Finite Group Databasebeta

← #7750

Subgroups — #7750

Order 256.

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

normal subgroups only

This group has 370 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#2679C211yesmaximal, characteristic
2#2023C212yesmaximal, complemented
3#2013C211yesmaximal, characteristic, complemented
4#1270C212yesmaximal, complemented
5#1270C211yesmaximal, characteristic, complemented
6C22×SD16—21no—
7C2×C4⋊Q8C2211yescharacteristic, complemented
8C2×#120—21no—
9C2×#120—22no—
10C2×#103—21no—
11C2×#102—22no—
12#394—22no—
13#394—21no—
14#390C2211yescharacteristic
15#390C2212yes—
16#340—22no—
17#336C2211yescharacteristic, complemented
18#336C2212yescomplemented
19C25—21no—
20C22×Q8—21no—
21C22×D4—22no—
22C22×D4—21no—
23C22×D4D411yescharacteristic
24C22×D4D412yes—
25C2×SD16—42no—
26C2×SD16—42no—
27C2×SD16—42no—
28C22×C8—21no—
29C22×C8—22no—
30C4⋊Q8—42no—
31#120—42no—
32#120—44no—
33#120—42no—
34#120—44no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4D412yes—
37C2×C4⋊C4D411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C42C2311yescharacteristic
44#103—42no—
45#102—44no—
46#95—21no—
47C24—21no—
48C24—22no—
49C24—42no—
50C24—44no—
51C24—81no—
52C24—82no—
53C24—81no—
54C2×Q8—22no—
55C2×Q8—41no—
56C2×Q8—41no—
57C2×Q8—41no—
58C2×Q8—41no—
59C2×Q8—41no—
60C2×D4—21no—
61C2×D4—22no—
62C2×D4—42no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4—42no—
66C2×D4—44no—
67C2×D4—44no—
68C2×D4—44no—
69C2×D4—81no—
70C2×D4—82no—
71C2×D4D812yes—
72C2×D4SD1614yescomplemented
73C22×C4—22no—
74C22×C4—21no—
75C22×C4—41no—
76C22×C4C2×D411yescharacteristic
77C22×C4C2×D412yes—
78SD16—48no—
79C2×C8—22no—
80C2×C8—41no—
81C2×C8—41no—
82C2×C8—42no—
83C2×C8—42no—
84C2×C8—42no—
85C4⋊C4—24no—
86C4⋊C4—41no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C22⋊C4—42no—
90C22⋊C4—44no—
91C22⋊C4—42no—
92C22⋊C4—42no—
93C22⋊C4—42no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C42—41no—
98C23—22no—
99C23—24no—
100C23—42no—
101C23—42no—
102C23—42no—
103C23—41no—
104C23—42no—
105C23—42no—
106C23—42no—
107C23—42no—
108C23—41no—
109C23—42no—
110C23—84no—
111C23—88no—
112C23—84no—
113C23—88no—
114C23—84no—
115C23#12011yescharacteristic
116Q8—42no—
117Q8—42no—
118Q8—81no—
119Q8—81no—
120D4—42no—
121D4—44no—
122D4—81no—
123D4—82no—
124D4—84no—
125D4—88no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—21no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—81no—
137C2×C4—81no—
138C2×C4C2×D811yescharacteristic
139C2×C4C2×SD1612yes—
140C2×C4AGL(1,ℤ/8)11yescharacteristic
141C2×C4AGL(1,ℤ/8)12yes—
142C8—42no—
143C8—82no—
144C22—42no—
145C22—44no—
146C22—44no—
147C22—44no—
148C22—44no—
149C22—816no—
150C22—82no—
151C22—82no—
152C22—82no—
153C22—81no—
154C22—82no—
155C22—81no—
156C22—82no—
157C22#44711yescharacteristic
158C22#45012yes—
159C22#45311yescharacteristic
160C22#45512yes—
161C22B(4,𝔽2)11yescharacteristic
162C4—21no—
163C4—21no—
164C4—22no—
165C4—22no—
166C4—82no—
167C4—81no—
168C4—82no—
169C2—82no—
170C2—84no—
171C2#159511yescharacteristic
172C2#160112yes—
173C2#162911yescharacteristic
174C2#163912yes—
175C2#165911yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radical#2679C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2023C2semidirect
3. #2013C2semidirect
4. #1270C2semidirect
5. #1270C2semidirect
7. C2×C4⋊Q8C22semidirect
17. #336C22semidirect
18. #336C22semidirect
72. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#26792
#20232
#20132
#12702