Finite Group Databasebeta

← #12107

Subgroups — #12107

Order 256.

214 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 39 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#3191C211yesmaximal, characteristic
2#2188C211yesmaximal, characteristic, complemented
3#2185C211yesmaximal, characteristic, complemented
4#2001C211yesmaximal, characteristic, complemented
5#1348C211yesmaximal, characteristic, complemented
6#1347C211yesmaximal, characteristic, complemented
7#1314C211yesmaximal, characteristic, complemented
8C22×Q16C2211yescharacteristic
9#525—21no—
10C2×C22⋊Q8—21no—
11C2×Q32—21no—
12C2×SD32—21no—
13#485—21no—
14#482—21no—
15#467C2211yescharacteristic
16#466C2211yescharacteristic
17#465—21no—
18#452—21no—
19#449—21no—
20C2×#103—21no—
21C2×C22⋊C8C2211yescharacteristic
22#358—21no—
23#358—21no—
24C22⋊C16C2211yescharacteristic
25C22⋊C16C2211yescharacteristic
26#340C2211yescharacteristic
27C2×#41—21no—
28C22×Q8—21no—
29C23×C4D411yescharacteristic
30C2×Q16—21no—
31C2×Q16—21no—
32C2×Q16—42no—
33C2×Q16D411yescharacteristic
34C2×SD16—21no—
35C2×D8D411yescharacteristic
36C22×C8D411yescharacteristic
37C22×C8C2311yescharacteristic
38#123—41no—
39C22⋊Q8—21no—
40C22⋊Q8—21no—
41C22⋊Q8—42no—
42C22⋊Q8—42no—
43#121—21no—
44#121—21no—
45#120—41no—
46C4×D4—41no—
47C4×D4—41no—
48#117—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—41no—
51C2×C22⋊C4—41no—
52Q32—42no—
53SD32—42no—
54C2×C16—21no—
55C2×C16—21no—
56#107D411yescharacteristic
57#107D411yescharacteristic
58#106—21no—
59#103—21no—
60#103—21no—
61#103—42no—
62#102—21no—
63#102—21no—
64C22⋊C8—21no—
65C22⋊C8—21no—
66C24D811yescharacteristic
67#41—81no—
68#41—81no—
69C2×Q8—22no—
70C2×Q8—21no—
71C2×Q8—41no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×D4—21no—
77C2×D4—41no—
78C2×D4—41no—
79C2×D4—41no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4D811yescharacteristic
88C22×C4C2×D411yescharacteristic
89Q16—22no—
90Q16—22no—
91Q16—41no—
92Q16—41no—
93Q16—41no—
94SD16—81no—
95D8—41no—
96C2×C8—21no—
97C2×C8—41no—
98C2×C8—41no—
99C2×C8C2×D411yescharacteristic
100C2×C8C2×D411yescharacteristic
101C4⋊C4—22no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—44no—
107C4⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—44no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C42—41no—
115C16—41no—
116C16—41no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23C2×D811yescharacteristic
124Q8—42no—
125Q8—41no—
126Q8—42no—
127Q8—41no—
128Q8—81no—
129Q8—81no—
130D4—41no—
131D4—41no—
132D4—82no—
133D4—82no—
134D4—81no—
135D4—81no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—81no—
148C2×C4—82no—
149C2×C4—81no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4#12011yescharacteristic
154C2×C4AGL(1,ℤ/8)11yescharacteristic
155C8—21no—
156C8—21no—
157C8—41no—
158C22—21no—
159C22—21no—
160C22—42no—
161C22—41no—
162C22—41no—
163C22—41no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22#44711yescharacteristic
169C22#50811yescharacteristic
170C22#51011yescharacteristic
171C4—21no—
172C4—41no—
173C4—82no—
174C4—81no—
175C4—82no—
176C4—81no—
177C4—81no—
178C4#45311yescharacteristic
179C4B(4,𝔽2)11yescharacteristic
180C2—21no—
181C2—21no—
182C2—81no—
183C2—161no—
184C2#159511yescharacteristic
185C2#218911yescharacteristic
186C2#219111yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
prime radical#3191C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2188C2semidirect
3. #2185C2semidirect
4. #2001C2semidirect
5. #1348C2semidirect
6. #1347C2semidirect
7. #1314C2semidirect

Maximal subgroups

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

subgroupindex
#31912
#21882
#21852
#20012
#13482
#13472
#13142