Finite Group Databasebeta

← #9333

Subgroups — #9333

Order 256.

219 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 57 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#3213C211yesmaximal, characteristic
2#2980C211yesmaximal, characteristic, complemented
3#1553C211yesmaximal, characteristic, complemented
4#1533C211yesmaximal, characteristic, complemented
5#1469C211yesmaximal, characteristic, complemented
6#1308C211yesmaximal, characteristic, complemented
7#1287C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×#124C2211yescharacteristic
10#496—21no—
11#489—21no—
12#463—22no—
13#448C412yes—
14#447C412yes—
15#435—22no—
16#435—22no—
17#432C2211yescharacteristic
18#432C2211yescharacteristic
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#431C2211yescharacteristic
22#408—22no—
23#332—21no—
24#331—21no—
25#329—22no—
26#323—22no—
27C2×#41—21no—
28C22×Q8C2×C411yescharacteristic
29C22×D4C2×C411yescharacteristic
30C2×SD16—22no—
31C2×D8—22no—
32C2×M16—22no—
33C22×C8—22no—
34C4⋊Q8D411yescharacteristic
35#127D411yescharacteristic
36#124—21no—
37#124—21no—
38#124—21no—
39#124—21no—
40#124—21no—
41#121—21no—
42#121C2×C412yes—
43C4×D4—21no—
44C4×D4D412yes—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C42C2311yescharacteristic
48C4⋊C8—21no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53#103—22no—
54#102—22no—
55C22⋊C8—24no—
56C22⋊C8—21no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59C22⋊C8C2×C412yes—
60C22⋊C8D412yes—
61#97—21no—
62#97—21no—
63#97—21no—
64#97D412yes—
65C4×C8—21no—
66C24—21no—
67#41—81no—
68#41—81no—
69C2×Q8—21no—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—81no—
73C2×D4—22no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—81no—
79C2×D4C22⋊C412yes—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4C22×C411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4#4111yescharacteristic
85SD16—82no—
86D8—82no—
87M16—82no—
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—22no—
91C2×C8—41no—
92C2×C8—41no—
93C2×C8—82no—
94C4⋊C4—21no—
95C4⋊C4C22⋊C412yes—
96C22⋊C4—22no—
97C22⋊C4—22no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C42C2×D411yescharacteristic
103C42C2×D411yescharacteristic
104C42C2×D411yescharacteristic
105C42#4111yescharacteristic
106C23—22no—
107C23—42no—
108C23—41no—
109C23—81no—
110C23—81no—
111C23C4×D411yescharacteristic
112Q8—42no—
113Q8—81no—
114Q8—81no—
115D4—42no—
116D4—42no—
117D4—81no—
118D4—81no—
119D4—81no—
120D4—82no—
121C2×C4—22no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—41no—
128C2×C4—81no—
129C2×C4C2×C22⋊C411yescharacteristic
130C2×C4C4×D411yescharacteristic
131C2×C4#12011yescharacteristic
132C2×C4#12111yescharacteristic
133C2×C4#12111yescharacteristic
134C2×C4#12311yescharacteristic
135C8—42no—
136C8—42no—
137C8—42no—
138C8—81no—
139C8—81no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22—42no—
144C22—42no—
145C22—42no—
146C22—81no—
147C22—81no—
148C22—81no—
149C22—162no—
150C22#38611yescharacteristic
151C4—21no—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—42no—
158C4—81no—
159C2—41no—
160C2—82no—
161C2—161no—
162C2#188111yescharacteristic
163C2#190411yescharacteristic
164C2#191411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#3213C2
LCSG ▷ C22×C4 ▷ 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
2. #2980C2semidirect
3. #1553C2semidirect
4. #1533C2semidirect
5. #1469C2semidirect
6. #1308C2semidirect
7. #1287C2semidirect

Maximal subgroups

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

subgroupindex
#32132
#29802
#15532
#15332
#14692
#13082
#12872