Finite Group Databasebeta

← #10033

Subgroups — #10033

Order 256.

275 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 57 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#3135C211yesmaximal, characteristic, complemented
2#3101C211yesmaximal, characteristic, complemented
3#3100C211yesmaximal, characteristic
4#1955C211yesmaximal, characteristic, complemented
5#1481C211yesmaximal, characteristic, complemented
6#1480C211yesmaximal, characteristic, complemented
7#1275C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#491—21no—
12#490—21no—
13#489—21no—
14#489—21no—
15#483—21no—
16#480—21no—
17#476—21no—
18#474—21no—
19#461—21no—
20#459—21no—
21#450—21no—
22#447—21no—
23C4×SD16—22no—
24C4×D8—22no—
25#431C2211yescharacteristic
26#431C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28#416—22no—
29C2×C22⋊C8—22no—
30C2×#97C2211yescharacteristic
31Q8⋊C8—22no—
32D4⋊C8—22no—
33C2×#41—21no—
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C2×SD16—22no—
37C2×D8—22no—
38C22×C8—21no—
39C22×C8D412yes—
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43#127D411yescharacteristic
44#125—21no—
45#124—21no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48#121—21no—
49#121—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4—21no—
57C4×D4—21no—
58#117—21no—
59C2×C4⋊C4D411yescharacteristic
60C2×C22⋊C4—21no—
61C2×C42C2311yescharacteristic
62#107—22no—
63#106—22no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68#103—22no—
69#103—22no—
70#103—22no—
71#102—22no—
72#102—22no—
73#102—22no—
74#102—22no—
75#102—22no—
76C22⋊C8—21no—
77C22⋊C8—21no—
78C22⋊C8—44no—
79#97—21no—
80#97—21no—
81C4×C8D412yes—
82C4×C8D412yes—
83C24—21no—
84#41—42no—
85#41—42no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—42no—
95C2×D4—41no—
96C22×C4—21no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4#4111yescharacteristic
105SD16—44no—
106D8—44no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—41no—
113C2×C8—42no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C42—41no—
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C42C2×D411yescharacteristic
133C42C2×D411yescharacteristic
134C23—22no—
135C23—41no—
136C23—42no—
137C23—41no—
138C23—42no—
139C23—41no—
140C23#12111yescharacteristic
141Q8—42no—
142Q8—42no—
143Q8—42no—
144D4—22no—
145D4—22no—
146D4—41no—
147D4—41no—
148D4—42no—
149D4—42no—
150D4—42no—
151D4—82no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—82no—
160C2×C4—81no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4#12011yescharacteristic
165C2×C4#12011yescharacteristic
166C2×C4#12011yescharacteristic
167C2×C4#12111yescharacteristic
168C2×C4#12111yescharacteristic
169C2×C4#12711yescharacteristic
170C2×C4#13512yes—
171C2×C4AGL(1,ℤ/8)12yes—
172C8—42no—
173C8—42no—
174C8—42no—
175C8—81no—
176C22—21no—
177C22—21no—
178C22—44no—
179C22—42no—
180C22—82no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22#39211yescharacteristic
187C22#46312yes—
188C4—21no—
189C4—21no—
190C4—21no—
191C4—21no—
192C4—81no—
193C4—81no—
194C4—81no—
195C4—81no—
196C4#44912yes—
197C4#49511yescharacteristic
198C4#49611yescharacteristic
199C2—22no—
200C2—82no—
201C2—161no—
202C2#201011yescharacteristic
203C2#201211yescharacteristic
204C2#201711yescharacteristic
subgroupG/H
centerC2×C4#127
commutator, Frattini, cosocleC2×C42C23
socleC22#392
prime radical#3100C2
LCSG ▷ C2×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
1. #3135C2semidirect
2. #3101C2semidirect
4. #1955C2semidirect
5. #1481C2semidirect
6. #1480C2semidirect
7. #1275C2semidirect

Maximal subgroups

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

subgroupindex
#31352
#31012
#31002
#19552
#14812
#14802
#12752