Finite Group Databasebeta

← #12378

Subgroups — #12378

Order 256.

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

normal subgroups only

This group has 186 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#3273C211yesmaximal, characteristic
2#2243C211yesmaximal, characteristic, complemented
3#2226C211yesmaximal, characteristic, complemented
4#1693C211yesmaximal, characteristic, complemented
5#1360C211yesmaximal, characteristic, complemented
6#1333C211yesmaximal, characteristic, complemented
7#1330C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9#546—21no—
10#500C2211yescharacteristic, complemented
11#495C2211yescharacteristic
12#468—21no—
13#463—21no—
14#461—21no—
15#452—21no—
16#447—21no—
17C4×D8C2211yescharacteristic
18#436C2211yescharacteristic
19#367—21no—
20C4⋊C16C2211yescharacteristic, complemented
21#358—21no—
22#357—21no—
23#357—21no—
24#346C2211yescharacteristic, complemented
25#334C2211yescharacteristic, complemented
26#332—21no—
27C2×#41—21no—
28C22×D4—21no—
29AGL(1,ℤ/8)—42no—
30AGL(1,ℤ/8)—42no—
31C2×Q16D411yescharacteristic
32C2×SD16—21no—
33C2×SD16—21no—
34C2×SD16—41no—
35C2×D8—21no—
36C2×D8D411yescharacteristic
37C2×D8D411yescharacteristic
38C2×M16—21no—
39C4⋊Q8—21no—
40#124—21no—
41#123—41no—
42C22⋊Q8—21no—
43#121—21no—
44#121—41no—
45#120—41no—
46C4×D4—21no—
47C4×D4D411yescharacteristic
48C2×C22⋊C4—41no—
49C2×C16—21no—
50C2×C16—21no—
51#107—21no—
52#107—21no—
53#107D411yescharacteristic
54C4⋊C8—21no—
55C4⋊C8D411yescharacteristic
56#103—21no—
57#103—41no—
58#102—21no—
59#102—21no—
60C22⋊C8—21no—
61C4×C8C2311yescharacteristic
62C24—41no—
63#41—42no—
64#41—81no—
65C2×Q8—21no—
66C2×Q8—41no—
67C2×D4—21no—
68C2×D4—21no—
69C2×D4—41no—
70C2×D4—42no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—81no—
74C2×D4D811yescharacteristic
75C22×C4—21no—
76C22×C4—41no—
77C22×C4—41no—
78Q16—41no—
79SD16—42no—
80SD16—41no—
81SD16—41no—
82SD16—82no—
83D8—21no—
84D8—41no—
85D8—42no—
86D8SD1612yescomplemented
87M16—41no—
88M16—41no—
89C2×C8—21no—
90C2×C8—41no—
91C2×C8C2×D411yescharacteristic
92C2×C8C2×D411yescharacteristic
93C4⋊C4—21no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4D811yescharacteristic
98C22⋊C4—21no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C42C2×D411yescharacteristic
105C16—41no—
106C16—41no—
107C23—21no—
108C23—41no—
109C23—41no—
110C23—82no—
111C23—82no—
112C23—161no—
113Q8—41no—
114Q8—41no—
115Q8—81no—
116Q8—81no—
117D4—22no—
118D4—41no—
119D4—41no—
120D4—41no—
121D4—41no—
122D4—82no—
123D4—82no—
124D4—161no—
125C2×C4—21no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—82no—
131C2×C4—81no—
132C2×C4—161no—
133C2×C4#12011yescharacteristic
134C2×C4C2×D811yescharacteristic
135C2×C4AGL(1,ℤ/8)11yescharacteristic
136C8—21no—
137C8—41no—
138C8—81no—
139C8C2×SD1611yescharacteristic
140C8AGL(1,ℤ/8)11yescharacteristic
141C22—41no—
142C22—42no—
143C22—41no—
144C22—41no—
145C22—81no—
146C22—81no—
147C22—81no—
148C22—81no—
149C22—81no—
150C22—164no—
151C22#44711yescharacteristic
152C4—21no—
153C4—21no—
154C4—41no—
155C4—81no—
156C4—81no—
157C4—161no—
158C4#45011yescharacteristic
159C4#45511yescharacteristic
160C2—81no—
161C2—82no—
162C2—161no—
163C2#163911yescharacteristic
164C2#219011yescharacteristic
165C2#219411yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#3273C2
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. #2226C2semidirect
4. #1693C2semidirect
5. #1360C2semidirect
6. #1333C2semidirect
7. #1330C2semidirect
10. #500C22semidirect
20. C4⋊C16C22semidirect
24. #346C22semidirect
25. #334C22semidirect
86. D8SD16semidirect

Maximal subgroups

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

subgroupindex
#32732
#22432
#22262
#16932
#13602
#13332
#13302