Finite Group Databasebeta

← #18188

Subgroups — #18188

Order 256.

467 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 117 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#3502C211yesmaximal, characteristic, complemented
2#3110C212yesmaximal
3#2398C211yesmaximal, characteristic
4#1617C214yesmaximal, complemented
5#1599C214yesmaximal
6#1506C212yesmaximal, complemented
7#1467C211yesmaximal, characteristic, complemented
8Q82—22no—
9#557—22no—
10#557—24no—
11#544—21no—
12#544C2214yes—
13#533—22no—
14#533C2212yes—
15#519—21no—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18#484C2214yes—
19#474—22no—
20#461—22no—
21#450C2214yes—
22#440—24no—
23#432C2212yes—
24#400C2212yes—
25#397—21no—
26#394C2212yes—
27#393—21no—
28C4×C22⋊C4—22no—
29#376C2211yescharacteristic
30#328C2218yes—
31#324C2214yes—
32C22×Q8D411yescharacteristic
33C22×D4C2311yescharacteristic
34C23×C4—21no—
35C2×SD16—24no—
36C4⋊Q8—22no—
37C4⋊Q8—22no—
38C4⋊Q8—24no—
39C4⋊Q8—28no—
40C4⋊Q8D412yes—
41C4⋊Q8D412yes—
42#125—24no—
43#125—24no—
44#125—28no—
45#125—22no—
46C22⋊Q8—22no—
47C22⋊Q8—24no—
48C22⋊Q8C2314yes—
49C4×Q8—24no—
50C4×Q8—24no—
51C4×Q8—24no—
52C4×Q8—24no—
53C4×Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8D414yes—
57C4×D4—22no—
58C4×D4—22no—
59#117—24no—
60#117—22no—
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C42—21no—
68C2×C42D412yes—
69C2×C42C2311yescharacteristic
70#107—24no—
71C4⋊C8—22no—
72#103—24no—
73#102—24no—
74C22⋊C8C2314yes—
75#97—22no—
76#95—21no—
77#95—22no—
78#95—22no—
79#95C2312yes—
80C24—21no—
81C2×Q8—24no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—24no—
85C2×Q8C2×D414yes—
86C2×D4—22no—
87C2×D4—42no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—44no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D412yes—
96C22×C4C2×D412yes—
97C22×C4C2411yescharacteristic
98SD16—48no—
99C2×C8—24no—
100C4⋊C4—24no—
101C4⋊C4—24no—
102C4⋊C4—24no—
103C4⋊C4—28no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—24no—
109C4⋊C4—24no—
110C4⋊C4—24no—
111C4⋊C4—22no—
112C4⋊C4C2×D414yes—
113C22⋊C4—24no—
114C22⋊C4—44no—
115C22⋊C4—42no—
116C22⋊C4—44no—
117C42—28no—
118C42—22no—
119C42—22no—
120C42—24no—
121C42—42no—
122C42C2×D412yes—
123C42C2×D412yes—
124C23—22no—
125C23—42no—
126C23—41no—
127C23—41no—
128C23—42no—
129C23C22×D411yescharacteristic
130Q8—28no—
131Q8—44no—
132Q8—44no—
133Q8—84no—
134D4—24no—
135D4—42no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—24no—
141C2×C4—24no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4#12012yes—
153C2×C4#12011yescharacteristic
154C2×C4#12011yescharacteristic
155C2×C4#13714yes—
156C2×C4C22×D412yes—
157C8—44no—
158C22—21no—
159C22—21no—
160C22—44no—
161C22—42no—
162C22—82no—
163C22—81no—
164C22—81no—
165C22C2×#12011yescharacteristic
166C22C2×#13712yes—
167C4—22no—
168C4—24no—
169C4—21no—
170C4—22no—
171C4—44no—
172C4—42no—
173C4—44no—
174C4—82no—
175C4—81no—
176C4#45412yes—
177C2—22no—
178C2—82no—
179C2#300411yescharacteristic
180C2C2×#45411yescharacteristic
181C2#302511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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. #3502C2semidirect
4. #1617C2semidirect
6. #1506C2semidirect
7. #1467C2semidirect

Maximal subgroups

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

subgroupindex
#35022
#31102
#23982
#16172
#15992
#15062
#14672