Finite Group Databasebeta

← #10148

Subgroups — #10148

Order 256.

307 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 65 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, complemented
2#3151C211yesmaximal, characteristic
3C2×#486C211yesmaximal, characteristic, complemented
4#1997C211yesmaximal, characteristic, complemented
5#1545C211yesmaximal, characteristic, complemented
6#1492C211yesmaximal, characteristic, complemented
7#1278C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—22no—
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×#127C2211yescharacteristic
12#498—21no—
13#496—22no—
14#493—21no—
15#492—21no—
16#486—22no—
17#486—21no—
18#486—22no—
19#481—22no—
20#476—22no—
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23C2×#102—22no—
24C4×M16C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#343—22no—
27#331—22no—
28D4⋊C8—22no—
29C2×#41—21no—
30C22×Q8—21no—
31C22×D4—21no—
32C22×D4—21no—
33C22×D4—21no—
34AGL(1,ℤ/8)—44no—
35AGL(1,ℤ/8)—44no—
36C2×SD16—24no—
37C2×D8—24no—
38C2×M16—21no—
39C2×M16D412yes—
40C22×C8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8—21no—
45C4⋊Q8D411yescharacteristic
46#127—21no—
47#127—21no—
48#127—21no—
49#127D412yes—
50#127D411yescharacteristic
51#124—21no—
52#121—21no—
53#121—21no—
54C4×D4—21no—
55C4×D4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D412yes—
58C2×C42C2311yescharacteristic
59#106—24no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8D412yes—
63#102—28no—
64#102—24no—
65C22⋊C8—21no—
66#97—21no—
67C4×C8—21no—
68C4×C8—21no—
69C4×C8—21no—
70C4×C8D412yes—
71C24—41no—
72#41—42no—
73#41—42no—
74C2×Q8—21no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×Q8—82no—
78C2×D4—21no—
79C2×D4—22no—
80C2×D4—21no—
81C2×D4—22no—
82C2×D4—21no—
83C2×D4—22no—
84C2×D4—44no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—42no—
90C2×D4—42no—
91C2×D4—82no—
92C22×C4—22no—
93C22×C4—41no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4#4111yescharacteristic
97SD16—48no—
98D8—48no—
99M16—22no—
100M16—42no—
101M16SD1614yes—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—42no—
106C2×C8—41no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C42C2×D411yescharacteristic
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C42C2×D411yescharacteristic
119C23—41no—
120C23—42no—
121C23—82no—
122C23—82no—
123C23—82no—
124C23#12111yescharacteristic
125Q8—42no—
126Q8—82no—
127Q8—82no—
128D4—44no—
129D4—42no—
130D4—44no—
131D4—42no—
132D4—42no—
133D4—82no—
134D4—82no—
135D4—82no—
136D4—82no—
137D4—81no—
138D4—81no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—82no—
146C2×C4#12011yescharacteristic
147C2×C4#12011yescharacteristic
148C2×C4#12011yescharacteristic
149C2×C4#12111yescharacteristic
150C2×C4#12111yescharacteristic
151C2×C4#12711yescharacteristic
152C2×C4C2×SD1612yes—
153C2×C4AGL(1,ℤ/8)12yes—
154C8—24no—
155C8—42no—
156C8—42no—
157C8—81no—
158C22—21no—
159C22—21no—
160C22—82no—
161C22—81no—
162C22—81no—
163C22—82no—
164C22—81no—
165C22—82no—
166C22—164no—
167C22#39211yescharacteristic
168C22#46012yes—
169C4—21no—
170C4—21no—
171C4—81no—
172C4—82no—
173C4#45012yes—
174C4#45412yes—
175C4#47212yes—
176C4#49211yescharacteristic
177C4#49611yescharacteristic
178C2—22no—
179C2—161no—
180C2—162no—
181C2#200311yescharacteristic
182C2#201411yescharacteristic
183C2#201811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3151C2
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
1. #3227C2semidirect
3. C2×#486C2semidirect
4. #1997C2semidirect
5. #1545C2semidirect
6. #1492C2semidirect
7. #1278C2semidirect

Maximal subgroups

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

subgroupindex
#32272
#31512
C2×#4862
#19972
#15452
#14922
#12782