Finite Group Databasebeta

← #10252

Subgroups — #10252

Order 256.

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

normal subgroups only

This group has 250 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
2#3078C211yesmaximal, characteristic
3#2568C211yesmaximal, characteristic
4#1548C211yesmaximal, characteristic
5#1503C211yesmaximal, characteristic
6#1479C211yesmaximal, characteristic
7#1303C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11#494—21no—
12#487—21no—
13#479—22no—
14#477—21no—
15#475—21no—
16#462—22no—
17#451—22no—
18#431C2211yescharacteristic
19#431C2211yescharacteristic
20#428—21no—
21#423C2211yescharacteristic
22#419—21no—
23#400—22no—
24#391—22no—
25#391—21no—
26#382—21no—
27C4×C4⋊C4C2211yescharacteristic
28#333—22no—
29#328—22no—
30Q8⋊C8—22no—
31C22×Q8D411yescharacteristic
32C22×Q8D412yes—
33C2×Q16—24no—
34C2×M16—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41#125—21no—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44C4×Q8—21no—
45C4×Q8—21no—
46C4×Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50#117—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4D411yescharacteristic
56C2×C42—21no—
57C2×C42—21no—
58C2×C42C2311yescharacteristic
59#107—22no—
60#106—22no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8D412yes—
64C4⋊C8D412yes—
65#103—22no—
66#103—22no—
67#103—24no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70C4×C8—21no—
71C4×C8—21no—
72#95—24no—
73#95—24no—
74#95D412yes—
75C2×Q8—21no—
76C2×Q8—21no—
77C2×Q8—22no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—22no—
82C2×Q8—21no—
83C2×Q8—21no—
84C2×Q8—44no—
85C2×Q8Q1612yescomplemented
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93Q16—84no—
94M16—41no—
95M16—41no—
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—41no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—42no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C22⋊C4—41no—
115C42—41no—
116C42—41no—
117C42—41no—
118C42—41no—
119C42—41no—
120C42C2×D411yescharacteristic
121C42C2×D411yescharacteristic
122C42C2×D411yescharacteristic
123C42#4111yescharacteristic
124C23#12011yescharacteristic
125Q8—22no—
126Q8—22no—
127Q8—41no—
128Q8—41no—
129Q8—42no—
130Q8—44no—
131Q8—81no—
132Q8—82no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—44no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4#12011yescharacteristic
148C2×C4#12011yescharacteristic
149C2×C4#12111yescharacteristic
150C2×C4#12111yescharacteristic
151C2×C4#12111yescharacteristic
152C2×C4#12711yescharacteristic
153C2×C4C2×Q1611yescharacteristic
154C2×C4#13511yescharacteristic
155C2×C4#13711yescharacteristic
156C2×C4#13711yescharacteristic
157C8—42no—
158C8—42no—
159C8—81no—
160C22—21no—
161C22—21no—
162C22#39211yescharacteristic
163C22#45111yescharacteristic
164C22#45211yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—42no—
169C4—82no—
170C4—82no—
171C4—81no—
172C4—81no—
173C4—81no—
174C4#45111yescharacteristic
175C4#45211yescharacteristic
176C4#45512yes—
177C4#46711yescharacteristic
178C4#47011yescharacteristic
179C2—21no—
180C2—21no—
181C2#200111yescharacteristic
182C2#200411yescharacteristic
183C2#200911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC23#120
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
85. C2×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#32092
#30782
#25682
#15482
#15032
#14792
#13032