Finite Group Databasebeta

← #10013

Subgroups — #10013

Order 256.

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

normal subgroups only

This group has 299 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#3103C211yesmaximal, characteristic, complemented
2#3100C211yesmaximal, characteristic
3#2398C211yesmaximal, characteristic, complemented
4#1963C211yesmaximal, characteristic, complemented
5#1487C211yesmaximal, characteristic, complemented
6#1486C211yesmaximal, characteristic, complemented
7#1274C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#481—21no—
12#480—21no—
13#475—21no—
14#474—21no—
15#460—21no—
16#459—21no—
17#448—21no—
18#447—21no—
19C4×SD16—22no—
20C4×D8—22no—
21#431C2211yescharacteristic
22#431C2211yescharacteristic
23#423C2211yescharacteristic
24C2×#104—22no—
25#400—21no—
26#397—21no—
27#394—21no—
28#393—21no—
29C4×C22⋊C4—22no—
30#376C2211yescharacteristic
31Q8⋊C8—22no—
32D4⋊C8—22no—
33C2×#41—21no—
34C22×Q8D411yescharacteristic
35C22×D4D411yescharacteristic
36C23×C4—21no—
37C2×SD16—22no—
38C2×D8—22no—
39C2×M16—21no—
40C4⋊Q8D411yescharacteristic
41#127D411yescharacteristic
42#124—21no—
43#121—21no—
44#121—21no—
45C4×Q8—21no—
46C4×Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×D4—21no—
50C4×D4—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—21no—
61C2×C42—21no—
62C2×C42D412yes—
63C2×C42C2311yescharacteristic
64#107—22no—
65#106—22no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8—21no—
70#104—44no—
71#103—22no—
72#102—22no—
73#102—22no—
74#102—22no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C4×C8D412yes—
78C4×C8D412yes—
79#95—22no—
80#95—22no—
81#95—21no—
82#95—21no—
83C24—21no—
84#41—42no—
85#41—42no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—42no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—42no—
96C2×D4—41no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109SD16—44no—
110D8—44no—
111M16—42no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—41no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C22⋊C4—44no—
124C22⋊C4—44no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C42—22no—
130C42—22no—
131C42—41no—
132C42—41no—
133C42C2×D411yescharacteristic
134C42C2×D411yescharacteristic
135C42C2×D411yescharacteristic
136C42#4111yescharacteristic
137C23—22no—
138C23—42no—
139C23—41no—
140C23—41no—
141C23—42no—
142C23—41no—
143C23#12011yescharacteristic
144Q8—22no—
145Q8—22no—
146Q8—41no—
147Q8—41no—
148Q8—42no—
149D4—22no—
150D4—22no—
151D4—41no—
152D4—41no—
153D4—42no—
154D4—42no—
155D4—42no—
156D4—82no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—82no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4#12011yescharacteristic
176C2×C4#12011yescharacteristic
177C2×C4#12111yescharacteristic
178C2×C4#12111yescharacteristic
179C2×C4#12111yescharacteristic
180C2×C4#12711yescharacteristic
181C2×C4#13512yes—
182C2×C4AGL(1,ℤ/8)12yes—
183C8—42no—
184C8—42no—
185C8—81no—
186C22—21no—
187C22—21no—
188C22—44no—
189C22—42no—
190C22—82no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22#39211yescharacteristic
197C22#44912yes—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—42no—
203C4—82no—
204C4—81no—
205C4—81no—
206C4—81no—
207C4#46312yes—
208C4#49511yescharacteristic
209C4#49611yescharacteristic
210C2—22no—
211C2—82no—
212C2—161no—
213C2#201011yescharacteristic
214C2#201011yescharacteristic
215C2#201211yescharacteristic
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. #3103C2semidirect
3. #2398C2semidirect
4. #1963C2semidirect
5. #1487C2semidirect
6. #1486C2semidirect
7. #1274C2semidirect

Maximal subgroups

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

subgroupindex
#31032
#31002
#23982
#19632
#14872
#14862
#12742