Finite Group Databasebeta

← #10498

Subgroups — #10498

Order 256.

190 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 45 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#3215C211yesmaximal, characteristic
2#1967C211yesmaximal, characteristic
3#1569C211yesmaximal, characteristic, complemented
4#1559C211yesmaximal, characteristic, complemented
5#1527C211yesmaximal, characteristic
6#1526C211yesmaximal, characteristic, complemented
7#1291C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#124C2211yescharacteristic
10#497—21no—
11#489—21no—
12#464—21no—
13#464—21no—
14#451—21no—
15#450—21no—
16#432C2211yescharacteristic
17#432C2211yescharacteristic
18#423C2211yescharacteristic
19C2×#101—21no—
20C2×#100—21no—
21C4×M16C2211yescharacteristic
22C4×M16C2211yescharacteristic
23#335—21no—
24#334—21no—
25#333—21no—
26#332—21no—
27#330—21no—
28#330—21no—
29#329—21no—
30#329—21no—
31C22×Q8D411yescharacteristic
32C22×D4D411yescharacteristic
33C2×Q16—21no—
34C2×Q16—21no—
35C2×SD16—21no—
36C2×SD16—21no—
37C2×M16—21no—
38C2×M16—21no—
39C2×M16—21no—
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42#125—21no—
43#124—21no—
44#124—21no—
45#124—21no—
46#124—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51#117—21no—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C42C2311yescharacteristic
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8D411yescharacteristic
59C4⋊C8D411yescharacteristic
60#103—21no—
61#103—21no—
62#102—21no—
63#102—21no—
64#101—42no—
65#100—42no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68#97—21no—
69#97—21no—
70#97—21no—
71#97D411yescharacteristic
72#97D411yescharacteristic
73C4×C8—21no—
74C4×C8—21no—
75C24—21no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—41no—
81C2×Q8—41no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—41no—
85C2×D4—41no—
86C22×C4—21no—
87C22×C4C2×D411yescharacteristic
88C22×C4#4111yescharacteristic
89C22×C4#4111yescharacteristic
90Q16—81no—
91Q16—81no—
92SD16—81no—
93SD16—81no—
94M16—22no—
95M16—42no—
96M16—41no—
97M16—41no—
98M16—41no—
99C2×C8—21no—
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—41no—
104C2×C8—41no—
105C2×C8—41no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—41no—
117C42—41no—
118C42C2×D411yescharacteristic
119C42C2×D411yescharacteristic
120C42C2×D411yescharacteristic
121C42#4111yescharacteristic
122C23—42no—
123C23—41no—
124C23—81no—
125C23—81no—
126C23#12311yescharacteristic
127Q8—41no—
128Q8—41no—
129Q8—41no—
130Q8—41no—
131Q8—81no—
132Q8—81no—
133D4—41no—
134D4—41no—
135D4—81no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4#12011yescharacteristic
145C2×C4#12111yescharacteristic
146C2×C4#12111yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12311yescharacteristic
149C2×C4#12411yescharacteristic
150C8—22no—
151C8—41no—
152C8—42no—
153C8—41no—
154C8—41no—
155C8—81no—
156C8—81no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—81no—
161C22—81no—
162C22—81no—
163C22—162no—
164C22#39411yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—81no—
171C4—81no—
172C4—81no—
173C4#45311yescharacteristic
174C4#45511yescharacteristic
175C2—41no—
176C2—161no—
177C2#203711yescharacteristic
178C2#204911yescharacteristic
179C2#205011yescharacteristic
subgroupG/H
center, socleC22#394
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
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. #1569C2semidirect
4. #1559C2semidirect
6. #1526C2semidirect
7. #1291C2semidirect

Maximal subgroups

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

subgroupindex
#32152
#19672
#15692
#15592
#15272
#15262
#12912