Finite Group Databasebeta

← #8979

Subgroups — #8979

Order 256.

330 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 93 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×#460C211yesmaximal, characteristic
2Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#1982C211yesmaximal, characteristic
4#1957C211yesmaximal, characteristic
5C2×Q8⋊C8C211yesmaximal, characteristic, complemented
6#1296C211yesmaximal, characteristic, complemented
7#1279C211yesmaximal, characteristic, complemented
8C22×SD16—21no—
9C2×#127C2211yescharacteristic
10C2×C4×Q8—21no—
11C2×C4×Q8C2211yescharacteristic
12#460—22no—
13#460C414yescomplemented
14C2×C4⋊C8C2211yescharacteristic
15C2×C4⋊C8C2211yescharacteristic
16C2×#102—21no—
17C2×#102—22no—
18C2×#102—21no—
19C2×#102—21no—
20C2×C4×C8C2211yescharacteristic
21#389—21no—
22#389—22no—
23#384—21no—
24#384C2211yescharacteristic
25C4×C4⋊C4—21no—
26C4×C4⋊C4C2211yescharacteristic
27Q8⋊C8—22no—
28C22×Q8D411yescharacteristic
29C22×D4—21no—
30C22×D4—21no—
31C22×D4—21no—
32C2×SD16—24no—
33C2×SD16—42no—
34C2×SD16—42no—
35C22×C8—21no—
36C22×C8—21no—
37C22×C8—21no—
38#127—21no—
39#127—21no—
40#127—21no—
41#127C2×C412yes—
42C4×Q8—21no—
43C4×Q8—21no—
44C4×Q8—42no—
45C4×Q8—42no—
46C4×Q8C2×C412yes—
47C4×Q8D412yes—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4D411yescharacteristic
54C2×C4⋊C4D411yescharacteristic
55C2×C4⋊C4D411yescharacteristic
56C2×C4⋊C4D412yes—
57C2×C42—21no—
58C2×C42—22no—
59C2×C42—21no—
60C2×C42C2311yescharacteristic
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8C2×C412yes—
65#102—24no—
66#102—42no—
67#102—44no—
68#102—42no—
69C4×C8—21no—
70C4×C8—21no—
71#95—22no—
72#95—21no—
73C24—41no—
74C2×Q8—21no—
75C2×Q8—21no—
76C2×Q8—21no—
77C2×Q8C22⋊C412yes—
78C2×Q8D812yes—
79C2×Q8SD1612yes—
80C2×D4—22no—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4—42no—
84C2×D4—42no—
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—
94C2×D4—41no—
95C2×D4—41no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104SD16—48no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—41no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—42no—
121C4⋊C4—44no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C4⋊C4—41no—
125C4⋊C4—41no—
126C4⋊C4—82no—
127C4⋊C4C22⋊C412yes—
128C4⋊C4SD1612yescomplemented
129C42—22no—
130C42—22no—
131C42—44no—
132C42—41no—
133C42—41no—
134C42C22×C411yescharacteristic
135C42C2×D411yescharacteristic
136C42#4111yescharacteristic
137C42#4111yescharacteristic
138C23—42no—
139C23—81no—
140C23—81no—
141C23—81no—
142C23—81no—
143C23—81no—
144C23—81no—
145C23#12011yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—22no—
149Q8#10214yescomplemented
150D4—44no—
151D4—42no—
152D4—42no—
153D4—42no—
154D4—42no—
155D4—82no—
156D4—81no—
157D4—81no—
158D4—81no—
159D4—81no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4C2×C22⋊C411yescharacteristic
178C2×C4C4×D411yescharacteristic
179C2×C4C4×D411yescharacteristic
180C2×C4#12111yescharacteristic
181C2×C4#12111yescharacteristic
182C2×C4#12311yescharacteristic
183C2×C4C2×D811yescharacteristic
184C2×C4C2×SD1611yescharacteristic
185C2×C4C2×SD1611yescharacteristic
186C2×C4#13511yescharacteristic
187C2×C4AGL(1,ℤ/8)11yescharacteristic
188C2×C4AGL(1,ℤ/8)11yescharacteristic
189C2×C4#13711yescharacteristic
190C2×C4#13711yescharacteristic
191C8—42no—
192C8—42no—
193C8—81no—
194C22—82no—
195C22—82no—
196C22—82no—
197C22—161no—
198C22—161no—
199C22—161no—
200C22—161no—
201C22#38611yescharacteristic
202C22#44711yescharacteristic
203C22#44811yescharacteristic
204C22#44811yescharacteristic
205C22#44911yescharacteristic
206C22#45311yescharacteristic
207C22#45411yescharacteristic
208C4—24no—
209C4—21no—
210C4—21no—
211C4—42no—
212C4—42no—
213C4—81no—
214C4—82no—
215C4C2×#10211yescharacteristic
216C4#41711yescharacteristic
217C4C4×SD1611yescharacteristic
218C4#44011yescharacteristic
219C4#46011yescharacteristic
220C4#46411yescharacteristic
221C4#48011yescharacteristic
222C4#48111yescharacteristic
223C2—162no—
224C2#162111yescharacteristic
225C2#162311yescharacteristic
226C2#162611yescharacteristic
227C2#162711yescharacteristic
228C2#187411yescharacteristic
229C2#188011yescharacteristic
230C2#188311yescharacteristic
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. Q8×C4⋊C4C2semidirect
5. C2×Q8⋊C8C2semidirect
6. #1296C2semidirect
7. #1279C2semidirect
13. #460C4semidirect
128. C4⋊C4SD16semidirect
149. Q8#102semidirect

Maximal subgroups

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

subgroupindex
C2×#4602
Q8×C4⋊C42
#19822
#19572
C2×Q8⋊C82
#12962
#12792