Finite Group Databasebeta

← #12283

Subgroups — #12283

Order 256.

174 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 39 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#3191C211yesmaximal, characteristic
2#2236C211yesmaximal, characteristic, complemented
3#2233C211yesmaximal, characteristic, complemented
4#2076C211yesmaximal, characteristic, complemented
5#1348C211yesmaximal, characteristic, complemented
6#1347C211yesmaximal, characteristic, complemented
7#1314C211yesmaximal, characteristic, complemented
8#525—21no—
9#485—21no—
10#482—21no—
11#467C2211yescharacteristic
12#466C2211yescharacteristic
13#465—21no—
14#452—21no—
15#449—21no—
16C2×#107C2211yescharacteristic
17C2×C22⋊C8C2211yescharacteristic
18#380—21no—
19#367—21no—
20#366—21no—
21#358—21no—
22#357—21no—
23C22⋊C16C2211yescharacteristic
24C22⋊C16C2211yescharacteristic
25#340—21no—
26#340C2211yescharacteristic
27C2×#41—21no—
28C23×C4D411yescharacteristic
29C2×Q16D411yescharacteristic
30C2×SD16—21no—
31C2×D8D411yescharacteristic
32C22×C8D411yescharacteristic
33C22×C8C2311yescharacteristic
34#123—41no—
35C22⋊Q8—21no—
36C22⋊Q8—21no—
37#121—21no—
38#121—21no—
39#120—41no—
40C4×D4—41no—
41C4×D4—41no—
42#117—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C22⋊C4—41no—
46C2×C16—21no—
47C2×C16—21no—
48#107—21no—
49#107—21no—
50#107D411yescharacteristic
51#107D411yescharacteristic
52#106—21no—
53#103—21no—
54#103—21no—
55#102—21no—
56#102—21no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59#95—41no—
60C24D811yescharacteristic
61#41—81no—
62#41—81no—
63C2×Q8—21no—
64C2×D4—21no—
65C2×D4—41no—
66C2×D4—41no—
67C2×D4—41no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4D811yescharacteristic
76C22×C4C2×D411yescharacteristic
77Q16—41no—
78SD16—81no—
79D8—41no—
80C2×C8—21no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8C2×D411yescharacteristic
84C2×C8C2×D411yescharacteristic
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C22⋊C4—41no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—82no—
99C42—41no—
100C16—41no—
101C16—41no—
102C23—21no—
103C23—21no—
104C23—21no—
105C23—41no—
106C23—41no—
107C23—41no—
108C23C2×D811yescharacteristic
109Q8—41no—
110Q8—41no—
111D4—41no—
112D4—41no—
113D4—82no—
114D4—82no—
115D4—81no—
116D4—81no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—81no—
127C2×C4—82no—
128C2×C4—81no—
129C2×C4—81no—
130C2×C4—81no—
131C2×C4—81no—
132C2×C4—81no—
133C2×C4—81no—
134C2×C4#12011yescharacteristic
135C2×C4AGL(1,ℤ/8)11yescharacteristic
136C8—21no—
137C8—21no—
138C8—41no—
139C22—21no—
140C22—21no—
141C22—42no—
142C22—41no—
143C22—41no—
144C22—41no—
145C22—81no—
146C22—81no—
147C22—81no—
148C22—81no—
149C22#44711yescharacteristic
150C22#50811yescharacteristic
151C22#51011yescharacteristic
152C4—21no—
153C4—41no—
154C4—81no—
155C4—81no—
156C4—81no—
157C4—161no—
158C4—161no—
159C4#45511yescharacteristic
160C4#45811yescharacteristic
161C2—21no—
162C2—21no—
163C2—81no—
164C2—161no—
165C2#160711yescharacteristic
166C2#218911yescharacteristic
167C2#219011yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#3191C2
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. #2236C2semidirect
3. #2233C2semidirect
4. #2076C2semidirect
5. #1348C2semidirect
6. #1347C2semidirect
7. #1314C2semidirect

Maximal subgroups

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

subgroupindex
#31912
#22362
#22332
#20762
#13482
#13472
#13142