Finite Group Databasebeta

← #9321

Subgroups — #9321

Order 256.

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

normal subgroups only

This group has 237 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#3079C211yesmaximal, characteristic
2#2989C211yesmaximal, characteristic
3#1981C211yesmaximal, characteristic, complemented
4#1806C211yesmaximal, characteristic
5#1549C211yesmaximal, characteristic, complemented
6#1303C211yesmaximal, characteristic, complemented
7#1292C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8C2211yescharacteristic
9C4×#41C2211yescharacteristic
10#477C412yescomplemented
11#476C412yescomplemented
12#475—21no—
13#474—21no—
14#446—21no—
15#436—21no—
16#435—22no—
17#432—21no—
18#431C2211yescharacteristic
19#428—21no—
20#423—21no—
21#423C2211yescharacteristic
22#423C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#104—21no—
25#417—21no—
26C2×#103—22no—
27C4×C4⋊C4C2211yescharacteristic
28#339—21no—
29#332—21no—
30#332—21no—
31C2×#41D411yescharacteristic
32C22×Q8—21no—
33C2×M16—21no—
34C2×M16—22no—
35C2×M16—21no—
36C22×C8D412yes—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8C2×C412yes—
41C4×Q8C2×C411yescharacteristic
42C4×Q8D411yescharacteristic
43C4×D4—21no—
44C4×D4—21no—
45C4×D4C2×C411yescharacteristic
46C4×D4D411yescharacteristic
47#117—21no—
48#117D411yescharacteristic
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C42—21no—
53C2×C42—21no—
54C2×C42C2311yescharacteristic
55#107—22no—
56#106—22no—
57C4⋊C8—22no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8C2×C412yes—
64C4⋊C8D411yescharacteristic
65C4⋊C8D411yescharacteristic
66#104—42no—
67#103—22no—
68#103—44no—
69#102—22no—
70C22⋊C8—22no—
71C22⋊C8—21no—
72#97—22no—
73C4×C8—21no—
74#95—21no—
75#41—21no—
76#41—21no—
77#41—21no—
78#41—21no—
79C2×Q8—22no—
80C2×Q8—42no—
81C2×Q8—42no—
82C2×Q8—41no—
83C2×Q8C22⋊C411yescharacteristic
84C2×D4—21no—
85C2×D4C22⋊C411yescharacteristic
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4#4111yescharacteristic
94M16—41no—
95M16—41no—
96M16—41no—
97M16—41no—
98M16—82no—
99C2×C8—22no—
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—22no—
103C2×C8—42no—
104C2×C8—41no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4C22⋊C411yescharacteristic
114C4⋊C4C22⋊C411yescharacteristic
115C22⋊C4—21no—
116C22⋊C4—21no—
117C22⋊C4—21no—
118C42—21no—
119C42—21no—
120C42—21no—
121C42—41no—
122C42—41no—
123C42C22×C411yescharacteristic
124C42C2×D411yescharacteristic
125C42C2×D411yescharacteristic
126C42#4111yescharacteristic
127C23—21no—
128C23#12111yescharacteristic
129Q8—21no—
130Q8—21no—
131Q8—44no—
132Q8—81no—
133Q8—81no—
134D4—21no—
135D4—21no—
136D4—41no—
137D4—41no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—82no—
153C2×C4—81no—
154C2×C4C2×C22⋊C411yescharacteristic
155C2×C4C4×D411yescharacteristic
156C2×C4C4×D411yescharacteristic
157C2×C4#12011yescharacteristic
158C2×C4#12111yescharacteristic
159C2×C4#12311yescharacteristic
160C2×C4AGL(1,ℤ/8)11yescharacteristic
161C2×C4#13711yescharacteristic
162C2×C4#13711yescharacteristic
163C2×C4#13711yescharacteristic
164C8—42no—
165C8—42no—
166C8—41no—
167C8—41no—
168C8—81no—
169C22—21no—
170C22—21no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22#38611yescharacteristic
175C22#46811yescharacteristic
176C22#47011yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—82no—
184C4—82no—
185C4#41711yescharacteristic
186C4#41911yescharacteristic
187C4#45311yescharacteristic
188C4#45411yescharacteristic
189C4#46811yescharacteristic
190C4#47011yescharacteristic
191C2—21no—
192C2—21no—
193C2—81no—
194C2#188311yescharacteristic
195C2#189411yescharacteristic
196C2#190211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#41D4
LCSG ▷ 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
3. #1981C2semidirect
5. #1549C2semidirect
6. #1303C2semidirect
7. #1292C2semidirect
10. #477C4semidirect
11. #476C4semidirect

Maximal subgroups

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

subgroupindex
#30792
#29892
#19812
#18062
#15492
#13032
#12922