Finite Group Databasebeta

← #12377

Subgroups — #12377

Order 256.

172 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 41 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#3277C211yesmaximal, characteristic
2#2243C211yesmaximal, characteristic, complemented
3#2225C211yesmaximal, characteristic, complemented
4#1695C211yesmaximal, characteristic, complemented
5#1360C211yesmaximal, characteristic, complemented
6#1334C211yesmaximal, characteristic, complemented
7#1330C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#548—21no—
10#500C2211yescharacteristic
11#495C2211yescharacteristic
12#476—21no—
13#468—21no—
14#464—21no—
15#451—21no—
16#449—21no—
17C4×Q16C2211yescharacteristic
18#436C2211yescharacteristic
19#367—21no—
20C4⋊C16C2211yescharacteristic
21#358—21no—
22#358—21no—
23#357—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#333—21no—
27C2×#41—21no—
28C22×Q8—21no—
29#137—42no—
30#137—42no—
31C2×Q16—21no—
32C2×Q16D411yescharacteristic
33C2×Q16D411yescharacteristic
34C2×SD16—21no—
35C2×SD16—21no—
36C2×D8D411yescharacteristic
37C2×M16—21no—
38C4⋊Q8—21no—
39#124—21no—
40#124—41no—
41C22⋊Q8—21no—
42C22⋊Q8—41no—
43#121—21no—
44#121—41no—
45C4×Q8—21no—
46C4×D4—41no—
47C4×D4D411yescharacteristic
48C2×C16—21no—
49C2×C16—21no—
50#107—21no—
51#107—21no—
52#107D411yescharacteristic
53#106—41no—
54C4⋊C8—21no—
55C4⋊C8D411yescharacteristic
56#103—21no—
57#103—21no—
58#102—21no—
59#102—41no—
60C22⋊C8—21no—
61C4×C8C2311yescharacteristic
62#41—42no—
63#41—81no—
64C2×Q8—21no—
65C2×Q8—21no—
66C2×Q8—41no—
67C2×Q8—42no—
68C2×Q8—41no—
69C2×Q8—81no—
70C2×D4—21no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4D811yescharacteristic
74C22×C4—21no—
75C22×C4—41no—
76C22×C4—41no—
77Q16—21no—
78Q16—41no—
79Q16—42no—
80Q16SD1612yes—
81SD16—42no—
82SD16—41no—
83SD16—41no—
84D8—41no—
85M16—41no—
86M16—41no—
87C2×C8—21no—
88C2×C8—41no—
89C2×C8C2×D411yescharacteristic
90C2×C8C2×D411yescharacteristic
91C4⋊C4—21no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4D811yescharacteristic
97C22⋊C4—21no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C42—41no—
103C42C2×D411yescharacteristic
104C16—41no—
105C16—41no—
106C23—21no—
107C23—41no—
108Q8—22no—
109Q8—41no—
110Q8—41no—
111Q8—41no—
112Q8—82no—
113Q8—161no—
114D4—41no—
115D4—41no—
116D4—41no—
117D4—82no—
118D4—82no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—82no—
126C2×C4—82no—
127C2×C4—81no—
128C2×C4#12011yescharacteristic
129C2×C4C2×D811yescharacteristic
130C2×C4AGL(1,ℤ/8)11yescharacteristic
131C8—21no—
132C8—41no—
133C8—81no—
134C8C2×SD1611yescharacteristic
135C8AGL(1,ℤ/8)11yescharacteristic
136C22—41no—
137C22—41no—
138C22—41no—
139C22—81no—
140C22—81no—
141C22—81no—
142C22#44711yescharacteristic
143C4—21no—
144C4—21no—
145C4—41no—
146C4—42no—
147C4—81no—
148C4—81no—
149C4—161no—
150C4#45011yescharacteristic
151C4#45511yescharacteristic
152C2—81no—
153C2—161no—
154C2#163911yescharacteristic
155C2#219111yescharacteristic
156C2#219311yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#3277C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ #436 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2243C2semidirect
3. #2225C2semidirect
4. #1695C2semidirect
5. #1360C2semidirect
6. #1334C2semidirect
7. #1330C2semidirect

Maximal subgroups

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

subgroupindex
#32772
#22432
#22252
#16952
#13602
#13342
#13302