Finite Group Databasebeta

← #18373

Subgroups — #18373

Order 256.

551 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 115 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#3497C211yesmaximal, characteristic
2#3234C212yesmaximal, complemented
3#2634C211yesmaximal, characteristic, complemented
4#1616C214yesmaximal, complemented
5#1610C214yesmaximal, complemented
6#1514C212yesmaximal, complemented
7#1467C211yesmaximal, characteristic, complemented
8C2×2−1+4—21no—
9#550—22no—
10D4×Q8—22no—
11#548—24no—
12#537—21no—
13#537C2214yes—
14#532—22no—
15#532C2212yes—
16C2×#125C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18#501—22no—
19#490—22no—
20#481C2214yes—
21#480C2214yes—
22#478—24no—
23#432C2212yes—
24#400—21no—
25#400C2212yes—
26#398C2212yes—
27#395—21no—
28#387—22no—
29#383C2211yescharacteristic
30#327C2218yes—
31#324C2214yes—
322−1+4—48no—
33C2×#41—24no—
34C2×#41—24no—
35C2×#41—22no—
36C22×Q8—24no—
37C22×Q8D411yescharacteristic
38C4⋊Q8—22no—
39C4⋊Q8D412yes—
40#127—22no—
41#127D412yes—
42#125—22no—
43#125—22no—
44#124—24no—
45#124—22no—
46C22⋊Q8—24no—
47C22⋊Q8—28no—
48#121—24no—
49#121—28no—
50#121—22no—
51#121—24no—
52#121C2314yes—
53C4×Q8—22no—
54C4×Q8—22no—
55C4×D4—24no—
56C4×D4—24no—
57C4×D4—22no—
58C4×D4D414yes—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4D411yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C42D412yes—
67C2×C42C2311yescharacteristic
68#107—24no—
69#106—24no—
70C4⋊C8—22no—
71#102—24no—
72#102—24no—
73C22⋊C8C2314yes—
74#97—22no—
75#95—21no—
76#95—21no—
77#95—22no—
78#95—22no—
79#95—22no—
80#95C2312yes—
81#41—48no—
82#41—48no—
83#41—416no—
84#41—48no—
85C2×Q8—24no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—24no—
89C2×Q8—416no—
90C2×D4—24no—
91C2×D4—24no—
92C2×D4—24no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4C2×D414yes—
96C22×C4—28no—
97C22×C4—24no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D412yes—
103C22×C4C2×D412yes—
104C22×C4C2×D411yescharacteristic
105C22×C4C2411yescharacteristic
106C2×C8—24no—
107C4⋊C4—24no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—24no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—42no—
116C4⋊C4—44no—
117C4⋊C4C2×D414yes—
118C22⋊C4—28no—
119C22⋊C4—24no—
120C22⋊C4—24no—
121C42—22no—
122C42—22no—
123C42C2×D412yes—
124C42C2×D412yes—
125C23—24no—
126C23C22×D411yescharacteristic
127Q8—28no—
128Q8—44no—
129Q8—48no—
130D4—44no—
131D4—44no—
132D4—44no—
133D4—48no—
134D4—48no—
135D4—84no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—24no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—48no—
144C2×C4—416no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4#12011yescharacteristic
154C2×C4#12011yescharacteristic
155C2×C4#12012yes—
156C2×C4AGL(1,ℤ/8)14yes—
157C2×C4C22×D412yes—
158C8—44no—
159C22—21no—
160C22—21no—
161C22—44no—
162C22—44no—
163C22—44no—
164C22C2×#12011yescharacteristic
165C22C2×AGL(1,ℤ/8)12yes—
166C4—22no—
167C4—24no—
168C4—21no—
169C4—21no—
170C4—22no—
171C4—44no—
172C4—42no—
173C4—82no—
174C4—81no—
175C4—81no—
176C2—22no—
177C2—84no—
178C2#300311yescharacteristic
179C2#302011yescharacteristic
180C2#302811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3497C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3234C2semidirect
3. #2634C2semidirect
4. #1616C2semidirect
5. #1610C2semidirect
6. #1514C2semidirect
7. #1467C2semidirect

Maximal subgroups

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

subgroupindex
#34972
#32342
#26342
#16162
#16102
#15142
#14672