Finite Group Databasebeta

← #10388

Subgroups — #10388

Order 256.

297 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 45 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#2608C211yesmaximal, characteristic, complemented
2#1789C212yesmaximal
3#1782C211yesmaximal, characteristic
4#1306C211yesmaximal, characteristic, complemented
5#1284C212yesmaximal, complemented
6C22×#41C2211yescharacteristic
7C2×#104—22no—
8C2×#104—22no—
9C2×#99—22no—
10#407C2212yes—
11#396—22no—
12#389—22no—
13#387—22no—
14#381C2211yescharacteristic
15#381C2211yescharacteristic
16C4×C22⋊C4C2212yes—
17C2×#41—22no—
18C2×#41—22no—
19C2×#41—44no—
20C22×Q8Q811yescharacteristic
21C22×D4—21no—
22C22×D4Q811yescharacteristic
23C23×C4—21no—
24C23×C4C2311yescharacteristic
25C2×M16—22no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4—22no—
28C2×C4⋊C4—22no—
29C2×C22⋊C4—22no—
30C2×C22⋊C4—21no—
31C2×C22⋊C4D412yes—
32C2×C42D412yes—
33C2×C42D412yes—
34#104—44no—
35#104—44no—
36#99—44no—
37C22⋊C8—22no—
38C22⋊C8—22no—
39#95—22no—
40#95—22no—
41#95—22no—
42#95—22no—
43C24—21no—
44C24C2×D411yescharacteristic
45#41—24no—
46#41—24no—
47#41—42no—
48#41—42no—
49#41—44no—
50#41—44no—
51C2×Q8—22no—
52C2×Q8—42no—
53C2×D4—24no—
54C2×D4—22no—
55C2×D4—42no—
56C2×D4—42no—
57C2×D4—41no—
58C2×D4—41no—
59C22×C4—22no—
60C22×C4—21no—
61C22×C4—22no—
62C22×C4—22no—
63C22×C4—42no—
64C22×C4—42no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4—41no—
68C22×C4C2×D412yes—
69C22×C4C2×Q811yescharacteristic
70C22×C4#4112yes—
71C22×C4#4111yescharacteristic
72M16—42no—
73M16—42no—
74C2×C8—42no—
75C4⋊C4—44no—
76C4⋊C4—44no—
77C4⋊C4—82no—
78C22⋊C4—44no—
79C22⋊C4—42no—
80C22⋊C4—42no—
81C42—42no—
82C42—42no—
83C23—22no—
84C23—22no—
85C23—42no—
86C23—41no—
87C23—41no—
88C23—82no—
89C23#12011yescharacteristic
90C23C22⋊Q811yescharacteristic
91C23#12411yescharacteristic
92Q8—24no—
93Q8—42no—
94D4—24no—
95D4—42no—
96D4—44no—
97D4—44no—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—24no—
101C2×C4—22no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—44no—
106C2×C4—44no—
107C2×C4—42no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—81no—
118C2×C4—82no—
119C2×C4C22⋊Q812yes—
120C2×C4#12412yes—
121C8—82no—
122C22—22no—
123C22—21no—
124C22—21no—
125C22—22no—
126C22—22no—
127C22—42no—
128C22—82no—
129C22—82no—
130C22—82no—
131C22—82no—
132C22#39311yescharacteristic
133C22B(4,𝔽2)12yes—
134C4—22no—
135C4—22no—
136C4—44no—
137C4—42no—
138C4—82no—
139C4—82no—
140C4—161no—
141C2—22no—
142C2—41no—
143C2—41no—
144C2—82no—
145C2#202911yescharacteristic
146C2#203112yes—
subgroupG/H
center, socleC22#393
commutator, Frattini, cosocleC23×C4C23
prime radicalC22×#41C22
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. #2608C2semidirect
4. #1306C2semidirect
5. #1284C2semidirect

Maximal subgroups

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

subgroupindex
#26082
#17892
#17822
#13062
#12842