Finite Group Databasebeta

← #10286

Subgroups — #10286

Order 256.

271 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 61 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#3227C211yesmaximal, characteristic
2#3226C211yesmaximal, characteristic
3#2862C211yesmaximal, characteristic, complemented
4#1962C211yesmaximal, characteristic
5#1549C211yesmaximal, characteristic, complemented
6#1546C211yesmaximal, characteristic, complemented
7#1276C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#500—21no—
12#498—21no—
13#486—21no—
14#486—21no—
15#481—22no—
16#480—22no—
17#476—22no—
18#474—22no—
19#431C2211yescharacteristic
20#431C2211yescharacteristic
21#423C2211yescharacteristic
22C2×#104—22no—
23#391—22no—
24#391—22no—
25#382—22no—
26C43C2211yescharacteristic
27#332—22no—
28#332—22no—
29C2×#41—21no—
30C22×Q8—21no—
31C22×Q8—21no—
32C2×M16—21no—
33C4⋊Q8—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8D411yescharacteristic
42#127D411yescharacteristic
43#124—21no—
44#121—21no—
45#121—21no—
46C4×D4—21no—
47C4×D4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4D412yes—
51C2×C4⋊C4D412yes—
52C2×C42—21no—
53C2×C42—21no—
54C2×C42D412yes—
55C2×C42C2311yescharacteristic
56#107—24no—
57#106—24no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8D412yes—
61C4⋊C8D412yes—
62#104—44no—
63#102—24no—
64#102—24no—
65C22⋊C8—21no—
66C22⋊C8—21no—
67C4×C8—21no—
68C4×C8—21no—
69#95—28no—
70#41—42no—
71#41—42no—
72C2×Q8—21no—
73C2×Q8—44no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×Q8—41no—
78C2×Q8—42no—
79C2×Q8—42no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—41no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—41no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91M16—42no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—41no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C42—24no—
108C42—22no—
109C42—22no—
110C42—42no—
111C42C2×D411yescharacteristic
112C42C2×D411yescharacteristic
113C42C2×D411yescharacteristic
114C42#4111yescharacteristic
115C23—41no—
116C23#12011yescharacteristic
117Q8—42no—
118Q8—82no—
119Q8—82no—
120Q8—82no—
121Q8—81no—
122Q8—81no—
123D4—42no—
124D4—42no—
125D4—42no—
126D4—82no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—24no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—44no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—44no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4#12011yescharacteristic
140C2×C4#12011yescharacteristic
141C2×C4#12111yescharacteristic
142C2×C4#12111yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12711yescharacteristic
145C2×C4#13512yes—
146C2×C4AGL(1,ℤ/8)12yes—
147C8—42no—
148C8—42no—
149C8—81no—
150C22—21no—
151C22—21no—
152C22—81no—
153C22—81no—
154C22—81no—
155C22#39211yescharacteristic
156C22#44912yes—
157C4—21no—
158C4—21no—
159C4—44no—
160C4—82no—
161C4—82no—
162C4—81no—
163C4#45412yes—
164C4#45612yes—
165C4#46312yes—
166C4#49511yescharacteristic
167C4#49611yescharacteristic
168C2—22no—
169C2—161no—
170C2#200311yescharacteristic
171C2#200511yescharacteristic
172C2#201211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2862C2semidirect
5. #1549C2semidirect
6. #1546C2semidirect
7. #1276C2semidirect

Maximal subgroups

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

subgroupindex
#32272
#32262
#28622
#19622
#15492
#15462
#12762