Finite Group Databasebeta

← #12104

Subgroups — #12104

Order 256.

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

normal subgroups only

This group has 240 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, complemented
2#2187C211yesmaximal, characteristic, complemented
3#2184C211yesmaximal, characteristic, complemented
4#1999C211yesmaximal, characteristic, complemented
5#1348C211yesmaximal, characteristic, complemented
6#1347C211yesmaximal, characteristic, complemented
7#1314C211yesmaximal, characteristic, complemented
8C22×D8C2211yescharacteristic
9#525—21no—
10C2×#121—21no—
11C2×SD32—21no—
12C2×D16—21no—
13#485—21no—
14#482—21no—
15#467C2211yescharacteristic
16#466C2211yescharacteristic
17#465—21no—
18#452—21no—
19#449—21no—
20C2×#102—21no—
21C2×C22⋊C8C2211yescharacteristic
22#357—21no—
23#357—21no—
24C22⋊C16C2211yescharacteristic
25C22⋊C16C2211yescharacteristic
26#340C2211yescharacteristic
27C2×#41—21no—
28C22×D4—21no—
29C22×D4—41no—
30C23×C4D411yescharacteristic
31C2×Q16D411yescharacteristic, complemented
32C2×SD16—21no—
33C2×D8—21no—
34C2×D8—21no—
35C2×D8—42no—
36C2×D8D411yescharacteristic, complemented
37C22×C8D411yescharacteristic
38C22×C8C2311yescharacteristic
39#123—41no—
40C22⋊Q8—21no—
41C22⋊Q8—21no—
42#121—21no—
43#121—21no—
44#121—42no—
45#121—42no—
46#120—41no—
47C4×D4—41no—
48C4×D4—41no—
49#117—21no—
50C2×C4⋊C4—21no—
51C2×C22⋊C4—41no—
52SD32—42no—
53D16—42no—
54C2×C16—21no—
55C2×C16—21no—
56#107D411yescharacteristic, complemented
57#107D411yescharacteristic, complemented
58#106—21no—
59#103—21no—
60#103—21no—
61#102—21no—
62#102—21no—
63#102—42no—
64C22⋊C8—21no—
65C22⋊C8—21no—
66C24—41no—
67C24D811yescharacteristic
68#41—81no—
69#41—81no—
70C2×Q8—21no—
71C2×D4—22no—
72C2×D4—21no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4—41no—
77C2×D4—41no—
78C2×D4—44no—
79C2×D4—41no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—82no—
83C2×D4—82no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4D811yescharacteristic
91C22×C4C2×D411yescharacteristic
92Q16—41no—
93SD16—81no—
94D8—22no—
95D8—22no—
96D8—41no—
97D8—41no—
98D8—41no—
99C2×C8—21no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8C2×D411yescharacteristic
103C2×C8C2×D411yescharacteristic
104C4⋊C4—22no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—44no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C42—41no—
117C16—41no—
118C16—41no—
119C23—21no—
120C23—21no—
121C23—21no—
122C23—41no—
123C23—41no—
124C23—42no—
125C23—41no—
126C23—81no—
127C23—81no—
128C23—81no—
129C23—81no—
130C23—81no—
131C23—81no—
132C23C2×D811yescharacteristic
133Q8—41no—
134Q8—41no—
135D4—42no—
136D4—41no—
137D4—42no—
138D4—41no—
139D4—81no—
140D4—82no—
141D4—88no—
142D4—82no—
143D4—81no—
144D4—81no—
145D4—81no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—81no—
157C2×C4—82no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4#12011yescharacteristic
161C2×C4AGL(1,ℤ/8)11yescharacteristic
162C8—21no—
163C8—21no—
164C8—41no—
165C22—21no—
166C22—21no—
167C22—42no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—82no—
173C22—82no—
174C22—81no—
175C22—82no—
176C22—81no—
177C22—81no—
178C22—161no—
179C22—161no—
180C22—161no—
181C22—161no—
182C22#44711yescharacteristic
183C22#50811yescharacteristic
184C22#50911yescharacteristic
185C4—21no—
186C4—41no—
187C4—82no—
188C4—81no—
189C4—81no—
190C4—81no—
191C4#45311yescharacteristic
192C4B(4,𝔽2)11yescharacteristic
193C2—21no—
194C2—21no—
195C2—81no—
196C2—161no—
197C2—162no—
198C2#159511yescharacteristic
199C2#218611yescharacteristic
200C2#219011yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC22×C8C23
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
1. #3191C2semidirect
2. #2187C2semidirect
3. #2184C2semidirect
4. #1999C2semidirect
5. #1348C2semidirect
6. #1347C2semidirect
7. #1314C2semidirect
31. C2×Q16D4semidirect
36. C2×D8D4semidirect
56. #107D4semidirect
57. #107D4semidirect

Maximal subgroups

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

subgroupindex
#31912
#21872
#21842
#19992
#13482
#13472
#13142