Finite Group Databasebeta

← #19065

Subgroups — #19065

Order 256.

460 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 111 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#3548C211yesmaximal, characteristic
2#3216C212yesmaximal, complemented
3#3162C211yesmaximal, characteristic, complemented
4#1661C218yesmaximal, complemented
5#1537C211yesmaximal, characteristic, complemented
6#1530C212yesmaximal, complemented
7#562C2214yes—
8#554—22no—
9#550—22no—
10#547—22no—
11#540—22no—
12#537—24no—
13#533C2211yescharacteristic
14#533C2211yescharacteristic
15#532C2212yes—
16#501—22no—
17#501C2214yes—
18#497C2214yes—
19#489C2214yes—
20#462—22no—
21#460—22no—
22#452—24no—
23#432C2212yes—
24#427—22no—
25C2×#97C2211yescharacteristic
26#330C2214yes—
27#329C2218yes—
28C2×#41—24no—
29C2×#41D414yes—
30C2×Q16—24no—
31C2×SD16—24no—
32C22×C8—21no—
33C4⋊Q8D412yes—
34C4⋊Q8D412yes—
35C4⋊Q8C2312yes—
36#127—22no—
37#127D412yes—
38#125—22no—
39#125—21no—
40#125C2312yes—
41#124C2314yes—
42#123—28no—
43C22⋊Q8—24no—
44C22⋊Q8—28no—
45C22⋊Q8—22no—
46C22⋊Q8—22no—
47#121—28no—
48#121—24no—
49#121—28no—
50#121—22no—
51#121—22no—
52C4×Q8—22no—
53C4×Q8—22no—
54C4×D4—24no—
55C4×D4—22no—
56C4×D4—22no—
57#117—21no—
58#117D412yes—
59C2×C4⋊C4—24no—
60C2×C42C2311yescharacteristic
61#107—24no—
62#106—24no—
63C4⋊C8—22no—
64#103—24no—
65#102—24no—
66C22⋊C8—22no—
67#97—21no—
68#97C2312yes—
69#97C2314yes—
70#41—84no—
71#41—88no—
72C2×Q8—22no—
73C2×Q8—22no—
74C2×Q8C2×D414yes—
75C2×D4—28no—
76C2×D4—24no—
77C2×D4—22no—
78C2×D4—22no—
79C2×D4C2×D414yes—
80C22×C4—28no—
81C22×C4—24no—
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D412yes—
84Q16—84no—
85SD16—84no—
86C2×C8—24no—
87C2×C8—22no—
88C2×C8—24no—
89C4⋊C4—28no—
90C4⋊C4—22no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4—24no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—42no—
97C4⋊C4C2×D414yes—
98C22⋊C4—28no—
99C22⋊C4—22no—
100C22⋊C4—24no—
101C22⋊C4—24no—
102C22⋊C4—41no—
103C42—22no—
104C42—41no—
105C42C2×D411yescharacteristic
106C42C2×D412yes—
107C42C2411yescharacteristic
108C23—24no—
109C23#12011yescharacteristic
110Q8—44no—
111Q8—44no—
112Q8—42no—
113Q8—42no—
114D4—44no—
115D4—44no—
116D4—44no—
117D4—84no—
118D4—88no—
119C2×C4—22no—
120C2×C4—24no—
121C2×C4—24no—
122C2×C4—42no—
123C2×C4—48no—
124C2×C4—42no—
125C2×C4—88no—
126C2×C4#12012yes—
127C2×C4#12011yescharacteristic
128C2×C4C22×D411yescharacteristic
129C2×C4C22×D412yes—
130C8—24no—
131C8—44no—
132C22—21no—
133C22—22no—
134C22—44no—
135C22—44no—
136C22—44no—
137C22C2×#12011yescharacteristic
138C4—21no—
139C4—22no—
140C4—22no—
141C4—44no—
142C4—44no—
143C4—82no—
144C4#45612yes—
145C2—41no—
146C2—84no—
147C2C2×#45611yescharacteristic
148C2#301912yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3548C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3216C2semidirect
3. #3162C2semidirect
4. #1661C2semidirect
5. #1537C2semidirect
6. #1530C2semidirect

Maximal subgroups

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

subgroupindex
#35482
#32162
#31622
#16612
#15372
#15302