Finite Group Databasebeta

← #94237

Subgroups — #94237

Order 512.

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

normal subgroups only

This group has 205 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#13191C211yesmaximal, characteristic
2#12809C211yesmaximal, characteristic
3#12807C211yesmaximal, characteristic
4#2931C2211yescharacteristic
5#2150C411yescharacteristic, complemented
6#2132—21no—
7#2126—21no—
8C2×#350—21no—
9C2×#350—21no—
10C2×C4×C16C411yescharacteristic, complemented
11#1813—21no—
12#1812—21no—
13#1392C412yescomplemented
14#1391C412yes—
15C2×#131D411yescharacteristic
16C2×#131D411yescharacteristic
17#532D411yescharacteristic
18C2×M32—21no—
19C22×C16—21no—
20#436—41no—
21C8×D4—41no—
22C2×#108—41no—
23#427—41no—
24#423Q811yescharacteristic
25#413—22no—
26#410—22no—
27#408C2×C411yescharacteristic
28#408C2×C411yescharacteristic
29C2×C4×C8C2×C411yescharacteristic
30C4⋊C16—41no—
31#356—41no—
32#355—41no—
33#354—41no—
34#353—41no—
35#350—42no—
36#350—42no—
37C4×C16—41no—
38#338—41no—
39H(3,ℤ/4)—41no—
40C2×#41C22⋊C411yescharacteristic
41C2×#41C22⋊C411yescharacteristic
42#131—41no—
43#131—41no—
44#131—41no—
45#131—41no—
46C2×M16—21no—
47C2×M16—21no—
48C2×M16—42no—
49C2×M16C4⋊C411yescharacteristic
50C2×M16C4⋊C411yescharacteristic
51C22×C8—21no—
52C22×C8—21no—
53C22×C8C4211yescharacteristic
54C22×C8C22⋊C411yescharacteristic
55C4⋊Q8SD1611yescharacteristic
56#127SD1611yescharacteristic
57#124D811yescharacteristic
58#124Q1611yescharacteristic
59#121—41no—
60C4×D4—41no—
61#117—42no—
62C2×C42C4⋊C411yescharacteristic
63M32—41no—
64M32—41no—
65C2×C16—22no—
66C2×C16—41no—
67C2×C16—41no—
68C2×C16—41no—
69#108—81no—
70#108—81no—
71#107—81no—
72#106—81no—
73C4⋊C8—41no—
74#101—42no—
75#100—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#10211yescharacteristic
87C2×Q8#10311yescharacteristic
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—41no—
91C2×D4#10211yescharacteristic
92C2×D4#10311yescharacteristic
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4#9511yescharacteristic
97M16—41no—
98M16—41no—
99M16—81no—
100M16—81no—
101M16—82no—
102M16—82no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—82no—
114C2×C8#9911yescharacteristic
115C2×C8#9911yescharacteristic
116C4⋊C4—41no—
117C4⋊C4—82no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—82no—
121C42—21no—
122C42—82no—
123C42#10611yescharacteristic
124C42#10711yescharacteristic
125C16—42no—
126C16—81no—
127C23—21no—
128C23—21no—
129C23H(3,ℤ/4)11yescharacteristic
130Q8—41no—
131Q8—41no—
132D4—41no—
133D4—41no—
134D4—82no—
135D4—81no—
136D4—81no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—162no—
149C2×C4#34011yescharacteristic
150C2×C4#34211yescharacteristic
151C8—21no—
152C8—41no—
153C8—41no—
154C8—41no—
155C8—162no—
156C8#35311yescharacteristic
157C8#35411yescharacteristic
158C22—21no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22—81no—
163C22—81no—
164C22#128911yescharacteristic
165C4—22no—
166C4—21no—
167C4—41no—
168C4—41no—
169C4—41no—
170C4—322no—
171C4#139111yescharacteristic
172C4#139311yescharacteristic
173C2—21no—
174C2—41no—
175C2—81no—
176C2—81no—
177C2#723711yescharacteristic
subgroupG/H
centerC4#1391
commutatorC22×C8C42
Frattini, cosocle#2931C22
socleC2#7237
prime radical#532D4
LCSG ▷ C22×C8 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C8 ◁ 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
5. #2150C4semidirect
10. C2×C4×C16C4semidirect
13. #1392C4semidirect

Maximal subgroups

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

subgroupindex
#131912
#128092
#128072