Finite Group Databasebeta

← #12221

Subgroups — #12221

Order 256.

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

normal subgroups only

This group has 196 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#2804C211yesmaximal, characteristic
2#2027C211yesmaximal, characteristic
3#1617C212yesmaximal, complemented
4#1353C212yesmaximal, complemented
5#1318C211yesmaximal, characteristic
6#544C2211yescharacteristic
7#484—22no—
8#418—21no—
9#400C2212yes—
10#393—22no—
11#391—21no—
12#390C2211yescharacteristic
13#386—21no—
14#385—22no—
15#343C2211yescharacteristic
16#339—21no—
17#328—22no—
18#324C2212yes—
19C22×Q8—21no—
20C22×D4D411yescharacteristic
21C23×C4—21no—
22C2×M16—21no—
23C4⋊Q8—41no—
24#125—41no—
25C22⋊Q8—22no—
26#117—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4D412yes—
29C2×C4⋊C4C2311yescharacteristic
30C2×C22⋊C4—22no—
31C2×C22⋊C4—22no—
32C2×C22⋊C4—21no—
33C2×C42D411yescharacteristic
34#107—42no—
35#103—42no—
36#102—41no—
37#102—41no—
38C22⋊C8—22no—
39#95—22no—
40#95—22no—
41#95—22no—
42#95D412yes—
43C24—21no—
44C2×Q8—22no—
45C2×Q8—42no—
46C2×Q8—81no—
47C2×Q8—81no—
48C2×D4—21no—
49C2×D4—41no—
50C2×D4—41no—
51C2×D4Q1612yes—
52C22×C4—21no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—82no—
56C22×C4—81no—
57C22×C4C2×D411yescharacteristic
58C22×C4C2×D412yes—
59M16—41no—
60M16—41no—
61C2×C8—42no—
62C2×C8—41no—
63C4⋊C4—21no—
64C4⋊C4—21no—
65C4⋊C4—22no—
66C4⋊C4—41no—
67C4⋊C4—42no—
68C4⋊C4—42no—
69C4⋊C4—81no—
70C22⋊C4—44no—
71C22⋊C4—41no—
72C22⋊C4—82no—
73C22⋊C4—81no—
74C42—41no—
75C42—41no—
76C23—22no—
77C23—42no—
78C23—42no—
79C23—41no—
80C23—41no—
81C23#12011yescharacteristic
82Q8—82no—
83Q8—84no—
84D4—42no—
85D4—81no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—41no—
91C2×C4—41no—
92C2×C4—42no—
93C2×C4—42no—
94C2×C4—42no—
95C2×C4—42no—
96C2×C4—42no—
97C2×C4—42no—
98C2×C4—41no—
99C2×C4—41no—
100C2×C4—42no—
101C2×C4—84no—
102C2×C4C2×Q1611yescharacteristic
103C2×C4#13711yescharacteristic
104C8—81no—
105C8—82no—
106C22—21no—
107C22—21no—
108C22—44no—
109C22—42no—
110C22—82no—
111C22—81no—
112C22—81no—
113C22#45111yescharacteristic
114C22#45511yescharacteristic
115C22B(4,𝔽2)11yescharacteristic
116C4—21no—
117C4—21no—
118C4—42no—
119C4—42no—
120C4—82no—
121C4—81no—
122C4—81no—
123C4—82no—
124C2—21no—
125C2—21no—
126C2—82no—
127C2#160211yescharacteristic
128C2#219811yescharacteristic
129C2#220211yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radicalC22×D4D4
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
3. #1617C2semidirect
4. #1353C2semidirect

Maximal subgroups

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

subgroupindex
#28042
#20272
#16172
#13532
#13182