Finite Group Databasebeta

← #9373

Subgroups — #9373

Order 256.

210 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 59 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#3216C211yesmaximal, characteristic
2#2981C211yesmaximal, characteristic
3#1551C211yesmaximal, characteristic
4#1537C211yesmaximal, characteristic, complemented
5#1481C211yesmaximal, characteristic, complemented
6#1290C211yesmaximal, characteristic, complemented
7#1287C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10#497C412yes—
11#489C412yes—
12#462—21no—
13#460—21no—
14#452—22no—
15#436—21no—
16#435—22no—
17C8×D4—21no—
18#432C2211yescharacteristic
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#431C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23#408—22no—
24#333—21no—
25#331—21no—
26#330—22no—
27Q8⋊C8—21no—
28Q8⋊C8—21no—
29C2×#41D412yes—
30C2×Q16—22no—
31C2×SD16—22no—
32C2×M16—22no—
33C22×C8—22no—
34C22×C8—21no—
35C4⋊Q8C2×C411yescharacteristic
36C4⋊Q8C2×C411yescharacteristic
37C4⋊Q8D411yescharacteristic
38#127D411yescharacteristic
39#125—21no—
40#124C2×C412yes—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43#121—21no—
44#121—21no—
45C4×Q8—21no—
46C4×Q8—21no—
47C4×D4—21no—
48C4×D4—21no—
49#117—21no—
50C2×C42C2311yescharacteristic
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57#103—22no—
58#102—22no—
59C22⋊C8—22no—
60C22⋊C8—22no—
61C22⋊C8—21no—
62C22⋊C8—21no—
63C22⋊C8—21no—
64C22⋊C8—21no—
65#97—21no—
66#97C2×C412yes—
67#97D412yes—
68C4×C8D411yescharacteristic
69C4×C8D411yescharacteristic
70#41—82no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8C22⋊C412yes—
74C2×D4—22no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4C22⋊C412yes—
78C22×C4—22no—
79C22×C4C2×D411yescharacteristic
80C22×C4#4111yescharacteristic
81C22×C4#4111yescharacteristic
82Q16—82no—
83SD16—82no—
84M16—82no—
85C2×C8—22no—
86C2×C8—21no—
87C2×C8—21no—
88C2×C8—22no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8—82no—
92C2×C8—81no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C22⋊C4—41no—
104C42—41no—
105C42C22×C411yescharacteristic
106C42C2×D411yescharacteristic
107C42C2×D411yescharacteristic
108C42C2×D411yescharacteristic
109C23—22no—
110C23#12311yescharacteristic
111Q8—42no—
112Q8—41no—
113Q8—41no—
114Q8—42no—
115D4—42no—
116D4—41no—
117D4—41no—
118D4—81no—
119D4—82no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—22no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4C2×C22⋊C411yescharacteristic
130C2×C4C4×D411yescharacteristic
131C2×C4C4×D411yescharacteristic
132C2×C4#12011yescharacteristic
133C2×C4#12111yescharacteristic
134C2×C4#12111yescharacteristic
135C8—41no—
136C8—42no—
137C8—41no—
138C8—42no—
139C8—81no—
140C8—81no—
141C22—21no—
142C22—21no—
143C22—21no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22#38611yescharacteristic
148C4—21no—
149C4—21no—
150C4—21no—
151C4—21no—
152C4—21no—
153C4—42no—
154C4—81no—
155C4—81no—
156C4#47111yescharacteristic
157C4#47311yescharacteristic
158C2—41no—
159C2—82no—
160C2#188111yescharacteristic
161C2#190811yescharacteristic
162C2#191111yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1537C2semidirect
5. #1481C2semidirect
6. #1290C2semidirect
7. #1287C2semidirect

Maximal subgroups

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

subgroupindex
#32162
#29812
#15512
#15372
#14812
#12902
#12872