Finite Group Databasebeta

← #10291

Subgroups — #10291

Order 256.

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

normal subgroups only

This group has 326 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#3209C211yesmaximal, characteristic, complemented
2#3208C211yesmaximal, characteristic
3#2858C211yesmaximal, characteristic
4#1994C211yesmaximal, characteristic, complemented
5#1548C211yesmaximal, characteristic, complemented
6#1545C211yesmaximal, characteristic
7#1276C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#127C2211yescharacteristic
11C2×#124—22no—
12#494—21no—
13#492—21no—
14#487—21no—
15#486—21no—
16#462—22no—
17#460—22no—
18#451—22no—
19#450—22no—
20#431C2211yescharacteristic
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23#390—22no—
24#390—22no—
25C43C2211yescharacteristic
26#339—22no—
27#333—22no—
28#331—22no—
29C22×Q8D412yes—
30C22×D4—21no—
31C22×D4—21no—
32C22×D4D412yes—
33C2×Q16—24no—
34C2×SD16—24no—
35C2×M16—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42#127—21no—
43#127—21no—
44#127—21no—
45#127—21no—
46#125—21no—
47#124—48no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52#117—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C22⋊C4—28no—
56C2×C42—21no—
57C2×C42—21no—
58C2×C42D412yes—
59C2×C42C2311yescharacteristic
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8D412yes—
63C4⋊C8D412yes—
64#103—24no—
65#102—24no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68C4×C8—21no—
69C4×C8—21no—
70C24—22no—
71C2×Q8—22no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8—22no—
75C2×Q8—44no—
76C2×D4—22no—
77C2×D4—22no—
78C2×D4—44no—
79C2×D4—44no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4—41no—
84C2×D4—42no—
85C2×D4—42no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91Q16—84no—
92SD16—84no—
93M16—42no—
94C2×C8—22no—
95C2×C8—22no—
96C2×C8—41no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C22⋊C4—416no—
108C22⋊C4—41no—
109C42—24no—
110C42—22no—
111C42—22no—
112C42—42no—
113C42—41no—
114C42C2×D411yescharacteristic
115C42C2×D411yescharacteristic
116C42C2×D411yescharacteristic
117C42#4111yescharacteristic
118C23—44no—
119C23—44no—
120C23—42no—
121C23—44no—
122C23#12011yescharacteristic
123Q8—42no—
124Q8—44no—
125Q8—42no—
126Q8—82no—
127D4—44no—
128D4—82no—
129D4—82no—
130D4—82no—
131D4—82no—
132D4—81no—
133D4—81no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—24no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—44no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4#12011yescharacteristic
144C2×C4#12011yescharacteristic
145C2×C4#12111yescharacteristic
146C2×C4#12111yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12711yescharacteristic
149C2×C4#13512yes—
150C2×C4#13712yes—
151C8—42no—
152C8—42no—
153C8—81no—
154C22—21no—
155C22—21no—
156C22—88no—
157C22—82no—
158C22—82no—
159C22—82no—
160C22#39211yescharacteristic
161C22#45212yes—
162C4—21no—
163C4—21no—
164C4—44no—
165C4—82no—
166C4—81no—
167C4—81no—
168C4#45312yes—
169C4#45512yes—
170C4#46412yes—
171C4#49511yescharacteristic
172C4#49711yescharacteristic
173C2—22no—
174C2—162no—
175C2#200211yescharacteristic
176C2#200411yescharacteristic
177C2#201511yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
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
1. #3209C2semidirect
4. #1994C2semidirect
5. #1548C2semidirect
7. #1276C2semidirect

Maximal subgroups

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

subgroupindex
#32092
#32082
#28582
#19942
#15482
#15452
#12762