Finite Group Databasebeta

← #8507

Subgroups — #8507

Order 256.

368 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 79 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×#563C211yesmaximal, characteristic
2#1992C212yesmaximal, complemented
3#1959C211yesmaximal, characteristic
4C2×#330C212yesmaximal, complemented
5C2×#329C211yesmaximal, characteristic
6#563—24no—
7#563—22no—
8#563C414yescomplemented
9C2×C4⋊Q8—21no—
10C2×#126—22no—
11C2×#125C2212yes—
12C2×#124C2211yescharacteristic
13C2×#123—21no—
14C2×C22⋊Q8—22no—
15C2×#103—21no—
16C2×#102—21no—
17C2×#97C2211yescharacteristic
18C2×#97C2212yes—
19#384C2211yescharacteristic
20#340—22no—
21#340—22no—
22#330—24no—
23#329—22no—
24C22×Q8—21no—
25C22×Q8D411yescharacteristic
26C22×D4D411yescharacteristic
27C23×C4—21no—
28C22×C8—22no—
29C22×C8—21no—
30C4⋊Q8—24no—
31C4⋊Q8—24no—
32#126—28no—
33#126—28no—
34#125—22no—
35#125—22no—
36#125C2×C414yes—
37#125D414yes—
38#124—22no—
39#124C2×C412yes—
40#124D412yes—
41#123—24no—
42#123—24no—
43C22⋊Q8—28no—
44C22⋊Q8—28no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4D412yes—
51C2×C4⋊C4D412yes—
52C2×C22⋊C4—22no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—22no—
55C2×C42—21no—
56C2×C42C2311yescharacteristic
57#103—42no—
58#102—42no—
59#97—22no—
60#97—22no—
61#97—22no—
62C24—21no—
63C2×Q8—24no—
64C2×Q8—21no—
65C2×Q8—41no—
66C2×Q8—41no—
67C2×Q8—81no—
68C2×Q8—81no—
69C2×Q8C22⋊C412yes—
70C2×D4—21no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4C22⋊C412yes—
74C22×C4—22no—
75C22×C4—21no—
76C22×C4—22no—
77C22×C4—24no—
78C22×C4—41no—
79C22×C4—81no—
80C22×C4—81no—
81C22×C4C2×D412yes—
82C22×C4C2×D411yescharacteristic
83C2×C8—22no—
84C2×C8—24no—
85C2×C8—42no—
86C2×C8—41no—
87C2×C8—42no—
88C2×C8—41no—
89C4⋊C4—28no—
90C4⋊C4—24no—
91C4⋊C4—24no—
92C4⋊C4—24no—
93C4⋊C4—24no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—44no—
99C4⋊C4C22⋊C414yes—
100C4⋊C4C22⋊C414yes—
101C22⋊C4—28no—
102C22⋊C4—24no—
103C22⋊C4—24no—
104C22⋊C4—24no—
105C42—24no—
106C42C22×C411yescharacteristic
107C42C2×D412yes—
108C42C2×D411yescharacteristic
109C23—22no—
110C23—41no—
111C23—42no—
112C23—41no—
113C23—42no—
114C23#12011yescharacteristic
115Q8—42no—
116Q8—81no—
117Q8—84no—
118D4—42no—
119D4—81no—
120C2×C4—21no—
121C2×C4—22no—
122C2×C4—21no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—22no—
126C2×C4—24no—
127C2×C4—42no—
128C2×C4—41no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—84no—
134C2×C4—81no—
135C2×C4—82no—
136C2×C4C2×C22⋊C412yes—
137C2×C4C2×C22⋊C411yescharacteristic
138C2×C4#12012yes—
139C2×C4#12011yescharacteristic
140C8—42no—
141C8—44no—
142C22—42no—
143C22—44no—
144C22—81no—
145C22—81no—
146C22—82no—
147C22#37911yescharacteristic
148C22#45311yescharacteristic
149C22#45412yes—
150C22#45511yescharacteristic
151C22#45612yes—
152C4—21no—
153C4—21no—
154C4—22no—
155C4—22no—
156C4—44no—
157C4—42no—
158C4—44no—
159C4—161no—
160C2—82no—
161C2#165711yescharacteristic
162C2#166011yescharacteristic
163C2#166412yes—
164C2#179211yescharacteristic
165C2#179312yes—
subgroupG/H
center, socleC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1992C2semidirect
4. C2×#330C2semidirect
8. #563C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5632
#19922
#19592
C2×#3302
C2×#3292