C4—46no—
113C4⋊C4—46no—
114C4⋊C4—412no—
115C4⋊C4—46no—
116C4⋊C4—86no—
117C4⋊C4—163no—
118C4⋊C4#102112yes—
119C22⋊C4—44no—
120C42—412no—
121C42—86no—
122C42C2×C22⋊C416yes—
123C42C2×SD1616yes—
124C16—44no—
125C23C22×D811yescharacteristic
126Q8—44no—
127Q8—44no—
128Q8—166no—
129Q8#358116yes—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—43no—
133C2×C4—42no—
134C2×C4—44no—
135C2×C4—44no—
136C2×C4—81no—
137C2×C4—81no—
138C2×C4—83no—
139C2×C4—83no—
140C2×C4—83no—
141C2×C4C2×#10216yes—
142C2×C4C2×#10216yes—
143C2×C4C2×SD3216yes—
144C2×C4C2×Q3216yes—
145C2×C4C22×C22⋊C411yescharacteristic
146C2×C4C22×SD1611yescharacteristic
147C8—21no—
148C8—21no—
149C8—26no—
150C22—21no—
151C22—21no—
152C22C22×#10211yescharacteristic
153C22C22×SD3211yescharacteristic
154C22C22×Q3211yescharacteristic
155C4—26no—
156C4—21no—
157C4—82no—
158C4—84no—
159C4—84no—
160C4—163no—
161C4C2×#358112yes—
162C4#288411yescharacteristic
163C4#290111yescharacteristic
164C2—21no—
165C2—21no—
166C2#1761811yescharacteristic
167C2C22×#35811yescharacteristic
168C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC23C22×D8
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #3395C4semidirect
8. Q8×Q16C4semidirect

Maximal subgroups

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

subgroupindex
#623602
#334962
#227402
#118572
#117432