Finite Group Databasebeta

← #149704

Subgroups — #149704

Order 512.

285 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 70 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#21657C211yesmaximal, characteristic
2#11478C212yesmaximal
3#7367C212yesmaximal
4#7358C212yesmaximal
5C2×#491C411yescharacteristic
6C4×C4⋊Q8C2211yescharacteristic
7C4×#125C411yescharacteristic
8#1992C2211yescharacteristic
9#1960C411yescharacteristic
10#1933—22no—
11#1836C2211yescharacteristic
12#1763—22no—
13C4×#97C411yescharacteristic
14#1461—22no—
15#1325C2214yes—
16#1319—22no—
17C2×C4⋊Q8C2×C411yescharacteristic
18C2×#125C2×C411yescharacteristic
19C2×C4×Q8D412yes—
20#491—22no—
21#491C814yescomplemented
22#431—42no—
23C2×#103—22no—
24C2×#103—22no—
25C2×#97—21no—
26C2×#97C2×C411yescharacteristic
27C2×#97C2×C411yescharacteristic
28C2×C4×C8—21no—
29#391D412yes—
30#389—21no—
31#389—41no—
32#387—41no—
33#384C2311yescharacteristic
34#382C2×C411yescharacteristic
35C4×C4⋊C4—21no—
36C4×C4⋊C4—41no—
37C4×C4⋊C4—41no—
38C43C2×C411yescharacteristic
39#340—22no—
40#340—22no—
41#324—24no—
42#324—42no—
43C22×Q8C22⋊C412yes—
44C22×C8—21no—
45C22×C8—21no—
46C4⋊Q8—41no—
47C4⋊Q8C2×C812yes—
48C4⋊Q8M1612yes—
49#125—21no—
50#125—41no—
51#125C2×C812yes—
52C4×Q8—42no—
53C4×Q8—42no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4C2×D412yes—
62C2×C42—22no—
63C2×C42—21no—
64C2×C42—21no—
65C2×C42—41no—
66C2×C42C22⋊C411yescharacteristic
67C2×C42C22⋊C411yescharacteristic
68C2×C42C22×C411yescharacteristic
69C4⋊C8—82no—
70#103—28no—
71#103—44no—
72C22⋊C8—44no—
73#97—21no—
74#97—41no—
75#97—42no—
76#97C2×C812yes—
77C4×C8—42no—
78C4×C8—82no—
79#95—42no—
80#95—41no—
81#95—41no—
82#95—41no—
83C2×Q8—22no—
84C2×Q8—42no—
85C2×Q8—42no—
86C2×Q8C22⋊C814yes—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4#9911yescharacteristic
94C22×C4#9911yescharacteristic
95C22×C4C2×C22⋊C411yescharacteristic
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8—84no—
103C4⋊C4—22no—
104C4⋊C4—42no—
105C4⋊C4—42no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C4⋊C4—41no—
111C4⋊C4—82no—
112C4⋊C4—82no—
113C4⋊C4—82no—
114C42—22no—
115C42—21no—
116C42—41no—
117C42—42no—
118C42—42no—
119C42—42no—
120C42—81no—
121C42—82no—
122C42C22×C811yescharacteristic
123C42C2×M1611yescharacteristic
124C23C2×#9911yescharacteristic
125Q8—44no—
126Q8—82no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—22no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—24no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4—82no—
147C2×C4—81no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4#35412yes—
151C2×C4C2×C22⋊C811yescharacteristic
152C2×C4#41311yescharacteristic
153C8—42no—
154C8—42no—
155C8—164no—
156C22—21no—
157C22—21no—
158C22#145911yescharacteristic
159C22C2×#35411yescharacteristic
160C22#213211yescharacteristic
161C4—22no—
162C4—21no—
163C4—21no—
164C4—41no—
165C4—41no—
166C4—42no—
167C4—41no—
168C4—44no—
169C4—82no—
170C4—82no—
171C4—161no—
172C2—21no—
173C2—21no—
174C2#1147711yescharacteristic
175C2#1359611yescharacteristic
176C2#1359911yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC42C22×C8
Frattini, cosocle#384C23
prime radicalC23C2×#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #491C8semidirect

Maximal subgroups

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

subgroupindex
#216572
#114782
#73672
#73582