Finite Group Databasebeta

← #18389

Subgroups — #18389

Order 256.

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

normal subgroups only

This group has 585 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, complemented
2#3208C212yesmaximal, complemented
3#2541C211yesmaximal, characteristic
4#1596C218yesmaximal, complemented
5#1503C212yesmaximal, complemented
6#1460C211yesmaximal, characteristic, complemented
7#554—22no—
8#554—24no—
9D4×Q8—22no—
10#544—21no—
11#543C2214yes—
12#533C2212yes—
13C2×#127—21no—
14C2×#127C2211yescharacteristic
15C2×C22⋊Q8—22no—
16C2×#121—22no—
17#518—21no—
18C2×C4×D4C2211yescharacteristic
19#492—22no—
20#486—22no—
21#460—24no—
22#450C2218yes—
23#431C2212yes—
24#392C2214yes—
25#390—21no—
26C4×C4⋊C4C2211yescharacteristic
27#328C2218yes—
28#323C2214yes—
29C22×Q8—22no—
30C22×D4—24no—
31C22×D4—21no—
32C22×D4—21no—
33C22×D4D411yescharacteristic
34C22×D4C2312yes—
35C23×C4D412yes—
36C2×SD16—28no—
37C4⋊Q8—24no—
38C4⋊Q8D412yescomplemented
39C4⋊Q8D412yescomplemented
40#127—22no—
41#127—22no—
42#127—44no—
43#125—24no—
44#125—22no—
45C22⋊Q8—24no—
46C22⋊Q8—24no—
47C22⋊Q8—28no—
48C22⋊Q8—22no—
49C22⋊Q8—24no—
50C22⋊Q8—28no—
51C22⋊Q8C2314yes—
52#121—48no—
53C4×Q8—22no—
54C4×Q8D414yescomplemented
55C4×D4—24no—
56C4×D4—24no—
57C4×D4—22no—
58C4×D4—22no—
59#117—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4D411yescharacteristic
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4C2312yes—
67C2×C42—21no—
68C2×C42C2311yescharacteristic
69C4⋊C8—22no—
70#102—28no—
71C22⋊C8C2314yes—
72C4×C8—22no—
73#95C2312yes—
74C24—22no—
75C24C2×D412yes—
76C2×Q8—24no—
77C2×Q8—22no—
78C2×Q8—84no—
79C2×Q8C2×D414yes—
80C2×D4—22no—
81C2×D4—22no—
82C2×D4—24no—
83C2×D4—44no—
84C2×D4—48no—
85C2×D4—48no—
86C2×D4—41no—
87C2×D4—42no—
88C2×D4—41no—
89C2×D4—42no—
90C2×D4—42no—
91C2×D4—82no—
92C2×D4—84no—
93C2×D4SD1614yescomplemented
94C22×C4—22no—
95C22×C4—24no—
96C22×C4—24no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—44no—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D412yes—
102C22×C4C2×D411yescharacteristic
103C22×C4C2411yescharacteristic
104SD16—88no—
105C2×C8—24no—
106C4⋊C4—24no—
107C4⋊C4—24no—
108C4⋊C4—28no—
109C4⋊C4—22no—
110C4⋊C4—24no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—24no—
115C4⋊C4C2×D414yes—
116C22⋊C4—28no—
117C22⋊C4—24no—
118C22⋊C4—24no—
119C22⋊C4—48no—
120C42—24no—
121C42—42no—
122C42C2×D412yes—
123C42C2×D412yes—
124C23—22no—
125C23—22no—
126C23—42no—
127C23—42no—
128C23—44no—
129C23—44no—
130C23—42no—
131C23—42no—
132C23—42no—
133C23C2×SD1614yes—
134C23C22×D411yescharacteristic
135Q8—44no—
136Q8—44no—
137Q8—84no—
138D4—28no—
139D4—48no—
140D4—82no—
141D4—816no—
142D4—84no—
143D4—82no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—82no—
155C2×C4—84no—
156C2×C4—84no—
157C2×C4#12011yescharacteristic
158C2×C4#12012yes—
159C2×C4#12011yescharacteristic
160C2×C4C2×SD1612yes—
161C2×C4#13712yes—
162C2×C4C22×D412yes—
163C8—44no—
164C22—21no—
165C22—21no—
166C22—24no—
167C22—44no—
168C22—42no—
169C22—42no—
170C22—82no—
171C22—84no—
172C22—84no—
173C22—82no—
174C22—82no—
175C22—82no—
176C22C2×#12011yescharacteristic
177C22C22×SD1611yescharacteristic
178C22C2×#13711yescharacteristic
179C4—21no—
180C4—22no—
181C4—44no—
182C4—42no—
183C4—44no—
184C4—44no—
185C4#44814yes—
186C4#45312yes—
187C2—21no—
188C2—21no—
189C2—44no—
190C2—162no—
191C2C2×#44811yescharacteristic
192C2C2×#45311yescharacteristic
193C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#2541C2
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
1. #3501C2semidirect
2. #3208C2semidirect
4. #1596C2semidirect
5. #1503C2semidirect
6. #1460C2semidirect
38. C4⋊Q8D4semidirect
39. C4⋊Q8D4semidirect
54. C4×Q8D4semidirect
93. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#35012
#32082
#25412
#15962
#15032
#14602