Finite Group Databasebeta

← #26924

Subgroups — #26924

Order 256.

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

normal subgroups only

This group has 370 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#3082C212yesmaximal, complemented
2#2949C212yesmaximal
3#2046C212yesmaximal
4#2027C211yesmaximal, characteristic
5#2013C211yesmaximal, characteristic
6#1930C212yesmaximal, complemented
7#1922C212yesmaximal, complemented
8#1886C212yesmaximal, complemented
9#1856C211yesmaximal, characteristic
10C22×SD16C2212yes—
11#544C2211yescharacteristic
12C2×C4⋊Q8C2211yescharacteristic
13#519C2212yes—
14#518C2211yescharacteristic
15#479—22no—
16#479—22no—
17#477—22no—
18#475—22no—
19#440—22no—
20#440—22no—
21C4×SD16—22no—
22C4×SD16—22no—
23#428C2212yes—
24#427C2212yes—
25#424C2212yes—
26#419C2212yes—
27#419C2212yes—
28#418C2212yes—
29C2×#103C2212yes—
30C2×#101C2212yes—
31C2×#100C2211yescharacteristic
32#405C2212yes—
33#390C2211yescharacteristic
34#343C2211yescharacteristic
35#341C2212yes—
36#339C2212yes—
37#339C2212yes—
38#336C2211yescharacteristic
39C22×Q8C2312yes—
40C22×D4C2311yescharacteristic
41C2×SD16—24no—
42C2×SD16—24no—
43C2×SD16—22no—
44C2×SD16—22no—
45C2×M16C2312yes—
46C2×M16C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—22no—
49C4⋊Q8—24no—
50#125—22no—
51C22⋊Q8—41no—
52C4×Q8—22no—
53C4×Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×D4—22no—
57C4×D4—22no—
58#117—42no—
59#117C2312yes—
60#117C2312yes—
61C2×C4⋊C4—41no—
62C2×C4⋊C4C2311yescharacteristic
63C2×C22⋊C4—41no—
64C2×C22⋊C4—41no—
65C2×C42C2311yescharacteristic
66#107—22no—
67#107D412yes—
68#107D412yes—
69#106—22no—
70#106D412yes—
71#106D412yes—
72C4⋊C8—22no—
73C4⋊C8—22no—
74#103—22no—
75#103—22no—
76#103—22no—
77#103—22no—
78#103—22no—
79#103—22no—
80#102—22no—
81#102—22no—
82#101—24no—
83#100—22no—
84#97—22no—
85C4×C8—22no—
86C24—21no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—41no—
93C2×Q8#4114yes—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—21no—
97C2×D4#4112yes—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4C2411yescharacteristic
104SD16—28no—
105SD16—28no—
106M16—22no—
107M16—22no—
108M16—22no—
109M16—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8C2×D412yes—
115C2×C8C2×D412yes—
116C4⋊C4—22no—
117C4⋊C4—44no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—41no—
121C4⋊C4—42no—
122C4⋊C4C2×D412yes—
123C4⋊C4C2×D412yes—
124C4⋊C4C2×D412yes—
125C4⋊C4C2×D412yes—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—42no—
129C22⋊C4—82no—
130C42—22no—
131C42—22no—
132C42—22no—
133C42—42no—
134C42—42no—
135C23—22no—
136C23—41no—
137C23—41no—
138C23—81no—
139C23—81no—
140C23H(5,𝔽2)11yescharacteristic
141Q8—24no—
142Q8—24no—
143Q8—42no—
144Q8—42no—
145Q8—161no—
146D4—24no—
147D4—42no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—81no—
160C2×C4—161no—
161C2×C4—161no—
162C2×C4C22×D412yes—
163C2×C4C2×#4111yescharacteristic
164C2×C4C2×#4112yes—
165C2×C42−1+411yescharacteristic
166C8—22no—
167C8—22no—
168C8—42no—
169C8—42no—
170C22—21no—
171C22—21no—
172C22—42no—
173C22—82no—
174C22—81no—
175C22—81no—
176C22—161no—
177C22#53911yescharacteristic
178C22D4211yescharacteristic
179C22#55111yescharacteristic
180C4—42no—
181C4—44no—
182C4—42no—
183C4—42no—
184C4—42no—
185C4—42no—
186C4—81no—
187C4#52512yes—
188C4#54812yes—
189C2—21no—
190C2—21no—
191C2—82no—
192C2#246311yescharacteristic
193C2#328911yescharacteristic
194C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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. #3082C2semidirect
6. #1930C2semidirect
7. #1922C2semidirect
8. #1886C2semidirect

Maximal subgroups

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

subgroupindex
#30822
#29492
#20462
#20272
#20132
#19302
#19222
#18862
#18562