Finite Group Databasebeta

← #9050

Subgroups — #9050

Order 256.

238 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 69 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#3101C211yesmaximal, characteristic
2#2961C211yesmaximal, characteristic, complemented
3#1955C211yesmaximal, characteristic
4#1542C211yesmaximal, characteristic, complemented
5#1486C211yesmaximal, characteristic
6#1301C211yesmaximal, characteristic, complemented
7#1281C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10#483C412yes—
11#476—21no—
12#461—21no—
13#450C412yes—
14#446—22no—
15#446—22no—
16C4×SD16—22no—
17#431C2211yescharacteristic
18#431C2211yescharacteristic
19#424—21no—
20#423C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C22⋊C8—21no—
23C2×C22⋊C8—21no—
24#405—21no—
25C2×#97C2211yescharacteristic
26#333—21no—
27#332—21no—
28D4⋊C8—22no—
29C22×D4C2×C411yescharacteristic
30C23×C4—21no—
31C2×SD16—22no—
32C2×M16—21no—
33C22×C8—21no—
34C22×C8D411yescharacteristic
35C22×C8D411yescharacteristic
36C4⋊Q8D411yescharacteristic
37C4⋊Q8D411yescharacteristic
38#125—21no—
39C22⋊Q8—21no—
40C22⋊Q8C2×C412yes—
41C4×Q8—21no—
42C4×Q8D412yes—
43C4×D4—21no—
44C4×D4—21no—
45C4×D4—21no—
46C4×D4—21no—
47#117—21no—
48C2×C4⋊C4C2×C411yescharacteristic
49C2×C22⋊C4—21no—
50C2×C42C2311yescharacteristic
51#106—22no—
52C4⋊C8—24no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8D412yes—
59#103—22no—
60#102—22no—
61C22⋊C8—21no—
62C22⋊C8—42no—
63C22⋊C8—42no—
64C22⋊C8C2×C412yes—
65#97—22no—
66#97—21no—
67#97—21no—
68C4×C8—22no—
69C4×C8—21no—
70C4×C8—21no—
71C24—21no—
72C2×Q8—21no—
73C2×Q8C22⋊C412yes—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—42no—
77C22×C4—21no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4C22×C411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C22×C4#4111yescharacteristic
87SD16—44no—
88M16—41no—
89M16—41no—
90C2×C8—21no—
91C2×C8—21no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—21no—
95C2×C8—21no—
96C2×C8—21no—
97C2×C8—41no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C2×C8SD1612yescomplemented
102C4⋊C4—22no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—22no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4C22⋊C412yes—
110C22⋊C4—22no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C42—22no—
114C42C2×D411yescharacteristic
115C42C2×D411yescharacteristic
116C42C2×D411yescharacteristic
117C42#4111yescharacteristic
118C23—22no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23C4×D411yescharacteristic
126Q8—42no—
127Q8—81no—
128D4—22no—
129D4—22no—
130D4—41no—
131D4—41no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4C2×C22⋊C411yescharacteristic
144C2×C4C4×D411yescharacteristic
145C2×C4#12011yescharacteristic
146C2×C4#12111yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12311yescharacteristic
149C2×C4C2×SD1611yescharacteristic
150C2×C4#13511yescharacteristic
151C2×C4AGL(1,ℤ/8)11yescharacteristic
152C2×C4#13711yescharacteristic
153C8—22no—
154C8—42no—
155C8—41no—
156C8—42no—
157C8—81no—
158C8—81no—
159C22—21no—
160C22—21no—
161C22—42no—
162C22—42no—
163C22—42no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22#38611yescharacteristic
169C22C4×SD1611yescharacteristic
170C22#44011yescharacteristic
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—42no—
176C4—42no—
177C4—81no—
178C4#44811yescharacteristic
179C4#44911yescharacteristic
180C4#46011yescharacteristic
181C4#46411yescharacteristic
182C2—21no—
183C2—21no—
184C2—82no—
185C2#188011yescharacteristic
186C2#188111yescharacteristic
187C2#189111yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radicalC22×D4C2×C4
LCSG ▷ C22×C4 ▷ 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
2. #2961C2semidirect
4. #1542C2semidirect
6. #1301C2semidirect
7. #1281C2semidirect
101. C2×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#31012
#29612
#19552
#15422
#14862
#13012
#12812