Finite Group Databasebeta

← #13001

Subgroups — #13001

Order 256.

286 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 79 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#3050C211yesmaximal, characteristic
2C22×#106C211yesmaximal, characteristic, complemented
3C2×#360C212yesmaximal, complemented
4#2112C211yesmaximal, characteristic, complemented
5#1386C212yesmaximal, complemented
6C2×#135C2211yescharacteristic
7C23×C8C2211yescharacteristic
8#525—21no—
9C22×C4⋊C4—21no—
10C2×M32C2212yes—
11#467C412yescomplemented
12#466C412yescomplemented
13#465—22no—
14#427C2211yescharacteristic
15C2×#106—21no—
16C2×#106—24no—
17C2×#106C2212yes—
18#416—21no—
19#360—24no—
20C22⋊C16—22no—
21C2×#41—21no—
22C23×C4—41no—
23C23×C4D411yescharacteristic
24#135—22no—
25#135—22no—
26C2×Q16C2×C411yescharacteristic, complemented
27C2×SD16—21no—
28C2×D8C2×C411yescharacteristic, complemented
29C22×C8—22no—
30C22×C8—22no—
31C22×C8C2×C412yes—
32C22×C8D411yescharacteristic
33C22×C8D412yes—
34C22×C8C2311yescharacteristic
35#123—41no—
36C22⋊Q8—22no—
37#121—22no—
38#120—41no—
39C4×D4—42no—
40#117—21no—
41C2×C4⋊C4—24no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4—42no—
44C2×C4⋊C4—41no—
45M32—22no—
46M32—22no—
47C2×C16—22no—
48#107C2×C412yescomplemented
49#106—24no—
50#106—22no—
51#106—21no—
52#106D414yescomplemented
53#103—22no—
54#102—22no—
55C24D811yescharacteristic
56#41—41no—
57#41—41no—
58#41—41no—
59#41—41no—
60C2×Q8—21no—
61C2×D4—21no—
62C2×D4—42no—
63C2×D4—41no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—44no—
69C22×C4—42no—
70C22×C4—41no—
71C22×C4—84no—
72C22×C4C22⋊C412yes—
73C22×C4D811yescharacteristic
74C22×C4SD1612yes—
75C22×C4C2×D411yescharacteristic
76Q16—22no—
77SD16—42no—
78D8—22no—
79C2×C8—21no—
80C2×C8—21no—
81C2×C8—24no—
82C2×C8—24no—
83C2×C8C22⋊C412yes—
84C2×C8C22×C411yescharacteristic
85C2×C8C2×D412yes—
86C2×C8C2×D411yescharacteristic
87C2×C8#4112yes—
88C4⋊C4—22no—
89C4⋊C4—24no—
90C4⋊C4—44no—
91C4⋊C4—42no—
92C4⋊C4—42no—
93C22⋊C4—42no—
94C22⋊C4—41no—
95C22⋊C4—42no—
96C42—41no—
97C16—42no—
98C23—24no—
99C23—21no—
100C23—21no—
101C23—41no—
102C23#10212yes—
103C23C2×D811yescharacteristic
104Q8—41no—
105Q8—41no—
106D4—41no—
107D4—41no—
108D4—84no—
109D4—81no—
110D4—81no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—42no—
118C2×C4—41no—
119C2×C4—44no—
120C2×C4—88no—
121C2×C4—84no—
122C2×C4—81no—
123C2×C4—81no—
124C2×C4#10212yes—
125C2×C4C2×C22⋊C411yescharacteristic
126C2×C4#12011yescharacteristic
127C2×C4#12311yescharacteristic
128C2×C4C2×SD1611yescharacteristic
129C2×C4AGL(1,ℤ/8)11yescharacteristic
130C2×C4#13711yescharacteristic
131C8—22no—
132C8C4×D412yes—
133C8#12112yes—
134C22—24no—
135C22—24no—
136C22—22no—
137C22—22no—
138C22—22no—
139C22—81no—
140C22—82no—
141C22#36014yescomplemented
142C22C2×#10211yescharacteristic
143C22#44711yescharacteristic
144C22#48011yescharacteristic
145C4—22no—
146C4—84no—
147C4—81no—
148C4—82no—
149C4#38611yescharacteristic
150C4#41711yescharacteristic
151C4#44811yescharacteristic
152C4#48111yescharacteristic
153C2—21no—
154C2—24no—
155C2—21no—
156C2—161no—
157C2#187411yescharacteristic
158C2C2×#36012yes—
subgroupG/H
center, socleC22C2×#102
commutatorC2×C8C22×C4
Frattini, cosocleC22×C8C23
prime radical#3050C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#106C2semidirect
3. C2×#360C2semidirect
4. #2112C2semidirect
5. #1386C2semidirect
11. #467C4semidirect
12. #466C4semidirect
26. C2×Q16C2×C4semidirect
28. C2×D8C2×C4semidirect
48. #107C2×C4semidirect
52. #106D4semidirect
141. C22#360semidirect

Maximal subgroups

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

subgroupindex
#30502
C22×#1062
C2×#3602
#21122
#13862