Finite Group Databasebeta

← #28940

Subgroups — #28940

Order 256.

407 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 111 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#3136C212yesmaximal, complemented
2#3062C212yesmaximal, complemented
3#2061C211yesmaximal, characteristic, complemented
4#2040C211yesmaximal, characteristic, complemented
5#2002C212yesmaximal, complemented
6#1961C211yesmaximal, characteristic, complemented
7#1931C212yesmaximal, complemented
8#1907C212yesmaximal, complemented
9#1765C212yesmaximal, complemented
10C2×AGL(1,ℤ/8)C2212yes—
11C2×#131C2212yes—
12#544C2211yescharacteristic
13#537C2211yescharacteristic
14#535C2212yes—
15C2×#127C2211yescharacteristic
16#490—22no—
17#489—22no—
18#488—22no—
19#486—22no—
20#469—22no—
21#468—22no—
22#466—22no—
23#465—22no—
24#428C2212yes—
25C2×#104C2212yes—
26#418C2212yes—
27#417C2212yes—
28#416C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#100C2212yes—
32#405C2212yes—
33C2×#97C2211yescharacteristic
34#382C2211yescharacteristic
35#343C2211yescharacteristic
36#343C2211yescharacteristic
37#341C2212yes—
38#339C2212yes—
39C2×#41C2312yes—
40C22×D4—41no—
41C22×D4C2312yes—
42AGL(1,ℤ/8)—24no—
43AGL(1,ℤ/8)—42no—
44C2×SD16—22no—
45C2×D8—22no—
46#131—24no—
47#131—42no—
48C2×M16—22no—
49C2×M16C2312yes—
50C2×M16C2312yes—
51C22×C8—22no—
52C22×C8C2312yes—
53C4⋊Q8—22no—
54#127—22no—
55#127—22no—
56#127—41no—
57#125—22no—
58#124—22no—
59C22⋊Q8—22no—
60C22⋊Q8—41no—
61#121—22no—
62#121—22no—
63#121—22no—
64#121—41no—
65#120—42no—
66#117C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C42C2311yescharacteristic
70#107—22no—
71#106—22no—
72#104—24no—
73#103—22no—
74#103D412yescomplemented
75#103D412yescomplemented
76#102—22no—
77#102—22no—
78#102—22no—
79#102—22no—
80#102—22no—
81#102—22no—
82#102—22no—
83#102D412yescomplemented
84#102D412yescomplemented
85#100—24no—
86#97—22no—
87#97—41no—
88C4×C8—22no—
89#95—41no—
90C24—22no—
91#41—22no—
92#41—22no—
93#41—22no—
94#41—22no—
95C2×Q8—41no—
96C2×Q8C2×D412yes—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—44no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—41no—
108C2×D4—82no—
109C2×D4—82no—
110C2×D4C2×D412yes—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4C2411yescharacteristic
116SD16—44no—
117D8—44no—
118M16—22no—
119M16—22no—
120M16—22no—
121M16—44no—
122M16#4114yes—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—44no—
128C2×C8C2×D412yes—
129C2×C8C2×D412yes—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—42no—
135C4⋊C4C2×D412yes—
136C4⋊C4C2×D412yes—
137C22⋊C4—22no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—42no—
141C42—22no—
142C42—21no—
143C42#4112yes—
144C23—22no—
145C23—42no—
146C23—42no—
147C23—42no—
148C23—82no—
149C23—82no—
150C23H(5,𝔽2)11yescharacteristic
151Q8—22no—
152Q8—22no—
153Q8—161no—
154D4—22no—
155D4—22no—
156D4—42no—
157D4—42no—
158D4—42no—
159D4—42no—
160D4—42no—
161D4—42no—
162D4—88no—
163D4—162no—
164D4—162no—
165D4—161no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4C22×D412yes—
179C2×C4C2×#4111yescharacteristic
180C2×C4C2×#4112yes—
181C2×C4H(5,𝔽2)11yescharacteristic
182C8—22no—
183C8—22no—
184C8—24no—
185C8—42no—
186C22—21no—
187C22—21no—
188C22—42no—
189C22—42no—
190C22—42no—
191C22—84no—
192C22—82no—
193C22—82no—
194C22—82no—
195C22—162no—
196C22#53811yescharacteristic
197C22D4211yescharacteristic
198C22#55511yescharacteristic
199C4—42no—
200C4—42no—
201C4—42no—
202C4—42no—
203C4—81no—
204C4—81no—
205C4#53212yes—
206C4#54612yes—
207C2—21no—
208C2—21no—
209C2—82no—
210C2—162no—
211C2#255511yescharacteristic
212C2#328811yescharacteristic
213C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3136C2semidirect
2. #3062C2semidirect
3. #2061C2semidirect
4. #2040C2semidirect
5. #2002C2semidirect
6. #1961C2semidirect
7. #1931C2semidirect
8. #1907C2semidirect
9. #1765C2semidirect
74. #103D4semidirect
75. #103D4semidirect
83. #102D4semidirect
84. #102D4semidirect

Maximal subgroups

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

subgroupindex
#31362
#30622
#20612
#20402
#20022
#19612
#19312
#19072
#17652