Finite Group Databasebeta

← #9062

Subgroups — #9062

Order 256.

309 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#486C211yesmaximal, characteristic
2#2365C211yesmaximal, characteristic, complemented
3#1982C211yesmaximal, characteristic
4#1982C211yesmaximal, characteristic
5C2×#332C211yesmaximal, characteristic, complemented
6#1279C211yesmaximal, characteristic, complemented
7#1279C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8C2211yescharacteristic
9C2×#127C2211yescharacteristic
10C2×C4×Q8—21no—
11#486—22no—
12#486C414yescomplemented
13C2×C4⋊C8C2211yescharacteristic
14C2×C4⋊C8C2211yescharacteristic
15C2×#102—22no—
16C2×#102—21no—
17C2×#102—21no—
18C2×#102—21no—
19C2×#102—21no—
20C2×C4×C8C2211yescharacteristic
21#391—21no—
22#389—21no—
23#387—22no—
24C4×C4⋊C4—21no—
25C4×C4⋊C4C2211yescharacteristic
26C4×C4⋊C4C2211yescharacteristic
27#332—22no—
28C22×Q8—21no—
29C22×D4—21no—
30C22×D4—21no—
31C22×D4—21no—
32C22×C8—21no—
33C22×C8—21no—
34C22×C8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8C2×C412yes—
38C4⋊Q8D412yes—
39#127—21no—
40#127—21no—
41#127—21no—
42#127C2×C412yes—
43C4×Q8—42no—
44C4×Q8—42no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4D411yescharacteristic
48C2×C4⋊C4D411yescharacteristic
49C2×C4⋊C4D412yes—
50C2×C4⋊C4D411yescharacteristic
51C2×C4⋊C4D411yescharacteristic
52C2×C42—22no—
53C2×C42—21no—
54C2×C42—21no—
55C2×C42C2311yescharacteristic
56C4⋊C8—21no—
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60#102—28no—
61#102—42no—
62#102—42no—
63#102—42no—
64#102—42no—
65C4×C8—21no—
66C4×C8C2×C412yes—
67#95—22no—
68#95—22no—
69#95—21no—
70#95—21no—
71C24—41no—
72C2×Q8—22no—
73C2×Q8—22no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×Q8—41no—
78C2×D4—22no—
79C2×D4—22no—
80C2×D4—22no—
81C2×D4—42no—
82C2×D4—42no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—41no—
93C2×D4—41no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C2×C8—22no—
103C2×C8—41no—
104C2×C8—41no—
105C2×C8—41no—
106C2×C8—41no—
107C2×C8—41no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—41no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—44no—
121C4⋊C4C22⋊C414yes—
122C4⋊C4D812yes—
123C4⋊C4SD1612yes—
124C42—44no—
125C42—41no—
126C42—41no—
127C42—41no—
128C42—41no—
129C42C22×C411yescharacteristic
130C42C2×D411yescharacteristic
131C42#4111yescharacteristic
132C42#4111yescharacteristic
133C23—42no—
134C23—81no—
135C23—81no—
136C23—81no—
137C23—81no—
138C23—81no—
139C23—81no—
140C23#12011yescharacteristic
141Q8—42no—
142Q8—42no—
143Q8—42no—
144Q8—42no—
145D4—44no—
146D4—42no—
147D4—42no—
148D4—42no—
149D4—42no—
150D4—82no—
151D4—81no—
152D4—81no—
153D4—81no—
154D4—81no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4C2×C22⋊C411yescharacteristic
173C2×C4C4×D411yescharacteristic
174C2×C4C4×D411yescharacteristic
175C2×C4#12111yescharacteristic
176C2×C4#12111yescharacteristic
177C2×C4#12311yescharacteristic
178C2×C4C2×D811yescharacteristic
179C2×C4C2×SD1611yescharacteristic
180C2×C4#13511yescharacteristic
181C2×C4#13511yescharacteristic
182C2×C4AGL(1,ℤ/8)11yescharacteristic
183C2×C4AGL(1,ℤ/8)11yescharacteristic
184C2×C4AGL(1,ℤ/8)11yescharacteristic
185C2×C4#13711yescharacteristic
186C8—42no—
187C8—81no—
188C8—81no—
189C22—82no—
190C22—82no—
191C22—82no—
192C22—161no—
193C22—161no—
194C22—161no—
195C22—161no—
196C22#38611yescharacteristic
197C22#44711yescharacteristic
198C22#44811yescharacteristic
199C22#44911yescharacteristic
200C22#44911yescharacteristic
201C22#45411yescharacteristic
202C22#45411yescharacteristic
203C4—21no—
204C4—21no—
205C4—44no—
206C4—42no—
207C4—42no—
208C4—81no—
209C4—81no—
210C4C4×D811yescharacteristic
211C4C4×SD1611yescharacteristic
212C4#44011yescharacteristic
213C4#44211yescharacteristic
214C4#45911yescharacteristic
215C4#46011yescharacteristic
216C4#46311yescharacteristic
217C4#46411yescharacteristic
218C2—162no—
219C2#162511yescharacteristic
220C2#162611yescharacteristic
221C2#162711yescharacteristic
222C2#162811yescharacteristic
223C2#187911yescharacteristic
224C2#188011yescharacteristic
225C2#188811yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#127C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2365C2semidirect
5. C2×#332C2semidirect
6. #1279C2semidirect
7. #1279C2semidirect
12. #486C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4862
#23652
#19822
C2×#3322
#12792