Finite Group Databasebeta

← #10464

Subgroups — #10464

Order 256.

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

normal subgroups only

This group has 226 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#3036C211yesmaximal, characteristic
2#1804C211yesmaximal, characteristic, complemented
3#1571C211yesmaximal, characteristic, complemented
4#1540C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic, complemented
6#1460C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9#532C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#463—22no—
12#461—21no—
13#459—21no—
14#431C2211yescharacteristic
15#431C2211yescharacteristic
16#423C2211yescharacteristic
17C2×C4⋊C8C2211yescharacteristic
18#417—21no—
19C2×#102—21no—
20#382C2211yescharacteristic
21#340—21no—
22#335—22no—
23#332—21no—
24#331—21no—
25#327—22no—
26#323—22no—
27C2×#41—21no—
28C22×D4D411yescharacteristic
29C23×C4—21no—
30AGL(1,ℤ/8)—42no—
31AGL(1,ℤ/8)—42no—
32C2×SD16—22no—
33C2×D8—22no—
34C2×M16—21no—
35C22×C8—21no—
36C4⋊Q8D411yescharacteristic
37#127D411yescharacteristic
38#124—21no—
39#121—21no—
40#121—21no—
41C4×D4—21no—
42C4×D4—21no—
43C4×D4—21no—
44C4×D4—21no—
45C4×D4—21no—
46C4×D4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4D411yescharacteristic
49C2×C22⋊C4—21no—
50C2×C42C2311yescharacteristic
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8D412yes—
57#103—22no—
58#102—22no—
59#102—42no—
60C22⋊C8—21no—
61C22⋊C8D412yes—
62C4×C8—21no—
63C4×C8—21no—
64#95—21no—
65#95—21no—
66C24—21no—
67#41—41no—
68#41—41no—
69#41—41no—
70#41—41no—
71C2×Q8—21no—
72C2×D4—21no—
73C2×D4—21no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4—21no—
80C2×D4—41no—
81C2×D4D812yes—
82C22×C4—21no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—82no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4#4111yescharacteristic
91SD16—44no—
92D8—44no—
93M16—41no—
94M16—41no—
95C2×C8—22no—
96C2×C8—22no—
97C2×C8—41no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C42C2×D411yescharacteristic
112C42C2×D411yescharacteristic
113C42#4111yescharacteristic
114C42#4111yescharacteristic
115C23—22no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—81no—
120C23—81no—
121C23#12111yescharacteristic
122Q8—41no—
123Q8—41no—
124D4—22no—
125D4—22no—
126D4—41no—
127D4—41no—
128D4—41no—
129D4—41no—
130D4—81no—
131D4—81no—
132D4—81no—
133D4—81no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—82no—
139C2×C4—82no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4#12011yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4#12311yescharacteristic
150C2×C4#12311yescharacteristic
151C2×C4#12411yescharacteristic
152C2×C4C2×D811yescharacteristic
153C2×C4#13511yescharacteristic
154C2×C4AGL(1,ℤ/8)11yescharacteristic
155C2×C4#13711yescharacteristic
156C8—42no—
157C8—42no—
158C8—81no—
159C8—81no—
160C22—21no—
161C22—21no—
162C22—42no—
163C22—82no—
164C22—82no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22#39411yescharacteristic
171C22#46611yescharacteristic
172C22#46811yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—81no—
178C4—81no—
179C4—161no—
180C4#44711yescharacteristic
181C4#45211yescharacteristic
182C4#48011yescharacteristic
183C4#48211yescharacteristic
184C2—21no—
185C2—21no—
186C2—82no—
187C2—161no—
188C2#203311yescharacteristic
189C2#204811yescharacteristic
190C2#205311yescharacteristic
subgroupG/H
center, socleC22#394
commutator, Frattini, cosocleC2×C42C23
prime radical#3036C2
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. #1804C2semidirect
3. #1571C2semidirect
4. #1540C2semidirect
5. #1517C2semidirect
6. #1460C2semidirect
7. #1286C2semidirect

Maximal subgroups

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

subgroupindex
#30362
#18042
#15712
#15402
#15172
#14602
#12862