Finite Group Databasebeta

← #12776

Subgroups — #12776

Order 256.

424 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 71 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×#555C211yesmaximal, characteristic
2C2×#356C211yesmaximal, characteristic, complemented
3C2×#355C212yesmaximal, complemented
4#1888C212yesmaximal, complemented
5#1886C211yesmaximal, characteristic, complemented
6#555—22no—
7#555—24no—
8#555C414yescomplemented
9C2×C4⋊Q8C2211yescharacteristic
10C2×#124—21no—
11C2×#124C2212yes—
12C2×#121—22no—
13#519C2211yescharacteristic
14C2×C4×D4—21no—
15C2×#117—22no—
16#419—21no—
17#418—22no—
18C2×#101C2211yescharacteristic
19C2×#101C2212yes—
20#356—22no—
21#355—24no—
22#339—21no—
23#339—22no—
24C22×Q8D411yescharacteristic
25C22×Q8C2311yescharacteristic
26C22×D4—21no—
27C22×D4—21no—
28C22×D4D412yes—
29C23×C4—22no—
30C2×M16—22no—
31C2×M16—21no—
32C4⋊Q8—22no—
33C4⋊Q8C2×C412yes—
34C4⋊Q8D412yes—
35#124—24no—
36#124—24no—
37#124—22no—
38#124—22no—
39#124C2×C414yes—
40#124D414yes—
41#121—28no—
42#121—28no—
43C4×Q8—42no—
44C4×D4—24no—
45C4×D4—24no—
46#117—28no—
47#117—28no—
48#117—22no—
49#117—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—22no—
52C2×C22⋊C4—22no—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—22no—
55C2×C42—21no—
56C2×C42D412yes—
57C2×C42D411yescharacteristic
58#103—42no—
59#102—44no—
60#101—22no—
61#101—22no—
62#101—22no—
63C24—22no—
64C2×Q8—21no—
65C2×Q8—41no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×Q8—41no—
69C2×Q8C22⋊C412yes—
70C2×Q8C22×C411yescharacteristic
71C2×Q8C2×D411yescharacteristic
72C2×Q8C2×D412yes—
73C2×D4—24no—
74C2×D4—24no—
75C2×D4—22no—
76C2×D4—44no—
77C2×D4—42no—
78C2×D4—42no—
79C2×D4—81no—
80C2×D4—81no—
81C2×D4C22⋊C414yes—
82C22×C4—21no—
83C22×C4—28no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—48no—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D412yes—
89M16—42no—
90M16—41no—
91M16—41no—
92M16—42no—
93C2×C8—42no—
94C2×C8—41no—
95C4⋊C4—24no—
96C4⋊C4—24no—
97C4⋊C4—24no—
98C4⋊C4—24no—
99C4⋊C4—22no—
100C22⋊C4—28no—
101C22⋊C4—24no—
102C22⋊C4—24no—
103C22⋊C4—24no—
104C22⋊C4—24no—
105C22⋊C4—41no—
106C42—24no—
107C42—22no—
108C42—21no—
109C42—42no—
110C42—41no—
111C42C22⋊C414yes—
112C42C22⋊C412yes—
113C23—24no—
114C23—44no—
115C23—44no—
116C23—42no—
117C23—42no—
118C23#12011yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—42no—
122Q8—42no—
123Q8—81no—
124D4—44no—
125D4—48no—
126D4—82no—
127D4—84no—
128C2×C4—24no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—24no—
135C2×C4—22no—
136C2×C4—416no—
137C2×C4—44no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4C2×C22⋊C411yescharacteristic
142C2×C4C2×C22⋊C412yes—
143C2×C4#12011yescharacteristic
144C2×C4#12012yes—
145C8—82no—
146C8—81no—
147C22—22no—
148C22—44no—
149C22—44no—
150C22—44no—
151C22—84no—
152C22—84no—
153C22#37911yescharacteristic
154C22B(4,𝔽2)11yescharacteristic
155C22#45811yescharacteristic
156C4—24no—
157C4—22no—
158C4—21no—
159C4—22no—
160C4—21no—
161C4—44no—
162C4—42no—
163C4—82no—
164C2—21no—
165C2—21no—
166C2—84no—
167C2#179011yescharacteristic
168C2#220412yes—
subgroupG/H
center, socleC22B(4,𝔽2)
commutatorC2×Q8C22×C4
Frattini, cosocleC22×Q8C23
prime radicalC2×#555C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#356C2semidirect
3. C2×#355C2semidirect
4. #1888C2semidirect
5. #1886C2semidirect
8. #555C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#3562
C2×#3552
#18882
#18862