Finite Group Databasebeta

← #10492

Subgroups — #10492

Order 256.

212 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 49 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#3208C211yesmaximal, characteristic
2#1965C211yesmaximal, characteristic
3#1574C211yesmaximal, characteristic, complemented
4#1561C211yesmaximal, characteristic, complemented
5#1545C211yesmaximal, characteristic
6#1485C211yesmaximal, characteristic, complemented
7#1299C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#127C2211yescharacteristic
10#492—21no—
11#486—21no—
12#460—21no—
13#460—21no—
14#450—22no—
15#431C2211yescharacteristic
16#431C2211yescharacteristic
17#423C2211yescharacteristic
18#423C2211yescharacteristic
19C2×#100—22no—
20C4×M16C2211yescharacteristic
21#335—21no—
22#334—21no—
23#332—21no—
24#332—21no—
25#331—22no—
26Q8⋊C8—22no—
27C22×D4—21no—
28C22×D4—21no—
29C22×D4D412yes—
30C2×SD16—22no—
31C2×SD16—22no—
32C2×M16—21no—
33C2×M16—21no—
34C2×M16—21no—
35C4⋊Q8D411yescharacteristic
36C4⋊Q8D411yescharacteristic
37#127—21no—
38#127—21no—
39#127—21no—
40#127—21no—
41#125—21no—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44C4×Q8—21no—
45C4×Q8—21no—
46#117—21no—
47C2×C42C2311yescharacteristic
48C4⋊C8—21no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52C4⋊C8D412yes—
53C4⋊C8D411yescharacteristic
54C4⋊C8D411yescharacteristic
55#102—22no—
56#102—22no—
57#100—44no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60#97—21no—
61C4×C8—21no—
62C4×C8—21no—
63C4×C8—21no—
64C24—22no—
65C2×Q8—21no—
66C2×Q8—21no—
67C2×D4—22no—
68C2×D4—22no—
69C2×D4—42no—
70C2×D4—42no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—81no—
76C2×D4—82no—
77C2×D4—81no—
78C22×C4C2×D411yescharacteristic
79C22×C4#4111yescharacteristic
80C22×C4#4111yescharacteristic
81SD16—82no—
82SD16—82no—
83M16—22no—
84M16—41no—
85M16—42no—
86M16—81no—
87C2×C8—21no—
88C2×C8—21no—
89C2×C8—22no—
90C2×C8—41no—
91C2×C8—41no—
92C2×C8—41no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C22⋊C4—41no—
100C42—41no—
101C42C2×D411yescharacteristic
102C42C2×D411yescharacteristic
103C42C2×D411yescharacteristic
104C42#4111yescharacteristic
105C23—44no—
106C23—42no—
107C23—82no—
108C23—82no—
109C23#12311yescharacteristic
110Q8—41no—
111Q8—41no—
112Q8—41no—
113Q8—41no—
114D4—42no—
115D4—42no—
116D4—82no—
117D4—82no—
118D4—82no—
119D4—81no—
120D4—82no—
121D4—81no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—22no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4#12011yescharacteristic
130C2×C4#12111yescharacteristic
131C2×C4#12111yescharacteristic
132C2×C4#12111yescharacteristic
133C2×C4#12311yescharacteristic
134C2×C4#12411yescharacteristic
135C8—42no—
136C8—41no—
137C8—41no—
138C8—42no—
139C8—81no—
140C8—81no—
141C22—21no—
142C22—21no—
143C22—21no—
144C22—82no—
145C22—82no—
146C22—82no—
147C22—164no—
148C22#39411yescharacteristic
149C4—21no—
150C4—21no—
151C4—21no—
152C4—81no—
153C4—81no—
154C4#45311yescharacteristic
155C4#45511yescharacteristic
156C4#47111yescharacteristic
157C4#47112yes—
158C4#47211yescharacteristic
159C2—41no—
160C2—162no—
161C2#203711yescharacteristic
162C2#204611yescharacteristic
163C2#205111yescharacteristic
subgroupG/H
center, socleC22#394
commutator, Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
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
3. #1574C2semidirect
4. #1561C2semidirect
6. #1485C2semidirect
7. #1299C2semidirect

Maximal subgroups

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

subgroupindex
#32082
#19652
#15742
#15612
#15452
#14852
#12992