Finite Group Databasebeta

← #10466

Subgroups — #10466

Order 256.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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#3037C211yesmaximal, characteristic
2#1805C211yesmaximal, characteristic, complemented
3#1571C211yesmaximal, characteristic, complemented
4#1540C211yesmaximal, characteristic
5#1517C211yesmaximal, characteristic
6#1461C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#532C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11#464—22no—
12#462—21no—
13#460—21no—
14#431C2211yescharacteristic
15#431C2211yescharacteristic
16#423C2211yescharacteristic
17C2×C4⋊C8C2211yescharacteristic
18#417—21no—
19C2×#103—21no—
20#382C2211yescharacteristic
21#340—21no—
22#335—22no—
23#332—21no—
24#331—21no—
25#327—22no—
26#324—22no—
27C2×#41—21no—
28C22×Q8D411yescharacteristic
29#137—42no—
30#137—42no—
31C2×Q16—22no—
32C2×SD16—22no—
33C2×M16—21no—
34C22×C8—21no—
35C4⋊Q8D411yescharacteristic
36#127D411yescharacteristic
37#124—21no—
38#121—21no—
39#121—21no—
40C4×Q8—21no—
41C4×Q8—21no—
42C4×Q8—21no—
43C4×Q8—21no—
44C4×D4—21no—
45C4×D4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4D411yescharacteristic
49C2×C42—21no—
50C2×C42C2311yescharacteristic
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8D412yes—
57#103—22no—
58#103—42no—
59#102—22no—
60C22⋊C8—21no—
61C22⋊C8D412yes—
62C4×C8—21no—
63C4×C8—21no—
64#95—21no—
65#95—21no—
66#41—41no—
67#41—41no—
68#41—41no—
69#41—41no—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8—21no—
75C2×Q8—21no—
76C2×Q8D812yes—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4—21no—
80C2×D4—41no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4#4111yescharacteristic
88Q16—44no—
89SD16—44no—
90M16—41no—
91M16—41no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—41no—
95C2×C8—41no—
96C2×C8—41no—
97C2×C8—41no—
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—
108C42—41no—
109C42—41no—
110C42C2×D411yescharacteristic
111C42C2×D411yescharacteristic
112C42#4111yescharacteristic
113C42#4111yescharacteristic
114C23—41no—
115C23#12111yescharacteristic
116Q8—22no—
117Q8—22no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—41no—
122D4—41no—
123D4—41no—
124D4—81no—
125D4—81no—
126D4—81no—
127D4—81no—
128C2×C4—22no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—81no—
136C2×C4—81no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4#12011yescharacteristic
142C2×C4#12111yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12311yescharacteristic
145C2×C4#12311yescharacteristic
146C2×C4#12411yescharacteristic
147C2×C4C2×D811yescharacteristic
148C2×C4#13511yescharacteristic
149C2×C4AGL(1,ℤ/8)11yescharacteristic
150C2×C4#13711yescharacteristic
151C8—42no—
152C8—42no—
153C8—81no—
154C8—81no—
155C22—21no—
156C22—21no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22#39411yescharacteristic
161C22#46611yescharacteristic
162C22#46811yescharacteristic
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—42no—
168C4—81no—
169C4—81no—
170C4—161no—
171C4#44711yescharacteristic
172C4#45211yescharacteristic
173C4#48011yescharacteristic
174C4#48211yescharacteristic
175C2—21no—
176C2—21no—
177C2—161no—
178C2#203311yescharacteristic
179C2#204911yescharacteristic
180C2#205311yescharacteristic
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. #1805C2semidirect
3. #1571C2semidirect
6. #1461C2semidirect
7. #1286C2semidirect

Maximal subgroups

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

subgroupindex
#30372
#18052
#15712
#15402
#15172
#14612
#12862