Finite Group Databasebeta

← #12707

Subgroups — #12707

Order 256.

608 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 73 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#3484C211yesmaximal, characteristic
2#2127C214yesmaximal, complemented
3#1781C212yesmaximal, complemented
4C2×H(5,𝔽2)C411yescharacteristic, complemented
5C23×D4C2211yescharacteristic
6#546—24no—
7D42—24no—
8#535C2212yes—
9#534—22no—
10#525C412yescomplemented
11C2×#120—22no—
12#518C411yescharacteristic, complemented
13C2×#99—22no—
14C2×#99C2214yescomplemented
15#379—22no—
16#354—24no—
17#353—24no—
18C25D412yes—
19H(5,𝔽2)—44no—
20C2×#41—22no—
21C2×#41C2×C412yes—
22C22×D4—22no—
23C22×D4—24no—
24C22×D4—22no—
25C22×D4—24no—
26C22×D4—44no—
27C22×D4C2×C411yescharacteristic
28C22×D4D412yes—
29C22×D4C2311yescharacteristic
30C23×C4C2×C411yescharacteristic
31#127D414yes—
32#124D414yes—
33#123—24no—
34#123—22no—
35C22⋊Q8—22no—
36#121—28no—
37#121—22no—
38#121—24no—
39#120—28no—
40#120—24no—
41#120—22no—
42#120—24no—
43C4×D4—24no—
44C4×D4—24no—
45#117C2×C412yes—
46C2×C22⋊C4—24no—
47C2×C22⋊C4—22no—
48C2×C22⋊C4—24no—
49C2×C22⋊C4—44no—
50#99—28no—
51#99—44no—
52C24—22no—
53C24—24no—
54C24—24no—
55C24—42no—
56C24—84no—
57C24C22⋊C411yescharacteristic
58C24C22⋊C412yes—
59C24C2×D412yes—
60#41—44no—
61#41—48no—
62C2×Q8C22⋊C412yescomplemented
63C2×D4—24no—
64C2×D4—24no—
65C2×D4—24no—
66C2×D4—24no—
67C2×D4—22no—
68C2×D4—24no—
69C2×D4—28no—
70C2×D4—22no—
71C2×D4—42no—
72C2×D4—42no—
73C2×D4—416no—
74C2×D4—48no—
75C2×D4—44no—
76C2×D4—44no—
77C2×D4—88no—
78C2×D4C22⋊C412yescomplemented
79C2×D4C2×D414yes—
80C22×C4—22no—
81C22×C4—24no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—42no—
85C22×C4—44no—
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22×C411yescharacteristic
88C4⋊C4—24no—
89C4⋊C4—22no—
90C22⋊C4—24no—
91C22⋊C4—24no—
92C22⋊C4—24no—
93C22⋊C4—22no—
94C22⋊C4—28no—
95C22⋊C4—44no—
96C22⋊C4—44no—
97C22⋊C4—88no—
98C42C22⋊C414yescomplemented
99C23—21no—
100C23—22no—
101C23—22no—
102C23—22no—
103C23—22no—
104C23—22no—
105C23—22no—
106C23—42no—
107C23—48no—
108C23—48no—
109C23—48no—
110C23—44no—
111C23—84no—
112C23—82no—
113C23—82no—
114C23—84no—
115C23—84no—
116C23—84no—
117C23—84no—
118C23#9914yes—
119C23C2×C22⋊C411yescharacteristic
120C23#12014yes—
121Q8—42no—
122D4—28no—
123D4—44no—
124D4—42no—
125D4—48no—
126D4—48no—
127D4—48no—
128D4—44no—
129D4—88no—
130C2×C4—22no—
131C2×C4—24no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—44no—
136C2×C4—44no—
137C2×C4—44no—
138C2×C4—84no—
139C2×C4—84no—
140C2×C4—88no—
141C2×C4C2×C22⋊C412yes—
142C22—22no—
143C22—24no—
144C22—24no—
145C22—24no—
146C22—41no—
147C22—44no—
148C22—44no—
149C22—44no—
150C22—44no—
151C22—41no—
152C22—42no—
153C22—44no—
154C22—41no—
155C22—42no—
156C22—82no—
157C22—82no—
158C22—816no—
159C22—164no—
160C22#37911yescharacteristic
161C22C2×#9912yes—
162C4—22no—
163C4—44no—
164C4—41no—
165C4—42no—
166C4—164no—
167C2—21no—
168C2—22no—
169C2—44no—
170C2—42no—
171C2—44no—
172C2—82no—
173C2#178111yescharacteristic
subgroupG/H
center, socleC2#1781
commutatorC22×C4C22×C4
Frattini, cosocleC22×D4C23
prime radical#3484C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2127C2semidirect
3. #1781C2semidirect
4. C2×H(5,𝔽2)C4semidirect
10. #525C4semidirect
12. #518C4semidirect
14. C2×#99C22semidirect
62. C2×Q8C22⋊C4semidirect
78. C2×D4C22⋊C4semidirect
98. C42C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#34842
#21272
#17812