Finite Group Databasebeta

← #149639

Subgroups — #149639

Order 512.

243 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 72 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#21995C211yesmaximal, characteristic
2#11456C212yesmaximal
3#7374C212yesmaximal, complemented
4#7368C212yesmaximal, complemented
5#2921C411yescharacteristic
6C2×#433C411yescharacteristic
7#2905C2211yescharacteristic
8C2×#349—22no—
9#1950—22no—
10#1855—22no—
11#1849C2211yescharacteristic
12#1848C411yescharacteristic
13#1838C411yescharacteristic
14#1837C2211yescharacteristic
15#1327—22no—
16#1327C2214yes—
17C22×M16D412yes—
18C2×#117C2×C411yescharacteristic
19C2×M32—42no—
20#433—22no—
21#433—42no—
22#433C814yes—
23C2×#108—22no—
24C2×#108—22no—
25#424—42no—
26#423C2×C411yescharacteristic
27#423C2×C411yescharacteristic
28#423C2×C411yescharacteristic
29#423D412yes—
30#423C2311yescharacteristic
31C2×C4⋊C8—21no—
32#407—21no—
33C2×C22⋊C8—21no—
34C4×M16—21no—
35C2×#97C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37#349—24no—
38#349—42no—
39#341—22no—
40#341—22no—
41C23×C4C22⋊C411yescharacteristic
42C2×M16—22no—
43C2×M16—21no—
44C2×M16—21no—
45C2×M16—82no—
46C2×M16C22⋊C412yes—
47C2×M16C2×D412yes—
48C22×C8—22no—
49C22×C8—21no—
50C22×C8—21no—
51#117—22no—
52#117C2×C812yes—
53#117M1612yes—
54C2×C4⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C42C22⋊C411yescharacteristic
57C2×C42C22×C411yescharacteristic
58M32—44no—
59M32—82no—
60C2×C16—82no—
61#108—82no—
62#108—82no—
63C4⋊C8—24no—
64C4⋊C8—41no—
65C4⋊C8—42no—
66C4⋊C8—41no—
67C4⋊C8—41no—
68C4⋊C8—41no—
69C22⋊C8—24no—
70C22⋊C8—42no—
71#97—21no—
72#97—41no—
73#97C2×C812yes—
74C4×C8—21no—
75C4×C8—41no—
76C4×C8C2×C812yes—
77C24#10111yescharacteristic
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4C22⋊C812yes—
83C22×C4#9912yes—
84C22×C4#10111yescharacteristic
85C22×C4C2×C22⋊C411yescharacteristic
86M16—42no—
87M16—42no—
88M16—42no—
89M16—42no—
90M16—82no—
91C2×C8—22no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—41no—
96C2×C8—41no—
97C2×C8—42no—
98C2×C8—42no—
99C2×C8—41no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8—41no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C42—21no—
108C42—21no—
109C42C22⋊C812yes—
110C42C22×C811yescharacteristic
111C42C2×M1611yescharacteristic
112C16—84no—
113C23—41no—
114C23—81no—
115C23#32412yes—
116C23C2×#10111yescharacteristic
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—81no—
127C2×C4—81no—
128C2×C4#32412yes—
129C2×C4C2×C22⋊C811yescharacteristic
130C2×C4C2×#9911yescharacteristic
131C8—42no—
132C8—42no—
133C8—42no—
134C8—42no—
135C8—81no—
136C8—81no—
137C22—22no—
138C22—21no—
139C22—41no—
140C22—42no—
141C22—161no—
142C22C2×#32411yescharacteristic
143C4—21no—
144C4—22no—
145C4—42no—
146C4—41no—
147C4—41no—
148C4#212711yescharacteristic
149C4#213311yescharacteristic
150C2—21no—
151C2—42no—
152C2—41no—
153C2#1144811yescharacteristic
subgroupG/H
centerC4#2127
commutatorC42C22×C8
Frattini, cosocle#423C23
socleC2#11448
prime radicalC24#101
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C23×C4 ◁ #2905 ◁ G
LECSG ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7374C2semidirect
4. #7368C2semidirect

Maximal subgroups

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

subgroupindex
#219952
#114562
#73742
#73682