Finite Group Databasebeta

← #12105

Subgroups — #12105

Order 256.

238 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 41 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#3186C211yesmaximal, characteristic
2#2185C212yesmaximal, complemented
3#2001C211yesmaximal, characteristic, complemented
4#1347C212yesmaximal, complemented
5#1314C211yesmaximal, characteristic, complemented
6C22×Q16C2211yescharacteristic, complemented
7C2×C22⋊Q8—21no—
8C2×#121—21no—
9C2×SD32—22no—
10#480—22no—
11#466—21no—
12#466C2212yes—
13#447—22no—
14C2×#103—21no—
15C2×C22⋊C8C2211yescharacteristic
16#358—22no—
17C22⋊C16C2212yescomplemented
18#340C2211yescharacteristic, complemented
19C22×Q8—21no—
20C22×D4—21no—
21C22×D4—41no—
22C23×C4D411yescharacteristic
23C2×Q16—22no—
24C2×Q16—42no—
25C2×D8—21no—
26C2×D8D412yes—
27C22×C8D411yescharacteristic
28C22×C8C2311yescharacteristic
29C22⋊Q8—42no—
30C22⋊Q8—42no—
31#121—22no—
32#121—22no—
33#121—41no—
34#121—41no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—41no—
38C2×C22⋊C4—41no—
39C2×C22⋊C4—41no—
40SD32—44no—
41C2×C16—22no—
42#107—21no—
43#107D412yes—
44#103—42no—
45#102—22no—
46#102—22no—
47C22⋊C8—22no—
48C24—41no—
49C24D811yescharacteristic
50C2×Q8—22no—
51C2×Q8—41no—
52C2×Q8—41no—
53C2×Q8—41no—
54C2×Q8—41no—
55C2×Q8—41no—
56C2×D4—22no—
57C2×D4—41no—
58C2×D4—42no—
59C2×D4—42no—
60C2×D4—81no—
61C2×D4—81no—
62C2×D4—82no—
63C2×D4—82no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4D811yescharacteristic
72C22×C4C2×D411yescharacteristic
73Q16—24no—
74Q16—42no—
75D8—42no—
76D8—81no—
77C2×C8—21no—
78C2×C8—41no—
79C2×C8—41no—
80C2×C8C2×D412yes—
81C4⋊C4—22no—
82C4⋊C4—22no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—44no—
86C22⋊C4—42no—
87C22⋊C4—44no—
88C22⋊C4—42no—
89C16—42no—
90C23—21no—
91C23—21no—
92C23—41no—
93C23—41no—
94C23—42no—
95C23—82no—
96C23—82no—
97C23—82no—
98C23SD3212yescomplemented
99C23C2×D811yescharacteristic
100Q8—42no—
101Q8—42no—
102Q8—81no—
103Q8—81no—
104D4—42no—
105D4—42no—
106D4—82no—
107D4—84no—
108D4—82no—
109D4—82no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—81no—
120C2×C4—81no—
121C2×C4—82no—
122C2×C4—81no—
123C2×C4—81no—
124C2×C4—81no—
125C2×C4#12011yescharacteristic
126C2×C4AGL(1,ℤ/8)11yescharacteristic
127C8—22no—
128C8—41no—
129C22—21no—
130C22—21no—
131C22—22no—
132C22—42no—
133C22—41no—
134C22—41no—
135C22—81no—
136C22—82no—
137C22—82no—
138C22—82no—
139C22—164no—
140C22#44711yescharacteristic
141C22C2×SD3211yescharacteristic
142C22#50911yescharacteristic
143C4—21no—
144C4—41no—
145C4—82no—
146C4—82no—
147C4—82no—
148C4#45311yescharacteristic
149C4B(4,𝔽2)11yescharacteristic
150C2—21no—
151C2—21no—
152C2—42no—
153C2—162no—
154C2#159511yescharacteristic
155C2#218711yescharacteristic
156C2#219111yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#3186C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2185C2semidirect
3. #2001C2semidirect
4. #1347C2semidirect
5. #1314C2semidirect
6. C22×Q16C22semidirect
17. C22⋊C16C22semidirect
18. #340C22semidirect
98. C23SD32semidirect

Maximal subgroups

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

subgroupindex
#31862
#21852
#20012
#13472
#13142