Finite Group Databasebeta

← #10379

Subgroups — #10379

Order 256.

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

normal subgroups only

This group has 180 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#2605C211yesmaximal, characteristic
2#1986C211yesmaximal, characteristic
3#1515C211yesmaximal, characteristic
4#1506C211yesmaximal, characteristic
5#1300C211yesmaximal, characteristic
6#1288C211yesmaximal, characteristic
7#1285C211yesmaximal, characteristic
8#533C2211yescharacteristic
9#432C2211yescharacteristic
10#432C2211yescharacteristic
11C4×M16C2211yescharacteristic
12#400—21no—
13#395—21no—
14#391—21no—
15#391—21no—
16#387—21no—
17#387—21no—
18#383C2211yescharacteristic
19C4×C4⋊C4C2211yescharacteristic
20#376C2211yescharacteristic
21#339—21no—
22#339—21no—
23#328—21no—
24#328—21no—
25#328—21no—
26#328—21no—
27C22×Q8—21no—
28C2×M16—21no—
29C4⋊Q8Q811yescharacteristic, complemented
30C4⋊Q8Q811yescharacteristic, complemented
31#125—21no—
32C22⋊Q8—21no—
33C22⋊Q8—21no—
34C4×Q8—21no—
35C4×Q8—21no—
36#117—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C42—21no—
42C2×C42D411yescharacteristic
43C2×C42D411yescharacteristic
44C2×C42C2311yescharacteristic
45C4⋊C8—21no—
46C4⋊C8—21no—
47C22⋊C8—21no—
48C22⋊C8—21no—
49#97—21no—
50#97—21no—
51#97—21no—
52C4×C8—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95D411yescharacteristic
62#95D411yescharacteristic
63#95D411yescharacteristic
64#95D411yescharacteristic
65C2×Q8—21no—
66C2×Q8—21no—
67C2×Q8—42no—
68C2×Q8—42no—
69C2×Q8—81no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79M16—22no—
80M16—41no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—41no—
84C4⋊C4—21no—
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—21no—
88C4⋊C4—42no—
89C4⋊C4—42no—
90C4⋊C4—42no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—81no—
94C22⋊C4—41no—
95C42—41no—
96C42—41no—
97C42—42no—
98C42—41no—
99C42C2×Q811yescharacteristic
100C42#4111yescharacteristic
101C42#4111yescharacteristic
102C42#4111yescharacteristic
103C23#12011yescharacteristic
104Q8—81no—
105Q8—81no—
106Q8—84no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4C22⋊Q811yescharacteristic
132C2×C4C22⋊Q811yescharacteristic
133C2×C4C22⋊Q811yescharacteristic
134C2×C4#12411yescharacteristic
135C2×C4#12411yescharacteristic
136C2×C4#12411yescharacteristic
137C8—42no—
138C8—81no—
139C8—81no—
140C22—21no—
141C22—21no—
142C22#39311yescharacteristic
143C22#45511yescharacteristic
144C22#45611yescharacteristic
145C4—21no—
146C4—21no—
147C4—21no—
148C4—21no—
149C4—21no—
150C4—21no—
151C4—42no—
152C4—42no—
153C4—81no—
154C4—81no—
155C4—81no—
156C4—81no—
157C4—81no—
158C4—81no—
159C2—21no—
160C2—21no—
161C2#202811yescharacteristic
162C2#203111yescharacteristic
163C2#203111yescharacteristic
subgroupG/H
center, socleC22#393
commutator, Frattini, cosocleC2×C42C23
prime radicalC23#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
29. C4⋊Q8Q8semidirect
30. C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#26052
#19862
#15152
#15062
#13002
#12882
#12852