Finite Group Databasebeta

← #18397

Subgroups — #18397

Order 256.

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

normal subgroups only

This group has 505 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#3230C212yesmaximal, complemented
3#2542C211yesmaximal, characteristic
4#1614C218yesmaximal, 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—
15C2×C22⋊Q8—22no—
16#518—21no—
17C2×C4×D4C2211yescharacteristic
18#500—22no—
19#487—22no—
20#484C2218yes—
21#477—24no—
22#431C2212yes—
23#399C2214yes—
24#390—21no—
25#384—21no—
26C4×C4⋊C4C2211yescharacteristic
27#328C2218yes—
28#323C2214yes—
29C22×Q8—22no—
30C22×Q8—21no—
31C22×D4D411yescharacteristic
32C23×C4D412yes—
33C4⋊Q8—24no—
34C4⋊Q8—22no—
35C4⋊Q8—22no—
36C4⋊Q8D412yes—
37C4⋊Q8D412yes—
38#125—24no—
39#125—22no—
40C22⋊Q8—24no—
41C22⋊Q8—24no—
42C22⋊Q8—28no—
43C22⋊Q8—22no—
44C22⋊Q8—24no—
45C22⋊Q8—28no—
46C22⋊Q8—48no—
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—24no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4C2312yes—
65C2×C42—21no—
66C2×C42C2311yescharacteristic
67#107—28no—
68C4⋊C8—22no—
69#103—28no—
70C22⋊C8C2314yes—
71C4×C8—22no—
72#95C2312yes—
73C24C2×D412yes—
74C2×Q8—24no—
75C2×Q8—22no—
76C2×Q8—44no—
77C2×Q8—41no—
78C2×Q8—41no—
79C2×Q8—84no—
80C2×Q8C2×D414yes—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4Q1614yes—
84C22×C4—22no—
85C22×C4—24no—
86C22×C4—24no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—44no—
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D412yes—
93C22×C4C2×D411yescharacteristic
94C22×C4C2411yescharacteristic
95C2×C8—24no—
96C4⋊C4—24no—
97C4⋊C4—28no—
98C4⋊C4—24no—
99C4⋊C4—22no—
100C4⋊C4—24no—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—24no—
105C4⋊C4—24no—
106C4⋊C4—48no—
107C4⋊C4—42no—
108C4⋊C4C2×D414yes—
109C22⋊C4—28no—
110C22⋊C4—24no—
111C22⋊C4—24no—
112C22⋊C4—48no—
113C42—24no—
114C42—42no—
115C42C2×D412yes—
116C42C2×D412yes—
117C23—22no—
118C23—22no—
119C23—42no—
120C23—42no—
121C23C2×Q1614yes—
122C23C22×D411yescharacteristic
123Q8—44no—
124Q8—44no—
125Q8—82no—
126Q8—84no—
127Q8—82no—
128D4—28no—
129C2×C4—22no—
130C2×C4—24no—
131C2×C4—24no—
132C2×C4—24no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—44no—
138C2×C4—42no—
139C2×C4—44no—
140C2×C4—42no—
141C2×C4—82no—
142C2×C4—84no—
143C2×C4—84no—
144C2×C4#12011yescharacteristic
145C2×C4#12012yes—
146C2×C4#12011yescharacteristic
147C2×C4C2×Q1612yes—
148C2×C4#13712yes—
149C2×C4C22×D412yes—
150C8—44no—
151C22—21no—
152C22—21no—
153C22—24no—
154C22—44no—
155C22—42no—
156C22—42no—
157C22—82no—
158C22C2×#12011yescharacteristic
159C22C22×Q1611yescharacteristic
160C22C2×#13711yescharacteristic
161C4—21no—
162C4—22no—
163C4—44no—
164C4—42no—
165C4—44no—
166C4—44no—
167C4—82no—
168C4#45114yes—
169C4#45312yes—
170C2—21no—
171C2—21no—
172C2—44no—
173C2C2×#45111yescharacteristic
174C2C2×#45311yescharacteristic
175C2#302811yescharacteristic
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. #3230C2semidirect
4. #1614C2semidirect
5. #1503C2semidirect

Maximal subgroups

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

subgroupindex
#35012
#32302
#25422
#16142
#15032
#14602