Finite Group Databasebeta

← #149634

Subgroups — #149634

Order 512.

327 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 80 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#20806C211yesmaximal, characteristic
2C2×#1327C212yesmaximal
3#7374C214yesmaximal, complemented
4C2×#430—22no—
5C2×#423C411yescharacteristic
6C2×#423C2211yescharacteristic
7C2×C4×M16C411yescharacteristic
8C2×#349—22no—
9#1849C412yes—
10#1849C2212yes—
11#1848—22no—
12#1327—22no—
13#1327C2214yes—
14C22×M16—21no—
15C22×M16—21no—
16C22×M16D412yes—
17C22×C42C2×C411yescharacteristic
18C2×M32—42no—
19#430—44no—
20#430—44no—
21C2×#108—24no—
22C2×#108—24no—
23#423—21no—
24#423—41no—
25#423—41no—
26#423C2×C412yes—
27#423C2×C411yescharacteristic
28#423D412yes—
29#423C2311yescharacteristic
30C2×C4⋊C8—41no—
31C2×C4⋊C8—41no—
32C4×M16—21no—
33C4×M16—21no—
34C4×M16C812yes—
35C4×M16C812yes—
36C2×#97C2×C412yes—
37C2×C4×C8—21no—
38#349—24no—
39#349—42no—
40C23×C4—21no—
41C23×C4C22⋊C411yescharacteristic
42C2×M16—24no—
43C2×M16—24no—
44C2×M16—22no—
45C2×M16—22no—
46C2×M16—22no—
47C2×M16—22no—
48C2×M16—41no—
49C2×M16—42no—
50C2×M16—42no—
51C2×M16—42no—
52C2×M16—42no—
53C2×M16C22⋊C412yes—
54C2×M16C2×D412yes—
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—41no—
58C2×C42—21no—
59C2×C42—21no—
60C2×C42—21no—
61C2×C42C22⋊C411yescharacteristic
62C2×C42C2×C811yescharacteristic
63C2×C42C2×C811yescharacteristic
64C2×C42M1611yescharacteristic
65C2×C42M1611yescharacteristic
66C2×C42C22×C411yescharacteristic
67M32—44no—
68M32—82no—
69C2×C16—82no—
70#108—48no—
71#108—48no—
72C4⋊C8—41no—
73C4⋊C8—42no—
74C4⋊C8—41no—
75C4⋊C8—42no—
76C4⋊C8—42no—
77#97—22no—
78#97C2×C814yes—
79C4×C8—22no—
80C4×C8—22no—
81C24#10111yescharacteristic
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—82no—
91C22×C4C22⋊C811yescharacteristic
92C22×C4C22⋊C811yescharacteristic
93C22×C4#9911yescharacteristic
94C22×C4#9911yescharacteristic
95C22×C4#10111yescharacteristic
96C22×C4C2×C22⋊C411yescharacteristic
97M16—28no—
98M16—28no—
99M16—42no—
100M16—42no—
101M16—44no—
102M16—82no—
103M16—82no—
104C2×C8—24no—
105C2×C8—24no—
106C2×C8—24no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—42no—
110C2×C8—42no—
111C2×C8—42no—
112C2×C8—81no—
113C2×C8—81no—
114C42—21no—
115C42—21no—
116C42—21no—
117C42—21no—
118C42—21no—
119C42—21no—
120C42C22⋊C811yescharacteristic
121C42C22⋊C811yescharacteristic
122C42C22×C811yescharacteristic
123C42C2×M1611yescharacteristic
124C16—84no—
125C23—22no—
126C23—81no—
127C23#32411yescharacteristic
128C23#32411yescharacteristic
129C23C2×#10111yescharacteristic
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4—81no—
147C2×C4#32411yescharacteristic
148C2×C4#32411yescharacteristic
149C2×C4#35312yes—
150C2×C4#35612yes—
151C2×C4C2×C22⋊C811yescharacteristic
152C2×C4C2×#9911yescharacteristic
153C8—28no—
154C8—42no—
155C8—42no—
156C8—82no—
157C22—21no—
158C22—21no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22—82no—
163C22#132512yes—
164C22C2×#32411yescharacteristic
165C4—22no—
166C4—21no—
167C4—21no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4#132512yes—
172C4C2×#35311yescharacteristic
173C4C2×#35611yescharacteristic
174C2—21no—
175C2—22no—
176C2—41no—
177C2—41no—
178C2C2×#132511yescharacteristic
subgroupG/H
centerC4C2×#353
commutatorC42C22×C8
Frattini, cosocle#423C23
socleC2C2×#1325
prime radicalC24#101
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#423 ◁ G
LECSG ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7374C2semidirect

Maximal subgroups

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

subgroupindex
#208062
C2×#13272
#73742