Finite Group Databasebeta

← #19001

Subgroups — #19001

Order 256.

480 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#562C211yesmaximal, characteristic, complemented
2#3213C212yesmaximal, complemented
3#3142C211yesmaximal, characteristic, complemented
4#1662C214yesmaximal, complemented
5#1656C214yesmaximal, complemented
6#1534C211yesmaximal, characteristic, complemented
7#1527C212yesmaximal, complemented
8#562—22no—
9#562—24no—
10#562C2214yes—
11#532C2211yescharacteristic
12C2×#125C2211yescharacteristic
13C2×#124C2212yes—
14C2×#123—22no—
15C2×C22⋊Q8—22no—
16C2×#121—22no—
17#496C2214yescomplemented
18#491C2212yescomplemented
19#490C2212yescomplemented
20#489C2214yescomplemented
21#488—22no—
22#463—24no—
23#448—22no—
24#447—22no—
25#432C2212yes—
26#417—22no—
27C4×M16C2211yescharacteristic
28#330C2214yescomplemented
29#329C2218yescomplemented
30C2×#41—21no—
31C22×Q8D412yes—
32C22×D4—22no—
33C22×D4D412yes—
34C23×C4—22no—
35C2×SD16—24no—
36C2×D8—24no—
37C2×M16—21no—
38C4⋊Q8C2311yescharacteristic
39#127C2311yescharacteristic
40#125—22no—
41#125D412yes—
42#125C2312yes—
43#124—22no—
44#124—22no—
45#124—21no—
46#124D414yes—
47#124C2314yes—
48#123—28no—
49#123—28no—
50C22⋊Q8—28no—
51C22⋊Q8—28no—
52#121—28no—
53#121—28no—
54#121—22no—
55C4×D4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4D412yes—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C42C2311yescharacteristic
63C4⋊C8—22no—
64#103—24no—
65#103—24no—
66#102—24no—
67#102—24no—
68C22⋊C8—22no—
69#97C2312yes—
70#97C2314yes—
71C4×C8—21no—
72C24—22no—
73#41—44no—
74C2×Q8—22no—
75C2×Q8—21no—
76C2×Q8—82no—
77C2×Q8C2×D414yes—
78C2×D4—28no—
79C2×D4—22no—
80C2×D4—21no—
81C2×D4—22no—
82C2×D4—41no—
83C2×D4—82no—
84C2×D4—84no—
85C2×D4C2×D414yes—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—28no—
89C22×C4—41no—
90C22×C4—84no—
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D412yes—
93SD16—84no—
94D8—84no—
95M16—42no—
96C2×C8—22no—
97C2×C8—24no—
98C4⋊C4—28no—
99C4⋊C4—24no—
100C4⋊C4—22no—
101C4⋊C4—24no—
102C4⋊C4—22no—
103C4⋊C4C2×D414yes—
104C22⋊C4—28no—
105C22⋊C4—24no—
106C22⋊C4—24no—
107C22⋊C4—24no—
108C22⋊C4—24no—
109C22⋊C4—42no—
110C42C2×D411yescharacteristic
111C42C2×D412yes—
112C42C2411yescharacteristic
113C23—24no—
114C23—44no—
115C23—44no—
116C23—44no—
117C23—41no—
118C23#12011yescharacteristic
119Q8—44no—
120Q8—42no—
121Q8—82no—
122D4—44no—
123D4—44no—
124D4—42no—
125D4—82no—
126D4—88no—
127D4—82no—
128C2×C4—22no—
129C2×C4—24no—
130C2×C4—24no—
131C2×C4—42no—
132C2×C4—44no—
133C2×C4—44no—
134C2×C4—41no—
135C2×C4—82no—
136C2×C4—88no—
137C2×C4#12011yescharacteristic
138C2×C4#12012yes—
139C2×C4C22×D411yescharacteristic
140C2×C4C22×D412yes—
141C8—42no—
142C8—44no—
143C22—22no—
144C22—44no—
145C22—44no—
146C22—44no—
147C22—88no—
148C22—81no—
149C22—82no—
150C22#45412yes—
151C22C2×#12011yescharacteristic
152C4—21no—
153C4—21no—
154C4—22no—
155C4—22no—
156C4—44no—
157C4—44no—
158C4—81no—
159C2—22no—
160C2—84no—
161C2—161no—
162C2C2×#45411yescharacteristic
163C2#301912yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#562C2semidirect
2. #3213C2semidirect
3. #3142C2semidirect
4. #1662C2semidirect
5. #1656C2semidirect
6. #1534C2semidirect
7. #1527C2semidirect
17. #496C22semidirect
18. #491C22semidirect
19. #490C22semidirect
20. #489C22semidirect
28. #330C22semidirect
29. #329C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5622
#32132
#31422
#16622
#16562
#15342
#15272