Finite Group Databasebeta

← #10333

Subgroups — #10333

Order 256.

457 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 457 conjugacy classes of subgroups — too many to draw. Only its 53 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#3195C211yesmaximal, characteristic, complemented
2#2435C212yesmaximal
3#1783C211yesmaximal, characteristic
4#1508C212yesmaximal, complemented
5#1297C211yesmaximal, characteristic, complemented
6C23×D4C2211yescharacteristic
7C2×#124—22no—
8C2×C22⋊Q8C2212yes—
9#483—22no—
10#470—22no—
11#451—22no—
12C2×#100—22no—
13#407C2211yescharacteristic
14#407C2211yescharacteristic
15#394—24no—
16#390—22no—
17#379—24no—
18C4×C22⋊C4C2212yes—
19#328—24no—
20C25D412yes—
21C22×Q8D412yes—
22C22×D4—22no—
23C22×D4—21no—
24C22×D4—22no—
25C22×D4—21no—
26C22×D4—44no—
27C23×C4C2311yescharacteristic
28C2×Q16—22no—
29C2×M16—21no—
30C2×M16D412yes—
31#124—48no—
32C22⋊Q8—22no—
33C22⋊Q8—22no—
34C22⋊Q8—22no—
35C22⋊Q8—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4D412yes—
38C2×C22⋊C4—28no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—28no—
42C2×C42—22no—
43#106—22no—
44#103—24no—
45#100—44no—
46C22⋊C8—22no—
47C22⋊C8—22no—
48#95D414yes—
49C24—24no—
50C24—22no—
51C24—44no—
52C24—48no—
53C24C2×D411yescharacteristic
54C2×Q8—22no—
55C2×Q8—22no—
56C2×Q8—44no—
57C2×D4—24no—
58C2×D4—28no—
59C2×D4—44no—
60C2×D4—42no—
61C2×D4—48no—
62C2×D4—44no—
63C2×D4—41no—
64C2×D4—42no—
65C2×D4—41no—
66C22×C4—24no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—42no—
70C22×C4C2×D412yes—
71C22×C4C2×D411yescharacteristic
72C22×C4C2×D412yes—
73C22×C4#4111yescharacteristic
74Q16—82no—
75M16—42no—
76M16—42no—
77C2×C8—22no—
78C2×C8—41no—
79C4⋊C4—22no—
80C4⋊C4—22no—
81C4⋊C4—42no—
82C4⋊C4—42no—
83C22⋊C4—44no—
84C22⋊C4—416no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C22⋊C4—88no—
88C42—44no—
89C23—24no—
90C23—28no—
91C23—24no—
92C23—41no—
93C23—44no—
94C23—42no—
95C23—48no—
96C23—42no—
97C23—48no—
98C23—84no—
99C23—816no—
100C23#12012yes—
101C23#12111yescharacteristic
102Q8—44no—
103Q8—82no—
104D4—44no—
105D4—48no—
106D4—44no—
107C2×C4—24no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—41no—
111C2×C4—44no—
112C2×C4—44no—
113C2×C4—44no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—44no—
117C2×C4—44no—
118C2×C4#12011yescharacteristic
119C2×C4#12112yes—
120C2×C4#12711yescharacteristic
121C8—42no—
122C8—81no—
123C22—22no—
124C22—22no—
125C22—22no—
126C22—21no—
127C22—44no—
128C22—416no—
129C22—44no—
130C22—88no—
131C22—84no—
132C22—84no—
133C22—84no—
134C22#39211yescharacteristic
135C22#45514yes—
136C4—21no—
137C4—22no—
138C4—84no—
139C4—82no—
140C4—82no—
141C4B(4,𝔽2)12yes—
142C2—24no—
143C2—41no—
144C2—84no—
145C2#200412yes—
146C2#202111yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radicalC23×D4C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3195C2semidirect
4. #1508C2semidirect
5. #1297C2semidirect

Maximal subgroups

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

subgroupindex
#31952
#24352
#17832
#15082
#12972