Finite Group Databasebeta

← #12100

Subgroups — #12100

Order 256.

261 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 37 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#2733C211yesmaximal, characteristic, complemented
2#2007C211yesmaximal, characteristic, complemented
3#1598C212yesmaximal, complemented
4#1345C212yesmaximal, complemented
5#1319C211yesmaximal, characteristic, complemented
6C2×#137—21no—
7#537C2211yescharacteristic
8C2×#124—21no—
9#447—22no—
10C2×#104—21no—
11#394C2212yes—
12#392—22no—
13#391C2211yescharacteristic
14#390—21no—
15#388—22no—
16#343C2211yescharacteristic
17#327—22no—
18#324C2212yes—
19C2×#41—21no—
20C22×Q8D411yescharacteristic
21C22×D4—21no—
22C22×D4—21no—
23C22×D4D412yes—
24#137—42no—
25#137—42no—
26C2×Q16—41no—
27C2×SD16—41no—
28C2×D8—42no—
29C2×M16—21no—
30#124—42no—
31#124—42no—
32C22⋊Q8—41no—
33#121—22no—
34#121—41no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4C2311yescharacteristic
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C42—21no—
42C2×C42D411yescharacteristic
43#104—42no—
44#102—42no—
45C22⋊C8—22no—
46#95—22no—
47#95D412yes—
48C24—22no—
49#41—41no—
50#41—41no—
51#41—41no—
52#41—41no—
53C2×Q8—21no—
54C2×Q8—21no—
55C2×Q8—41no—
56C2×Q8—41no—
57C2×Q8D812yescomplemented
58C2×D4—21no—
59C2×D4—22no—
60C2×D4—42no—
61C2×D4—42no—
62C2×D4—42no—
63C2×D4—41no—
64C2×D4—82no—
65C2×D4—81no—
66C2×D4—81no—
67C2×D4—81no—
68C2×D4—81no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—22no—
72C22×C4—41no—
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D412yes—
75Q16—44no—
76SD16—44no—
77D8—84no—
78M16—22no—
79M16—41no—
80C2×C8—41no—
81C2×C8—42no—
82C4⋊C4—22no—
83C4⋊C4—41no—
84C4⋊C4—81no—
85C22⋊C4—44no—
86C22⋊C4—44no—
87C22⋊C4—42no—
88C22⋊C4—42no—
89C22⋊C4—42no—
90C22⋊C4—82no—
91C42—41no—
92C42—42no—
93C23—42no—
94C23—42no—
95C23—42no—
96C23—42no—
97C23—42no—
98C23—42no—
99C23—41no—
100C23—42no—
101C23#12011yescharacteristic
102Q8—42no—
103Q8—41no—
104Q8—41no—
105Q8—81no—
106D4—41no—
107D4—41no—
108D4—44no—
109D4—82no—
110D4—82no—
111D4—84no—
112D4—84no—
113D4—81no—
114D4—81no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—81no—
130C2×C4—81no—
131C2×C4C2×D811yescharacteristic
132C2×C4AGL(1,ℤ/8)11yescharacteristic
133C8—42no—
134C8—82no—
135C22—21no—
136C22—21no—
137C22—82no—
138C22—82no—
139C22—82no—
140C22—82no—
141C22—81no—
142C22—82no—
143C22—82no—
144C22—81no—
145C22—81no—
146C22—82no—
147C22#44711yescharacteristic
148C22#45311yescharacteristic
149C22B(4,𝔽2)11yescharacteristic
150C4—21no—
151C4—21no—
152C4—42no—
153C4—42no—
154C4—42no—
155C4—82no—
156C4—81no—
157C2—21no—
158C2—21no—
159C2—161no—
160C2—162no—
161C2#159511yescharacteristic
162C2#219911yescharacteristic
163C2#220411yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #2733C2semidirect
2. #2007C2semidirect
3. #1598C2semidirect
4. #1345C2semidirect
5. #1319C2semidirect
57. C2×Q8D8semidirect

Maximal subgroups

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

subgroupindex
#27332
#20072
#15982
#13452
#13192