Finite Group Databasebeta

← #12325

Subgroups — #12325

Order 256.

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

normal subgroups only

This group has 192 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, complemented
2#2250C211yesmaximal, characteristic, complemented
3#2207C211yesmaximal, characteristic, complemented
4#1687C211yesmaximal, characteristic, complemented
5#1360C211yesmaximal, characteristic, complemented
6#1334C211yesmaximal, characteristic, complemented
7#1330C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#548—21no—
10C2×SD32—21no—
11C2×SD32—21no—
12C2×D16—21no—
13#495C2211yescharacteristic
14#493C2211yescharacteristic
15#468—21no—
16#464—21no—
17#459—21no—
18#451—21no—
19#449—21no—
20C4×Q16C2211yescharacteristic
21#436C2211yescharacteristic
22C4⋊C16C2211yescharacteristic
23#357—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#331—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×D8—21no—
37C2×D8—21no—
38C2×D8—41no—
39C2×D8D411yescharacteristic
40C2×M16—21no—
41#127—21no—
42#124—21no—
43#124—41no—
44C22⋊Q8—21no—
45C22⋊Q8—41no—
46#121—21no—
47#121—41no—
48C4×Q8—21no—
49C4×D4—41no—
50C4×D4D411yescharacteristic
51SD32—42no—
52SD32—42no—
53D16—42no—
54C2×C16—21no—
55C2×C16—21no—
56#107D411yescharacteristic
57C4⋊C8—21no—
58C4⋊C8D411yescharacteristic
59#103—21no—
60#103—21no—
61#102—21no—
62#102—41no—
63C22⋊C8—21no—
64C4×C8C2311yescharacteristic
65#41—42no—
66#41—81no—
67C2×Q8—21no—
68C2×Q8—21no—
69C2×Q8—41no—
70C2×Q8—42no—
71C2×Q8—81no—
72C2×D4—21no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4—41no—
77C2×D4—41no—
78C2×D4D811yescharacteristic
79C22×C4—21no—
80C22×C4—41no—
81C22×C4—41no—
82Q16—21no—
83Q16—41no—
84Q16—42no—
85Q16D812yescomplemented
86SD16—42no—
87SD16—41no—
88SD16—41no—
89D8—41no—
90D8—41no—
91D8—41no—
92D8—41no—
93D8—41no—
94D8—82no—
95M16—41no—
96M16—41no—
97C2×C8—21no—
98C2×C8—41no—
99C2×C8C2×D411yescharacteristic
100C2×C8C2×D411yescharacteristic
101C4⋊C4—21no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4D811yescharacteristic
105C22⋊C4—21no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C42—41no—
111C42C2×D411yescharacteristic
112C16—22no—
113C16—41no—
114C23—21no—
115C23—41no—
116C23—81no—
117Q8—22no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—82no—
122D4—41no—
123D4—41no—
124D4—41no—
125D4—81no—
126D4—81no—
127D4—82no—
128D4—82no—
129D4—81no—
130D4—81no—
131D4—161no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—82no—
139C2×C4—82no—
140C2×C4#12011yescharacteristic
141C2×C4C2×D811yescharacteristic
142C2×C4AGL(1,ℤ/8)11yescharacteristic
143C8—21no—
144C8—41no—
145C8—81no—
146C8C2×D811yescharacteristic
147C8AGL(1,ℤ/8)11yescharacteristic
148C22—41no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22—81no—
153C22—81no—
154C22—161no—
155C22—161no—
156C22—161no—
157C22#44711yescharacteristic
158C4—21no—
159C4—21no—
160C4—41no—
161C4—42no—
162C4—81no—
163C4—81no—
164C4#44711yescharacteristic
165C4#45311yescharacteristic
166C2—81no—
167C2—161no—
168C2—321no—
169C2#161911yescharacteristic
170C2#219011yescharacteristic
171C2#219311yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
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
1. #3277C2semidirect
2. #2250C2semidirect
3. #2207C2semidirect
4. #1687C2semidirect
5. #1360C2semidirect
6. #1334C2semidirect
7. #1330C2semidirect
85. Q16D8semidirect

Maximal subgroups

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

subgroupindex
#32772
#22502
#22072
#16872
#13602
#13342
#13302