Finite Group Databasebeta

← #94233

Subgroups — #94233

Order 512.

207 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 48 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#12982C211yesmaximal, characteristic
2#12807C211yesmaximal, characteristic
3#8636C211yesmaximal, characteristic
4#3407C411yescharacteristic, complemented
5#2931C2211yescharacteristic
6#2126—21no—
7#2117—21no—
8#2115C411yescharacteristic, complemented
9#1812—21no—
10#1566—21no—
11#1560—21no—
12#1391C412yes—
13#1376—21no—
14#1309C412yescomplemented
15C2×#131C2×C411yescharacteristic
16C2×#131D411yescharacteristic
17#532D411yescharacteristic
18#504—42no—
19C2×M32—21no—
20C2×M32—21no—
21C2×M32—41no—
22#436—41no—
23C8×D4—41no—
24#427—41no—
25#423—22no—
26#423C2×C411yescharacteristic
27#410—22no—
28#408C2×C411yescharacteristic
29#408Q811yescharacteristic
30C2×C4×C8D411yescharacteristic
31#354—41no—
32#353—41no—
33#350—42no—
34#349—41no—
35H(3,ℤ/4)—41no—
36#334—41no—
37#334—41no—
38#333—41no—
39#331—41no—
40C2×#41C22⋊C411yescharacteristic
41C2×#41C4⋊C411yescharacteristic
42#131—41no—
43#131—41no—
44#131—41no—
45#131—41no—
46C2×M16—21no—
47C2×M16—21no—
48C2×M16—42no—
49C2×M16C4211yescharacteristic
50C2×M16C4⋊C411yescharacteristic
51C22×C8—21no—
52C22×C8—21no—
53C22×C8C22⋊C411yescharacteristic
54C22×C8C4⋊C411yescharacteristic
55C4⋊Q8SD1611yescharacteristic
56#127D811yescharacteristic
57#124SD1611yescharacteristic
58#124Q1611yescharacteristic
59#121—41no—
60C4×D4—41no—
61#117—42no—
62C2×C42C22⋊C411yescharacteristic
63M32—22no—
64M32—41no—
65M32—42no—
66M32—41no—
67M32—41no—
68C2×C16—41no—
69C2×C16—42no—
70C2×C16—41no—
71#107—81no—
72#106—81no—
73C4⋊C8—41no—
74C4⋊C8—42no—
75C4⋊C8—42no—
76#99—42no—
77#99—42no—
78C22⋊C8—41no—
79C22⋊C8—41no—
80C4×C8—21no—
81C4×C8—21no—
82#41—41no—
83#41—41no—
84#41—41no—
85#41—41no—
86C2×Q8#10311yescharacteristic
87C2×Q8#10611yescharacteristic
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—41no—
91C2×D4#10211yescharacteristic
92C2×D4#10711yescharacteristic
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4#9511yescharacteristic
97M16—42no—
98M16—41no—
99M16—41no—
100M16—81no—
101M16—162no—
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—41no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—82no—
113C2×C8#9911yescharacteristic
114C2×C8#9911yescharacteristic
115C4⋊C4—41no—
116C4⋊C4—82no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—82no—
120C42—21no—
121C42—82no—
122C42#10211yescharacteristic
123C42#10311yescharacteristic
124C16—42no—
125C16—81no—
126C23—21no—
127C23—21no—
128C23H(3,ℤ/4)11yescharacteristic
129Q8—41no—
130Q8—41no—
131D4—41no—
132D4—41no—
133D4—42no—
134D4—82no—
135D4—81no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—42no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—162no—
148C2×C4#34011yescharacteristic
149C2×C4#34211yescharacteristic
150C8—21no—
151C8—41no—
152C8—41no—
153C8—41no—
154C8—162no—
155C8AΣL(1,𝔽16)11yescharacteristic
156C8#35211yescharacteristic
157C22—21no—
158C22—41no—
159C22—41no—
160C22—41no—
161C22—81no—
162C22—81no—
163C22#128911yescharacteristic
164C4—22no—
165C4—21no—
166C4—41no—
167C4—41no—
168C4—41no—
169C4—322no—
170C4#139011yescharacteristic
171C4#139511yescharacteristic
172C2—21no—
173C2—41no—
174C2—81no—
175C2—81no—
176C2#723611yescharacteristic
subgroupG/H
center, socleC2#7236
commutatorC2×M16C42
Frattini, cosocle#2931C22
prime radical#532D4
LCSG ▷ C2×M16 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G
LECSG ▷ #2931 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #3407C4semidirect
8. #2115C4semidirect
14. #1309C4semidirect

Maximal subgroups

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

subgroupindex
#129822
#128072
#86362