Finite Group Databasebeta

← #9236

Subgroups — #9236

Order 256.

237 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 71 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#3104C211yesmaximal, characteristic
2#2295C211yesmaximal, characteristic
3#1994C211yesmaximal, characteristic
4#1503C211yesmaximal, characteristic
5#1487C211yesmaximal, characteristic
6#1303C211yesmaximal, characteristic
7#1276C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10#484—21no—
11#477C412yescomplemented
12#462C412yescomplemented
13#451—21no—
14C4×Q16—22no—
15#431C2211yescharacteristic
16#431C2211yescharacteristic
17#423C2211yescharacteristic
18#391—21no—
19#389—21no—
20#387—22no—
21C4×C4⋊C4—21no—
22C4×C4⋊C4C2211yescharacteristic
23#376—21no—
24C43C2211yescharacteristic
25#339—21no—
26#339—21no—
27#328—22no—
28Q8⋊C8—22no—
29C22×Q8D411yescharacteristic
30C2×Q16—22no—
31C2×M16—21no—
32C4⋊Q8C2×C411yescharacteristic
33C4⋊Q8C2×C411yescharacteristic
34#125—21no—
35C22⋊Q8—21no—
36C22⋊Q8—21no—
37C4×Q8—21no—
38C4×Q8—21no—
39C4×Q8—21no—
40C4×Q8—21no—
41C4×Q8C2×C412yes—
42C4×Q8D412yes—
43#117—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4D411yescharacteristic
48C2×C42—21no—
49C2×C42—21no—
50C2×C42—21no—
51C2×C42—21no—
52C2×C42D411yescharacteristic
53C2×C42D411yescharacteristic
54C2×C42C2311yescharacteristic
55#107—22no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8C2×C412yes—
60#103—22no—
61#103—22no—
62C22⋊C8—21no—
63C22⋊C8—21no—
64C4×C8—21no—
65C4×C8—21no—
66#95—22no—
67#95—22no—
68#95—22no—
69#95D412yes—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8—42no—
74C2×Q8C22⋊C412yes—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83Q16—44no—
84M16—41no—
85M16—41no—
86C2×C8—22no—
87C2×C8—41no—
88C2×C8—41no—
89C4⋊C4—22no—
90C4⋊C4—22no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—42no—
97C4⋊C4—44no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4C22⋊C412yes—
101C22⋊C4—41no—
102C42—22no—
103C42—22no—
104C42—21no—
105C42—21no—
106C42—21no—
107C42—42no—
108C42—42no—
109C42—41no—
110C42—41no—
111C42—41no—
112C42Q1612yescomplemented
113C42C22×C411yescharacteristic
114C42C2×D411yescharacteristic
115C42#4111yescharacteristic
116C42#4111yescharacteristic
117C23#12011yescharacteristic
118Q8—22no—
119Q8—22no—
120Q8—42no—
121Q8—42no—
122Q8—81no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—41no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4C2×C22⋊C411yescharacteristic
141C2×C4C4×D411yescharacteristic
142C2×C4C4×D411yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12111yescharacteristic
145C2×C4#12311yescharacteristic
146C2×C4C2×Q1611yescharacteristic
147C2×C4#13511yescharacteristic
148C2×C4#13711yescharacteristic
149C2×C4#13711yescharacteristic
150C8—42no—
151C8—81no—
152C8—81no—
153C22—21no—
154C22—21no—
155C22#38611yescharacteristic
156C22#45111yescharacteristic
157C22#45211yescharacteristic
158C4—21no—
159C4—21no—
160C4—21no—
161C4—42no—
162C4—42no—
163C4—42no—
164C4—42no—
165C4—82no—
166C4—81no—
167C4—81no—
168C4#42112yes—
169C4C4×Q1611yescharacteristic
170C4#44111yescharacteristic
171C4#46211yescharacteristic
172C4#46411yescharacteristic
173C2—21no—
174C2—21no—
175C2#187811yescharacteristic
176C2#189711yescharacteristic
177C2#190611yescharacteristic
subgroupG/H
centerC2×C4#121
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radicalC23#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #477C4semidirect
12. #462C4semidirect
112. C42Q16semidirect

Maximal subgroups

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

subgroupindex
#31042
#22952
#19942
#15032
#14872
#13032
#12762