Finite Group Databasebeta

← #10491

Subgroups — #10491

Order 256.

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

normal subgroups only

This group has 183 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#3227C211yesmaximal, characteristic
2#1997C211yesmaximal, characteristic, complemented
3#1574C211yesmaximal, characteristic, complemented
4#1560C211yesmaximal, characteristic
5#1546C211yesmaximal, characteristic, complemented
6#1484C211yesmaximal, characteristic
7#1299C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10#498—21no—
11#486—21no—
12#481—22no—
13#476—21no—
14#476—21no—
15#431C2211yescharacteristic
16#431C2211yescharacteristic
17#423C2211yescharacteristic
18#423C2211yescharacteristic
19C4×M16C2211yescharacteristic
20#343—22no—
21#335—21no—
22#334—21no—
23#333—21no—
24#332—22no—
25#331—21no—
26D4⋊C8—22no—
27C2×#41—21no—
28C22×Q8—21no—
29C2×M16—21no—
30C2×M16—21no—
31C2×M16—21no—
32C4⋊Q8—21no—
33C4⋊Q8—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8D411yescharacteristic
37#127D411yescharacteristic
38#124—21no—
39#121—21no—
40#121—21no—
41C4×D4—21no—
42C4×D4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4D412yes—
45C2×C42C2311yescharacteristic
46#106—22no—
47#106—22no—
48C4⋊C8—21no—
49C4⋊C8—21no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52C4⋊C8D412yes—
53C4⋊C8D411yescharacteristic
54C4⋊C8D411yescharacteristic
55#102—22no—
56#102—22no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59#97—21no—
60C4×C8—21no—
61C4×C8—21no—
62C4×C8—21no—
63#41—82no—
64C2×Q8—21no—
65C2×Q8—41no—
66C2×Q8—41no—
67C2×Q8—81no—
68C2×Q8—81no—
69C2×D4—21no—
70C2×D4—21no—
71C2×D4—21no—
72C2×D4—41no—
73C22×C4—22no—
74C22×C4—41no—
75C22×C4C2×D411yescharacteristic
76C22×C4#4111yescharacteristic
77C22×C4#4111yescharacteristic
78M16—22no—
79M16—41no—
80M16—42no—
81M16—81no—
82C2×C8—21no—
83C2×C8—21no—
84C2×C8—22no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8—41no—
88C4⋊C4—22no—
89C4⋊C4—21no—
90C4⋊C4—22no—
91C4⋊C4—21no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C42C2×D411yescharacteristic
97C42C2×D411yescharacteristic
98C42C2×D411yescharacteristic
99C42#4111yescharacteristic
100C23—41no—
101C23#12311yescharacteristic
102Q8—81no—
103Q8—82no—
104Q8—81no—
105Q8—81no—
106D4—41no—
107D4—41no—
108D4—41no—
109D4—41no—
110D4—81no—
111D4—82no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—22no—
116C2×C4—21no—
117C2×C4—41no—
118C2×C4—42no—
119C2×C4—82no—
120C2×C4—82no—
121C2×C4#12011yescharacteristic
122C2×C4#12111yescharacteristic
123C2×C4#12111yescharacteristic
124C2×C4#12111yescharacteristic
125C2×C4#12311yescharacteristic
126C2×C4#12411yescharacteristic
127C8—42no—
128C8—41no—
129C8—41no—
130C8—42no—
131C8—81no—
132C8—81no—
133C22—21no—
134C22—21no—
135C22—21no—
136C22—81no—
137C22—81no—
138C22—81no—
139C22#39411yescharacteristic
140C4—21no—
141C4—21no—
142C4—21no—
143C4—82no—
144C4—81no—
145C4#45411yescharacteristic
146C4#45611yescharacteristic
147C4#47111yescharacteristic
148C4#47112yes—
149C4#47211yescharacteristic
150C2—41no—
151C2—161no—
152C2#203811yescharacteristic
153C2#204611yescharacteristic
154C2#205111yescharacteristic
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. #1997C2semidirect
3. #1574C2semidirect
5. #1546C2semidirect
7. #1299C2semidirect

Maximal subgroups

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

subgroupindex
#32272
#19972
#15742
#15602
#15462
#14842
#12992