Finite Group Databasebeta

← #18390

Subgroups — #18390

Order 256.

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

normal subgroups only

This group has 489 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#3501C211yesmaximal, characteristic
2#3209C212yesmaximal, complemented
3#2563C211yesmaximal, characteristic
4#1602C218yesmaximal, complemented
5#1503C212yesmaximal, complemented
6#1460C211yesmaximal, characteristic
7#554—22no—
8#554—24no—
9D4×Q8—22no—
10#544—21no—
11#543C2214yes—
12#533C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8—22no—
15#518—21no—
16C2×C4×D4C2211yescharacteristic
17#494—22no—
18#487—22no—
19#462—24no—
20#451C2218yes—
21#431C2212yes—
22#393C2214yes—
23#390—21no—
24#382—21no—
25#380—22no—
26C4×C4⋊C4C2211yescharacteristic
27#328C2218yes—
28#323C2214yes—
29C22×Q8—22no—
30C22×Q8C2312yes—
31C22×D4D411yescharacteristic
32C23×C4D412yes—
33C2×Q16—28no—
34C4⋊Q8—24no—
35C4⋊Q8—22no—
36C4⋊Q8—22no—
37C4⋊Q8D412yes—
38C4⋊Q8D412yes—
39#125—24no—
40#125—22no—
41C22⋊Q8—24no—
42C22⋊Q8—24no—
43C22⋊Q8—28no—
44C22⋊Q8—22no—
45C22⋊Q8—24no—
46C22⋊Q8—28no—
47C22⋊Q8C2314yes—
48C4×Q8—22no—
49C4×Q8D414yes—
50C4×D4—24no—
51C4×D4—24no—
52C4×D4—22no—
53C4×D4—22no—
54#117—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4D411yescharacteristic
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4C2312yes—
64C2×C42—21no—
65C2×C42C2311yescharacteristic
66C4⋊C8—22no—
67#103—28no—
68C22⋊C8C2314yes—
69C4×C8—22no—
70#95—24no—
71#95C2312yes—
72C24C2×D412yes—
73C2×Q8—24no—
74C2×Q8—22no—
75C2×Q8—24no—
76C2×Q8—82no—
77C2×Q8—84no—
78C2×Q8C2×D414yes—
79C2×D4—22no—
80C2×D4—22no—
81C2×D4Q1614yescomplemented
82C22×C4—24no—
83C22×C4—24no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—44no—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D412yes—
91C22×C4C2×D411yescharacteristic
92C22×C4C2411yescharacteristic
93Q16—88no—
94C2×C8—24no—
95C4⋊C4—24no—
96C4⋊C4—28no—
97C4⋊C4—22no—
98C4⋊C4—24no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—24no—
103C4⋊C4—42no—
104C4⋊C4—41no—
105C4⋊C4—42no—
106C4⋊C4—41no—
107C4⋊C4C2×D414yes—
108C22⋊C4—28no—
109C22⋊C4—24no—
110C22⋊C4—24no—
111C22⋊C4—84no—
112C42—24no—
113C42—42no—
114C42C2×D412yes—
115C42C2×D412yes—
116C23—22no—
117C23—22no—
118C23—42no—
119C23—42no—
120C23C2×Q1614yes—
121C23C22×D411yescharacteristic
122Q8—44no—
123Q8—48no—
124Q8—44no—
125Q8—84no—
126D4—28no—
127C2×C4—22no—
128C2×C4—24no—
129C2×C4—24no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—44no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—44no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—82no—
140C2×C4—84no—
141C2×C4—84no—
142C2×C4#12011yescharacteristic
143C2×C4#12012yes—
144C2×C4#12011yescharacteristic
145C2×C4C2×Q1612yes—
146C2×C4#13712yes—
147C2×C4C22×D412yes—
148C8—44no—
149C22—21no—
150C22—21no—
151C22—24no—
152C22—44no—
153C22—42no—
154C22—42no—
155C22—82no—
156C22C2×#12011yescharacteristic
157C22C22×Q1611yescharacteristic
158C22C2×#13711yescharacteristic
159C4—21no—
160C4—22no—
161C4—42no—
162C4—44no—
163C4—44no—
164C4—82no—
165C4—82no—
166C4#45114yes—
167C4#45512yes—
168C2—21no—
169C2—21no—
170C2—44no—
171C2C2×#45111yescharacteristic
172C2C2×#45511yescharacteristic
173C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
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. #3209C2semidirect
4. #1602C2semidirect
5. #1503C2semidirect
81. C2×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#35012
#32092
#25632
#16022
#15032
#14602