Finite Group Databasebeta

← #26920

Subgroups — #26920

Order 256.

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

normal subgroups only

This group has 404 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#3081C212yesmaximal, complemented
2#2948C212yesmaximal, complemented
3#2045C212yesmaximal, complemented
4#2027C211yesmaximal, characteristic, complemented
5#2011C211yesmaximal, characteristic
6#1931C212yesmaximal, complemented
7#1922C212yesmaximal, complemented
8#1887C212yesmaximal, complemented
9#1856C211yesmaximal, characteristic, complemented
10C22×D8C2212yes—
11#544C2211yescharacteristic
12C2×#127C2211yescharacteristic
13#518C2212yes—
14#518C2211yescharacteristic
15#478—22no—
16#478—22no—
17#476—22no—
18#474—22no—
19#442—22no—
20#442—22no—
21C4×D8—22no—
22C4×D8—22no—
23#428C2212yes—
24#427C2212yes—
25#424C2212yes—
26#418C2212yes—
27#418C2212yes—
28#418C2212yes—
29C2×#102C2212yes—
30C2×#100C2211yescharacteristic
31C2×#100C2212yes—
32#405C2212yes—
33#390C2211yescharacteristic
34#343C2211yescharacteristic
35#341C2212yes—
36#339C2212yes—
37#339C2212yes—
38#336C2211yescharacteristic
39C22×D4—41no—
40C22×D4C2312yes—
41C22×D4C2311yescharacteristic
42C2×D8—24no—
43C2×D8—24no—
44C2×D8—22no—
45C2×D8—22no—
46C2×M16C2312yes—
47C2×M16C2312yes—
48C22×C8C2312yes—
49C4⋊Q8—22no—
50#127—24no—
51#125—22no—
52C22⋊Q8—41no—
53C4×D4—22no—
54C4×D4—22no—
55C4×D4—22no—
56C4×D4—22no—
57C4×D4—22no—
58C4×D4—22no—
59#117C2312yes—
60#117C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C22⋊C4—41no—
63C2×C22⋊C4—42no—
64C2×C22⋊C4—41no—
65C2×C42C2311yescharacteristic
66#107—22no—
67#107D412yescomplemented
68#107D412yescomplemented
69#106—22no—
70#106D412yescomplemented
71#106D412yescomplemented
72C4⋊C8—22no—
73C4⋊C8—22no—
74#102—22no—
75#102—22no—
76#102—22no—
77#102—22no—
78#102—22no—
79#102—22no—
80#102—22no—
81#102—22no—
82#100—24no—
83#100—22no—
84#97—22no—
85C4×C8—22no—
86C24—21no—
87C24—22no—
88C2×Q8—41no—
89C2×D4—22no—
90C2×D4—22no—
91C2×D4—22no—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—21no—
97C2×D4—44no—
98C2×D4—82no—
99C2×D4—82no—
100C2×D4#4114yes—
101C2×D4#4112yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4C2411yescharacteristic
109D8—28no—
110D8—28no—
111M16—22no—
112M16—22no—
113M16—22no—
114M16—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8C2×D412yes—
120C2×C8C2×D412yes—
121C4⋊C4—22no—
122C4⋊C4—41no—
123C4⋊C4—42no—
124C4⋊C4C2×D412yes—
125C4⋊C4C2×D412yes—
126C4⋊C4C2×D412yes—
127C4⋊C4C2×D412yes—
128C22⋊C4—22no—
129C22⋊C4—22no—
130C22⋊C4—42no—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C22⋊C4—82no—
134C42—22no—
135C42—22no—
136C42—22no—
137C23—22no—
138C23—24no—
139C23—42no—
140C23—41no—
141C23—42no—
142C23—41no—
143C23—81no—
144C23—82no—
145C23—82no—
146C23—81no—
147C23H(5,𝔽2)11yescharacteristic
148Q8—161no—
149D4—24no—
150D4—24no—
151D4—24no—
152D4—42no—
153D4—42no—
154D4—42no—
155D4—88no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—81no—
165C2×C4—162no—
166C2×C4—161no—
167C2×C4—161no—
168C2×C4—162no—
169C2×C4C22×D412yes—
170C2×C4C2×#4111yescharacteristic
171C2×C4C2×#4112yes—
172C2×C42−1+411yescharacteristic
173C8—22no—
174C8—22no—
175C8—42no—
176C8—42no—
177C22—21no—
178C22—21no—
179C22—42no—
180C22—44no—
181C22—84no—
182C22—82no—
183C22—82no—
184C22—81no—
185C22—81no—
186C22—82no—
187C22—161no—
188C22—162no—
189C22#53911yescharacteristic
190C22D4211yescharacteristic
191C22#55111yescharacteristic
192C4—42no—
193C4—42no—
194C4—42no—
195C4—42no—
196C4—42no—
197C4—81no—
198C4#52512yes—
199C4#54812yes—
200C2—21no—
201C2—21no—
202C2—84no—
203C2—82no—
204C2#246311yescharacteristic
205C2#328811yescharacteristic
206C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#2011C2
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. #3081C2semidirect
2. #2948C2semidirect
3. #2045C2semidirect
4. #2027C2semidirect
6. #1931C2semidirect
7. #1922C2semidirect
8. #1887C2semidirect
9. #1856C2semidirect
67. #107D4semidirect
68. #107D4semidirect
70. #106D4semidirect
71. #106D4semidirect

Maximal subgroups

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

subgroupindex
#30812
#29482
#20452
#20272
#20112
#19312
#19222
#18872
#18562