Finite Group Databasebeta

← #12357

Subgroups — #12357

Order 256.

141 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 141 conjugacy classes of subgroups — too many to draw. Only its 43 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#3383C211yesmaximal, characteristic
2#2241C211yesmaximal, characteristic
3#2209C211yesmaximal, characteristic
4#1670C211yesmaximal, characteristic
5#1364C211yesmaximal, characteristic
6Q8⋊C16C211yesmaximal, characteristic
7#1332C211yesmaximal, characteristic
8#557—21no—
9C2×Q32—21no—
10#500C2211yescharacteristic
11#499—21no—
12#494C2211yescharacteristic
13#479—22no—
14#475—21no—
15#475—21no—
16C8×Q8C2211yescharacteristic
17C4×Q16—21no—
18C4×Q16C2211yescharacteristic
19C4⋊C16C2211yescharacteristic
20#358—21no—
21#358—22no—
22C4×C16C2211yescharacteristic
23#335C2211yescharacteristic
24#332—21no—
25C2×Q16—21no—
26C2×Q16—21no—
27C2×Q16D411yescharacteristic
28C4⋊Q8—21no—
29C4⋊Q8—21no—
30C4⋊Q8—41no—
31#125—22no—
32#125—42no—
33C4×Q8—22no—
34C4×Q8—21no—
35C4×Q8—41no—
36C4×Q8D411yescharacteristic
37Q32—42no—
38C2×C16—21no—
39C2×C16—21no—
40#107—21no—
41#107D411yescharacteristic
42#107D412yes—
43#106—22no—
44#106—41no—
45C4⋊C8—21no—
46C4⋊C8—21no—
47C4⋊C8D411yescharacteristic
48#103—22no—
49#103—21no—
50#103—41no—
51C4×C8—21no—
52C4×C8C2311yescharacteristic
53C2×Q8—21no—
54C2×Q8—41no—
55C2×Q8—41no—
56C2×Q8D811yescharacteristic
57Q16—21no—
58Q16—42no—
59Q16—41no—
60Q16—41no—
61Q16SD1612yes—
62C2×C8—21no—
63C2×C8—41no—
64C2×C8C2×D411yescharacteristic
65C2×C8C2×D411yescharacteristic
66C4⋊C4—21no—
67C4⋊C4—21no—
68C4⋊C4—22no—
69C4⋊C4—42no—
70C4⋊C4—42no—
71C4⋊C4—42no—
72C4⋊C4—41no—
73C4⋊C4—41no—
74C4⋊C4—41no—
75C4⋊C4—41no—
76C4⋊C4D811yescharacteristic
77C42—21no—
78C42—42no—
79C42—41no—
80C42C2×D411yescharacteristic
81C16—41no—
82C16—41no—
83Q8—22no—
84Q8—21no—
85Q8—21no—
86Q8—41no—
87Q8—81no—
88Q8—81no—
89Q8—81no—
90Q8—81no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—41no—
94C2×C4—41no—
95C2×C4—42no—
96C2×C4—81no—
97C2×C4#12011yescharacteristic
98C2×C4C2×D811yescharacteristic
99C2×C4AGL(1,ℤ/8)11yescharacteristic
100C8—21no—
101C8—41no—
102C8—81no—
103C8C2×SD1611yescharacteristic
104C8#13711yescharacteristic
105C22#44711yescharacteristic
106C4—21no—
107C4—42no—
108C4—41no—
109C4—41no—
110C4—82no—
111C4—81no—
112C4—161no—
113C4#44811yescharacteristic
114C4#45411yescharacteristic
115C4#50811yescharacteristic
116C4#51011yescharacteristic
117C2#162711yescharacteristic
118C2#218911yescharacteristic
119C2#219511yescharacteristic
subgroupG/H
center, socle, prime radicalC22#447
commutator, Frattini, cosocleC4×C8C23
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#33832
#22412
#22092
#16702
#13642
Q8⋊C162
#13322