Finite Group Databasebeta

← #10125

Subgroups — #10125

Order 256.

229 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 49 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#3237C211yesmaximal, characteristic, complemented
2#3215C211yesmaximal, characteristic
3#3140C211yesmaximal, characteristic
4#1497C211yesmaximal, characteristic, complemented
5#1485C211yesmaximal, characteristic, complemented
6#1469C211yesmaximal, characteristic
7#1291C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#533C2211yescharacteristic
10C2×#124C2211yescharacteristic
11#501—21no—
12#497—21no—
13#491—21no—
14#489—21no—
15#489—21no—
16#489—21no—
17#488—22no—
18#485—22no—
19#477—21no—
20#475—21no—
21#464—22no—
22#451—21no—
23#450—21no—
24#432C2211yescharacteristic
25#432C2211yescharacteristic
26#431C2211yescharacteristic
27#419—21no—
28#418—21no—
29C4×M16C2211yescharacteristic
30Q8⋊C8—22no—
31Q8⋊C8—22no—
32#323—22no—
33C22×Q8D411yescharacteristic
34C22×D4D411yescharacteristic
35C2×Q16—22no—
36C2×SD16—22no—
37C2×M16—21no—
38C4⋊Q8D411yescharacteristic
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42#125—21no—
43#125—21no—
44#124—21no—
45#124—21no—
46#124—21no—
47#124—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C22⋊Q8—21no—
51C22⋊Q8—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54C4×Q8—21no—
55C4×Q8—21no—
56#117—21no—
57#117—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C42C2311yescharacteristic
61#107—22no—
62#106—22no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8D412yes—
66#103—22no—
67#103—22no—
68#103—22no—
69#103—22no—
70#103—22no—
71#102—22no—
72#102—22no—
73#102—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8D412yes—
77#97—21no—
78#97—21no—
79#97—21no—
80C4×C8—21no—
81C4×C8D412yes—
82C24—21no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—42no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—42no—
93C22×C4—21no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4#4111yescharacteristic
97Q16—82no—
98SD16—82no—
99M16—42no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—41no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C42—41no—
123C42—41no—
124C42C2×D411yescharacteristic
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42C2×D411yescharacteristic
128C23—42no—
129C23—41no—
130C23—81no—
131C23—81no—
132C23#12111yescharacteristic
133Q8—42no—
134Q8—42no—
135Q8—42no—
136Q8—42no—
137Q8—42no—
138Q8—42no—
139D4—42no—
140D4—42no—
141C2×C4—22no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—81no—
150C2×C4#12011yescharacteristic
151C2×C4#12011yescharacteristic
152C2×C4#12011yescharacteristic
153C2×C4#12111yescharacteristic
154C2×C4#12111yescharacteristic
155C2×C4#12711yescharacteristic
156C8—42no—
157C8—42no—
158C8—42no—
159C8—81no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22—162no—
167C22#39211yescharacteristic
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—81no—
174C4—81no—
175C4—81no—
176C4—81no—
177C4—81no—
178C4#45512yes—
179C2—41no—
180C2—161no—
181C2#200411yescharacteristic
182C2#200911yescharacteristic
183C2#201711yescharacteristic
subgroupG/H
center, socleC22#392
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
1. #3237C2semidirect
4. #1497C2semidirect
5. #1485C2semidirect
7. #1291C2semidirect

Maximal subgroups

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

subgroupindex
#32372
#32152
#31402
#14972
#14852
#14692
#12912