Finite Group Databasebeta

← #24279

Subgroups — #24279

Order 256.

664 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22×#135C211yesmaximal, characteristic, complemented
2C2×#471C212yesmaximal, complemented
3#3005C212yesmaximal, complemented
4#2929C211yesmaximal, characteristic, complemented
5#2047C211yesmaximal, characteristic, complemented
6#2045C211yesmaximal, characteristic, complemented
7#2020C211yesmaximal, characteristic, complemented
8#2018C211yesmaximal, characteristic, complemented
9#1977C212yesmaximal, complemented
10#1810C211yesmaximal, characteristic, complemented
11#1785C212yesmaximal, complemented
12C22×#41C2212yes—
13C2×#137C2212yes—
14C2×AGL(1,ℤ/8)C2212yes—
15C2×#135—28no—
16C2×#135—22no—
17C2×#135—22no—
18C22×Q16C2211yescharacteristic
19C22×SD16C2212yes—
20C22×D8C2211yescharacteristic
21C2×#131C2211yescharacteristic
22C23×C8C2211yescharacteristic
23#525C2211yescharacteristic
24#471—28no—
25#452—24no—
26#449—24no—
27#435—22no—
28C8×D4—22no—
29C2×#108C2212yes—
30#424C2211yescharacteristic
31#419C2212yes—
32#418C2212yes—
33C2×#101C2212yes—
34C2×#101C2212yes—
35C2×#100C2212yes—
36C2×#100C2212yes—
37#408C2211yescharacteristic
38#407C2212yes—
39#407C2211yescharacteristic
40#405C2211yescharacteristic
41#341C2211yescharacteristic
42#341C2211yescharacteristic
43C2×#41—24no—
44C2×#41—24no—
45C2×#41—24no—
46C2×#41—24no—
47C2×#41—42no—
48C2×#41—42no—
49C2×#41C2311yescharacteristic
50C22×Q8C2312yes—
51C22×D4—22no—
52C22×D4C2312yes—
53C23×C4—22no—
54C23×C4C2311yescharacteristic
55#137—24no—
56#137—42no—
57AGL(1,ℤ/8)—24no—
58AGL(1,ℤ/8)—42no—
59#135—216no—
60#135—216no—
61C2×Q16—24no—
62C2×Q16—22no—
63C2×Q16—22no—
64C2×SD16—22no—
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16—22no—
68C2×SD16—24no—
69C2×SD16D418yescomplemented
70C2×D8—24no—
71C2×D8—22no—
72C2×D8—22no—
73#131—24no—
74C2×M16—21no—
75C2×M16C2311yescharacteristic
76C2×M16C2312yes—
77C2×M16C2312yes—
78C2×M16C2311yescharacteristic
79C22×C8—24no—
80C22×C8—21no—
81C22×C8D412yes—
82C22×C8D412yes—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85#123—21no—
86C22⋊Q8D412yes—
87#121D412yes—
88#120—21no—
89C4×D4—22no—
90#117C2311yescharacteristic
91#108—24no—
92C4⋊C8—22no—
93#103—24no—
94#102—24no—
95#101—24no—
96#101—42no—
97#100—24no—
98#100—42no—
99C22⋊C8—22no—
100C22⋊C8—24no—
101C22⋊C8—22no—
102#97—21no—
103C4×C8—21no—
104C24—22no—
105C24#4111yescharacteristic
106#41—216no—
107#41—22no—
108#41—22no—
109#41—48no—
110#41—48no—
111#41—48no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8C2×D414yes—
118C2×Q8C2×D411yescharacteristic
119C2×D4—22no—
120C2×D4—21no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—44no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4C2×D414yes—
130C2×D4C2×D411yescharacteristic
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4C2×D412yes—
139C22×C4C2×D412yes—
140C22×C4#4111yescharacteristic
141C22×C4C2411yescharacteristic
142Q16—28no—
143Q16—44no—
144SD16—28no—
145SD16—28no—
146SD16—44no—
147SD16—44no—
148D8—28no—
149D8—44no—
150M16—22no—
151M16—22no—
152M16—22no—
153M16—22no—
154M16—22no—
155M16—22no—
156M16—42no—
157C2×C8—22no—
158C2×C8—24no—
159C2×C8—24no—
160C2×C8—21no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—21no—
164C2×C8—42no—
165C2×C8C2×D412yes—
166C2×C8C2×D411yescharacteristic
167C2×C8C2×D411yescharacteristic
168C2×C8C2×D414yes—
169C4⋊C4—22no—
170C4⋊C4C2×D412yes—
171C22⋊C4—22no—
172C22⋊C4—21no—
173C22⋊C4—22no—
174C42—21no—
175C23—22no—
176C23—21no—
177C23—24no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—84no—
182C23#12112yes—
183C23C2×#4111yescharacteristic
184Q8—24no—
185Q8—22no—
186Q8—24no—
187Q8—42no—
188Q8—42no—
189D4—24no—
190D4—22no—
191D4—24no—
192D4—44no—
193D4—416no—
194D4—42no—
195D4—42no—
196D4—42no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—24no—
204C2×C4—416no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4#12112yes—
210C2×C4C22×D411yescharacteristic
211C2×C4C22×D412yes—
212C2×C4C22×D411yescharacteristic
213C2×C4H(5,𝔽2)12yes—
214C8—22no—
215C8—41no—
216C8—41no—
217C8—42no—
218C8—42no—
219C8#12012yes—
220C8#12012yes—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—44no—
226C22—44no—
227C22—41no—
228C22—42no—
229C22—84no—
230C22—84no—
231C22—84no—
232C22C2×#12111yescharacteristic
233C22#54612yes—
234C4—22no—
235C4—44no—
236C4—41no—
237C4—42no—
238C4C2×#12011yescharacteristic
239C4#53511yescharacteristic
240C4D4212yes—
241C2—22no—
242C2—42no—
243C2—81no—
244C2—84no—
245C2#240811yescharacteristic
246C2#306712yes—
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#135C2semidirect
2. C2×#471C2semidirect
3. #3005C2semidirect
4. #2929C2semidirect
5. #2047C2semidirect
6. #2045C2semidirect
7. #2020C2semidirect
8. #2018C2semidirect
9. #1977C2semidirect
10. #1810C2semidirect
11. #1785C2semidirect
69. C2×SD16D4semidirect

Maximal subgroups

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

subgroupindex
C22×#1352
C2×#4712
#30052
#29292
#20472
#20452
#20202
#20182
#19772
#18102
#17852