Finite Group Databasebeta

← #9527

Subgroups — #9527

Order 256.

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

normal subgroups only

This group has 218 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#3112C211yesmaximal, characteristic
2#2981C211yesmaximal, characteristic
3#1557C211yesmaximal, characteristic
4#1487C211yesmaximal, characteristic, complemented
5#1467C211yesmaximal, characteristic, complemented
6#1307C211yesmaximal, characteristic, complemented
7#1287C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10#480C412yes—
11#477—21no—
12#460—21no—
13#448C412yes—
14#440—22no—
15#436—21no—
16#435—22no—
17C8×D4—21no—
18#432C2211yescharacteristic
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23#408—22no—
24#332—21no—
25#331—21no—
26Q8⋊C8—21no—
27Q8⋊C8—21no—
28#324—22no—
29C2×#41—21no—
30C22×Q8C2×C411yescharacteristic
31C2×SD16—22no—
32C2×M16—22no—
33C2×M16—21no—
34C22×C8—22no—
35C4⋊Q8D411yescharacteristic
36#127D411yescharacteristic
37#124—21no—
38#121—21no—
39#121C2×C412yes—
40C4×Q8—21no—
41C4×Q8—21no—
42C4×Q8—21no—
43C4×Q8—21no—
44C4×D4—21no—
45C4×D4D412yes—
46C2×C4⋊C4—21no—
47C2×C4⋊C4C2×C411yescharacteristic
48C2×C42—21no—
49C2×C42C2311yescharacteristic
50#107—22no—
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⋊C8C2×C412yes—
64C22⋊C8D412yes—
65#97—21no—
66#97—21no—
67#97—21no—
68C4×C8D411yescharacteristic
69C4×C8D411yescharacteristic
70#41—82no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8—42no—
75C2×D4—22no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4C22⋊C412yes—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4C22×C411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C22×C4#4111yescharacteristic
85SD16—44no—
86M16—82no—
87M16—81no—
88C2×C8—22no—
89C2×C8—21no—
90C2×C8—21no—
91C2×C8—22no—
92C2×C8—41no—
93C2×C8—41no—
94C2×C8—82no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4C22⋊C412yes—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C42—41no—
104C42—41no—
105C42C2×D411yescharacteristic
106C42C2×D411yescharacteristic
107C42C2×D411yescharacteristic
108C42#4111yescharacteristic
109C23—22no—
110C23C4×D411yescharacteristic
111Q8—22no—
112Q8—22no—
113Q8—41no—
114Q8—41no—
115Q8—81no—
116D4—42no—
117D4—41no—
118D4—41no—
119D4—81no—
120D4—82no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—81no—
133C2×C4C2×C22⋊C411yescharacteristic
134C2×C4C4×D411yescharacteristic
135C2×C4#12011yescharacteristic
136C2×C4#12111yescharacteristic
137C2×C4#12111yescharacteristic
138C2×C4#12311yescharacteristic
139C8—41no—
140C8—42no—
141C8—41no—
142C8—42no—
143C8—81no—
144C8—81no—
145C22—21no—
146C22—21no—
147C22—21no—
148C22—42no—
149C22—42no—
150C22—42no—
151C22#38611yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—42no—
158C4—42no—
159C4—81no—
160C4#47111yescharacteristic
161C4#47311yescharacteristic
162C2—41no—
163C2—82no—
164C2#189111yescharacteristic
165C2#190411yescharacteristic
166C2#190811yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
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
4. #1487C2semidirect
5. #1467C2semidirect
6. #1307C2semidirect
7. #1287C2semidirect

Maximal subgroups

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

subgroupindex
#31122
#29812
#15572
#14872
#14672
#13072
#12872