Finite Group Databasebeta

← #18402

Subgroups — #18402

Order 256.

587 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 129 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#3206C212yesmaximal, complemented
3#2567C211yesmaximal, characteristic, complemented
4#1604C218yesmaximal, complemented
5#1502C212yesmaximal, complemented
6#1461C211yesmaximal, characteristic, complemented
7C2×2−1+4—21no—
8#550—22no—
9D4×Q8—22no—
10#548—24no—
11#537—21no—
12#537C2214yes—
13#532—22no—
14#532C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4×Q8C2211yescharacteristic
17#492—22no—
18#487—22no—
19#461—24no—
20#448C2218yes—
21#431C2212yes—
22#395C2214yes—
23#391—21no—
24#389—22no—
25#384—21no—
26C4×C4⋊C4C2211yescharacteristic
27#327C2218yes—
28#324C2214yes—
292−1+4—44no—
302−1+4—44no—
31C2×#41—24no—
32C2×#41—24no—
33C2×#41—22no—
34C22×Q8—24no—
35C22×Q8D411yescharacteristic
36C22×Q8C2312yes—
37C2×SD16—28no—
38C4⋊Q8—22no—
39C4⋊Q8—22no—
40C4⋊Q8—22no—
41C4⋊Q8D412yes—
42#127—22no—
43#127D412yes—
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—21no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C42—21no—
66C2×C42D412yes—
67C2×C42C2311yescharacteristic
68C4⋊C8—22no—
69#103—28no—
70C22⋊C8C2314yes—
71C4×C8—22no—
72#95—24no—
73#95C2312yes—
74#41—48no—
75#41—416no—
76#41—48no—
77#41—44no—
78#41—44no—
79C2×Q8—24no—
80C2×Q8—22no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—24no—
84C2×Q8—416no—
85C2×Q8—82no—
86C2×Q8SD1614yescomplemented
87C2×D4—24no—
88C2×D4—24no—
89C2×D4—24no—
90C2×D4—22no—
91C2×D4—22no—
92C2×D4C2×D414yes—
93C22×C4—28no—
94C22×C4—22no—
95C22×C4—24no—
96C22×C4—22no—
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D412yes—
99C22×C4C2×D412yes—
100C22×C4C2×D411yescharacteristic
101C22×C4C2411yescharacteristic
102SD16—88no—
103C2×C8—24no—
104C4⋊C4—24no—
105C4⋊C4—24no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—24no—
109C4⋊C4—41no—
110C4⋊C4—42no—
111C4⋊C4—41no—
112C4⋊C4—84no—
113C4⋊C4C2×D414yes—
114C22⋊C4—28no—
115C22⋊C4—24no—
116C22⋊C4—24no—
117C42—22no—
118C42—22no—
119C42—42no—
120C42C2×D412yes—
121C42C2×D412yes—
122C23—24no—
123C23C22×D411yescharacteristic
124Q8—28no—
125Q8—48no—
126Q8—48no—
127Q8—44no—
128D4—44no—
129D4—48no—
130D4—44no—
131D4—44no—
132D4—44no—
133D4—44no—
134D4—84no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—24no—
139C2×C4—22no—
140C2×C4—24no—
141C2×C4—41no—
142C2×C4—416no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—48no—
146C2×C4—42no—
147C2×C4—44no—
148C2×C4—42no—
149C2×C4#12011yescharacteristic
150C2×C4#12012yes—
151C2×C4#12011yescharacteristic
152C2×C4C2×SD1614yes—
153C2×C4C2×SD1612yes—
154C2×C4AGL(1,ℤ/8)12yes—
155C2×C4C22×D412yes—
156C8—44no—
157C22—21no—
158C22—21no—
159C22—44no—
160C22—44no—
161C22—44no—
162C22C2×#12011yescharacteristic
163C22C22×SD1611yescharacteristic
164C22C2×AGL(1,ℤ/8)11yescharacteristic
165C4—24no—
166C4—21no—
167C4—22no—
168C4—44no—
169C4—44no—
170C4—42no—
171C4—82no—
172C4#45014yes—
173C4#45612yes—
174C2—21no—
175C2—21no—
176C2—84no—
177C2C2×#45011yescharacteristic
178C2C2×#45611yescharacteristic
179C2#302711yescharacteristic
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. #3206C2semidirect
3. #2567C2semidirect
4. #1604C2semidirect
5. #1502C2semidirect
6. #1461C2semidirect
86. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#34972
#32062
#25672
#16042
#15022
#14612