Finite Group Databasebeta

← #9529

Subgroups — #9529

Order 256.

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

normal subgroups only

This group has 234 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#3109C211yesmaximal, characteristic
2#2981C211yesmaximal, characteristic, complemented
3#1557C211yesmaximal, characteristic, complemented
4#1486C211yesmaximal, characteristic, complemented
5#1466C211yesmaximal, characteristic, complemented
6#1307C211yesmaximal, characteristic, complemented
7#1287C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10#481C412yes—
11#476—21no—
12#459—21no—
13#447C412yes—
14#442—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—
26D4⋊C8—21no—
27D4⋊C8—21no—
28#323—22no—
29C2×#41—21no—
30C22×D4C2×C411yescharacteristic
31C23×C4—21no—
32C2×D8—22no—
33C2×M16—22no—
34C2×M16—21no—
35C22×C8—22no—
36C4⋊Q8D411yescharacteristic
37#127D411yescharacteristic
38#124—21no—
39#121—21no—
40#121C2×C412yes—
41C4×D4—21no—
42C4×D4—21no—
43C4×D4—21no—
44C4×D4—21no—
45C4×D4—21no—
46C4×D4D412yes—
47C2×C4⋊C4C2×C411yescharacteristic
48C2×C22⋊C4—21no—
49C2×C42C2311yescharacteristic
50#106—22no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57#102—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
70C24—21no—
71#41—82no—
72C2×Q8—21no—
73C2×D4—22no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—42no—
79C2×D4C22⋊C412yes—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4—81no—
85C22×C4—81no—
86C22×C4C22×C411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4#4111yescharacteristic
89D8—44no—
90M16—82no—
91M16—81no—
92C2×C8—22no—
93C2×C8—21no—
94C2×C8—21no—
95C2×C8—22no—
96C2×C8—41no—
97C2×C8—41no—
98C2×C8—82no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4C22⋊C412yes—
103C22⋊C4—22no—
104C22⋊C4—22no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C42C2×D411yescharacteristic
108C42C2×D411yescharacteristic
109C42C2×D411yescharacteristic
110C42#4111yescharacteristic
111C23—22no—
112C23—22no—
113C23—41no—
114C23—41no—
115C23—81no—
116C23—81no—
117C23C4×D411yescharacteristic
118Q8—81no—
119D4—22no—
120D4—22no—
121D4—42no—
122D4—41no—
123D4—41no—
124D4—41no—
125D4—41no—
126D4—81no—
127D4—82no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4—82no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—81no—
140C2×C4C2×C22⋊C411yescharacteristic
141C2×C4C4×D411yescharacteristic
142C2×C4#12011yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12111yescharacteristic
145C2×C4#12311yescharacteristic
146C8—41no—
147C8—42no—
148C8—41no—
149C8—42no—
150C8—81no—
151C8—81no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—42no—
156C22—42no—
157C22—42no—
158C22—42no—
159C22—82no—
160C22—82no—
161C22—81no—
162C22—81no—
163C22#38611yescharacteristic
164C4—21no—
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—42no—
170C4—81no—
171C4#47111yescharacteristic
172C4#47211yescharacteristic
173C2—41no—
174C2—82no—
175C2—82no—
176C2#189111yescharacteristic
177C2#190411yescharacteristic
178C2#190711yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radical#3109C2
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. #2981C2semidirect
3. #1557C2semidirect
4. #1486C2semidirect
5. #1466C2semidirect
6. #1307C2semidirect
7. #1287C2semidirect

Maximal subgroups

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

subgroupindex
#31092
#29812
#15572
#14862
#14662
#13072
#12872