Finite Group Databasebeta

← #12285

Subgroups — #12285

Order 256.

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

normal subgroups only

This group has 159 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#3203C211yesmaximal, characteristic
2#2235C211yesmaximal, characteristic, complemented
3#2233C211yesmaximal, characteristic, complemented
4#2078C211yesmaximal, characteristic, complemented
5#1350C211yesmaximal, characteristic, complemented
6#1349C211yesmaximal, characteristic, complemented
7#1315C211yesmaximal, characteristic, complemented
8#537—21no—
9#484—21no—
10#480—21no—
11#468—21no—
12#467C2211yescharacteristic
13#466C2211yescharacteristic
14#452—21no—
15#449—21no—
16C2×#107C2211yescharacteristic
17#417—21no—
18#408C2211yescharacteristic
19#367—21no—
20#367—21no—
21#358—21no—
22#357—21no—
23C22⋊C16C2211yescharacteristic
24C22⋊C16C2211yescharacteristic
25#343—21no—
26#341C2211yescharacteristic
27C2×#41—21no—
28C2×#41D411yescharacteristic
29C2×Q16D411yescharacteristic
30C2×SD16—21no—
31C2×D8D411yescharacteristic
32C2×M16—21no—
33C2×M16D411yescharacteristic
34C22×C8C2311yescharacteristic
35C22⋊Q8—21no—
36C22⋊Q8—21no—
37C22⋊Q8—41no—
38#121—21no—
39#121—21no—
40#121—41no—
41#121—41no—
42#121—41no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C16—21no—
46C2×C16—21no—
47#107—21no—
48#107—21no—
49#107—21no—
50#107D411yescharacteristic
51#107D411yescharacteristic
52#103—21no—
53#103—21no—
54#103—41no—
55#102—21no—
56#102—21no—
57#102—41no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60#41—41no—
61#41—41no—
62#41—81no—
63#41—81no—
64C2×Q8—21no—
65C2×Q8D811yescharacteristic
66C2×D4—21no—
67C2×D4—21no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4D811yescharacteristic
72C22×C4—21no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4C2×D411yescharacteristic
77Q16—41no—
78SD16—81no—
79D8—41no—
80M16—41no—
81M16—41no—
82M16—41no—
83C2×C8—21no—
84C2×C8—41no—
85C2×C8—41no—
86C2×C8C2×D411yescharacteristic
87C2×C8C2×D411yescharacteristic
88C4⋊C4—21no—
89C4⋊C4—21no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C16—41no—
100C16—41no—
101C23—21no—
102C23—41no—
103C23AGL(1,ℤ/8)11yescharacteristic
104Q8—41no—
105Q8—41no—
106Q8—41no—
107D4—41no—
108D4—41no—
109D4—41no—
110D4—81no—
111D4—82no—
112D4—161no—
113D4—161no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—82no—
123C2×C4—81no—
124C2×C4—161no—
125C2×C4—161no—
126C2×C4#12011yescharacteristic
127C2×C4C2×D811yescharacteristic
128C8—21no—
129C8—21no—
130C8—41no—
131C8—81no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22—41no—
136C22—41no—
137C22—41no—
138C22—81no—
139C22—81no—
140C22—81no—
141C22#44711yescharacteristic
142C4—21no—
143C4—41no—
144C4—81no—
145C4—81no—
146C4—81no—
147C4—161no—
148C4#45511yescharacteristic
149C4#45811yescharacteristic
150C2—41no—
151C2—81no—
152C2—161no—
153C2#160711yescharacteristic
154C2#219311yescharacteristic
155C2#219411yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#3203C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2235C2semidirect
3. #2233C2semidirect
4. #2078C2semidirect
5. #1350C2semidirect
6. #1349C2semidirect
7. #1315C2semidirect

Maximal subgroups

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

subgroupindex
#32032
#22352
#22332
#20782
#13502
#13492
#13152