Finite Group Databasebeta

← #10497

Subgroups — #10497

Order 256.

174 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 45 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#3234C211yesmaximal, characteristic
2#1998C211yesmaximal, characteristic, complemented
3#1569C211yesmaximal, characteristic, complemented
4#1557C211yesmaximal, characteristic
5#1528C211yesmaximal, characteristic, complemented
6#1525C211yesmaximal, characteristic
7#1291C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×#125C2211yescharacteristic
10#501—21no—
11#490—21no—
12#481—21no—
13#480—21no—
14#478—21no—
15#478—21no—
16#432C2211yescharacteristic
17#432C2211yescharacteristic
18#423C2211yescharacteristic
19C4×M16C2211yescharacteristic
20C4×M16C2211yescharacteristic
21#343—21no—
22#343—21no—
23#335—21no—
24#334—21no—
25#332—21no—
26#331—21no—
27#330—21no—
28#330—21no—
29#329—21no—
30#329—21no—
31C2×#41—21no—
32C2×M16—21no—
33C2×M16—21no—
34C2×M16—21no—
35C4⋊Q8D411yescharacteristic
36#127D411yescharacteristic
37#125—21no—
38#125—21no—
39#125—21no—
40#125—21no—
41#124—21no—
42#121—21no—
43#121—21no—
44C4×D4—21no—
45C4×D4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4D411yescharacteristic
49C2×C4⋊C4D411yescharacteristic
50C2×C42C2311yescharacteristic
51#107—21no—
52#107—21no—
53#106—21no—
54#106—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8D411yescharacteristic
59C4⋊C8D411yescharacteristic
60#102—21no—
61#102—21no—
62#102—21no—
63#102—21no—
64C22⋊C8—21no—
65C22⋊C8—21no—
66#97—21no—
67#97—21no—
68#97—21no—
69#97D411yescharacteristic
70#97D411yescharacteristic
71C4×C8—21no—
72C4×C8—21no—
73#41—82no—
74C2×Q8—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—41no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—41no—
82C22×C4C2×D411yescharacteristic
83C22×C4#4111yescharacteristic
84C22×C4#4111yescharacteristic
85M16—22no—
86M16—42no—
87M16—41no—
88M16—42no—
89C2×C8—21no—
90C2×C8—21no—
91C2×C8—21no—
92C2×C8—21no—
93C2×C8—41no—
94C2×C8—41no—
95C2×C8—41no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C42C2×D411yescharacteristic
109C42C2×D411yescharacteristic
110C42C2×D411yescharacteristic
111C42#4111yescharacteristic
112C23—41no—
113C23#12311yescharacteristic
114Q8—81no—
115D4—41no—
116D4—41no—
117D4—81no—
118D4—81no—
119D4—82no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—82no—
128C2×C4—81no—
129C2×C4—81no—
130C2×C4#12011yescharacteristic
131C2×C4#12111yescharacteristic
132C2×C4#12111yescharacteristic
133C2×C4#12111yescharacteristic
134C2×C4#12311yescharacteristic
135C2×C4#12411yescharacteristic
136C8—22no—
137C8—41no—
138C8—42no—
139C8—41no—
140C8—41no—
141C8—81no—
142C8—81no—
143C22—21no—
144C22—21no—
145C22—21no—
146C22—81no—
147C22—81no—
148C22—81no—
149C22#39411yescharacteristic
150C4—21no—
151C4—21no—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—81no—
156C4—81no—
157C4—81no—
158C4#45411yescharacteristic
159C4#45611yescharacteristic
160C2—41no—
161C2—161no—
162C2#203811yescharacteristic
163C2#204811yescharacteristic
164C2#205011yescharacteristic
subgroupG/H
center, socleC22#394
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1998C2semidirect
3. #1569C2semidirect
5. #1528C2semidirect
7. #1291C2semidirect

Maximal subgroups

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

subgroupindex
#32342
#19982
#15692
#15572
#15282
#15252
#12912