Finite Group Databasebeta

← #31072

Subgroups — #31072

Order 256.

544 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 115 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×#454C211yesmaximal, characteristic, complemented
2C2×#455C211yesmaximal, characteristic, complemented
3#3013C212yesmaximal, complemented
4#3012C212yesmaximal, complemented
5#2052C212yesmaximal, complemented
6#2051C212yesmaximal, complemented
7#2040C212yesmaximal, complemented
8#1974C212yesmaximal, complemented
9#1958C211yesmaximal, characteristic, complemented
10C2×2−1+4C2211yescharacteristic, complemented
11C2×H(5,𝔽2)C2211yescharacteristic
12C2×#137C2212yes—
13C2×AGL(1,ℤ/8)C2212yes—
14C2×#135C2212yes—
15#537C2212yes—
16C2×#124C2211yescharacteristic
17#455—24no—
18#454—24no—
19#452—22no—
20#452—22no—
21#451—22no—
22#450—22no—
23#449—22no—
24#449—22no—
25#448—22no—
26#447—22no—
27C2×#108C2212yescomplemented
28C2×#104C2212yes—
29C2×#104C2212yescomplemented
30#417C2212yes—
31#417C2212yescomplemented
32C2×#103C2212yescomplemented
33C2×#102C2212yes—
34C2×#101C2211yescharacteristic, complemented
35C2×#100C2211yescharacteristic
36#408C2212yes—
37#408C2212yescomplemented
38C2×C4×C8C2211yescharacteristic, complemented
39#382C2211yescharacteristic, complemented
40#343C2212yescomplemented
412−1+4—22no—
422−1+4—41no—
432−1+4—42no—
44H(5,𝔽2)—22no—
45H(5,𝔽2)—42no—
46H(5,𝔽2)—41no—
47C2×#41—22no—
48C2×#41—22no—
49C2×#41—41no—
50C2×#41C2312yes—
51C2×#41C2312yes—
52C22×Q8—22no—
53C22×Q8C2311yescharacteristic
54C22×D4—22no—
55C22×D4—41no—
56C22×D4C2311yescharacteristic
57#137—24no—
58#137—42no—
59AGL(1,ℤ/8)—24no—
60AGL(1,ℤ/8)—42no—
61#135—42no—
62#135—42no—
63C2×Q16—22no—
64C2×Q16D412yes—
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16D412yes—
68C2×SD16D412yes—
69C2×D8—22no—
70C2×D8D412yes—
71C2×M16C2312yes—
72C2×M16C2312yes—
73C22×C8C2312yes—
74#124—41no—
75#124D414yescomplemented
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78#121—22no—
79#121—22no—
80#121—42no—
81C2×C4⋊C4C2312yes—
82C2×C22⋊C4—41no—
83C2×C42C2311yescharacteristic
84#108—42no—
85#104—24no—
86#104—24no—
87#103—22no—
88#103—22no—
89#103—22no—
90#103—22no—
91#102—22no—
92#102—22no—
93#102—22no—
94#102—22no—
95#101—22no—
96#100—22no—
97C22⋊C8—22no—
98C22⋊C8—22no—
99C22⋊C8—22no—
100C22⋊C8—22no—
101C4×C8—41no—
102#95—41no—
103C24—21no—
104C24—41no—
105#41—22no—
106#41—22no—
107#41—22no—
108#41—22no—
109#41—22no—
110#41—22no—
111#41—22no—
112#41—22no—
113#41—44no—
114#41—44no—
115#41—44no—
116#41—44no—
117#41—44no—
118#41—44no—
119C2×Q8—21no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—41no—
123C2×Q8—44no—
124C2×Q8—44no—
125C2×Q8C2×D412yes—
126C2×Q8C2×D412yes—
127C2×Q8C2×D412yes—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—21no—
133C2×D4—42no—
134C2×D4—44no—
135C2×D4—44no—
136C2×D4—41no—
137C2×D4—44no—
138C2×D4—44no—
139C2×D4—42no—
140C2×D4C2×D412yes—
141C2×D4C2×D412yes—
142C2×D4C2×D412yes—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—41no—
149C22×C4—42no—
150C22×C4C2411yescharacteristic
151Q16—42no—
152Q16—44no—
153SD16—42no—
154SD16—42no—
155SD16—44no—
156SD16—44no—
157D8—42no—
158D8—44no—
159M16—22no—
160M16—22no—
161M16—22no—
162M16—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8C2×D412yes—
168C2×C8C2×D412yes—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—42no—
172C22⋊C4—44no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C42—21no—
176C42C2×D412yes—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—44no—
181C23—41no—
182C23—41no—
183C23—82no—
184C23—162no—
185C23H(5,𝔽2)11yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—22no—
189Q8—24no—
190Q8—22no—
191Q8—44no—
192Q8—42no—
193D4—22no—
194D4—22no—
195D4—22no—
196D4—24no—
197D4—22no—
198D4—48no—
199D4—44no—
200D4—42no—
201D4—42no—
202D4—42no—
203D4—42no—
204D4—42no—
205D4—162no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—48no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—82no—
221C2×C4C22×D412yes—
222C2×C4C22×D411yescharacteristic
223C2×C4H(5,𝔽2)11yescharacteristic
224C2×C4H(5,𝔽2)12yes—
225C8—22no—
226C8—22no—
227C8—42no—
228C8—42no—
229C22—22no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—82no—
238C22—82no—
239C22—161no—
240C22—164no—
241C22#53411yescharacteristic
242C22D4211yescharacteristic
243C22H(3,𝔽4)11yescharacteristic
244C4—42no—
245C4—42no—
246C4—42no—
247C4—42no—
248C4—82no—
249C4#53412yes—
250C4D4212yes—
251C2—21no—
252C2—21no—
253C2—82no—
254C2—82no—
255C2—82no—
256C2#267911yescharacteristic
257C2#329012yes—
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#454C2semidirect
2. C2×#455C2semidirect
3. #3013C2semidirect
4. #3012C2semidirect
5. #2052C2semidirect
6. #2051C2semidirect
7. #2040C2semidirect
8. #1974C2semidirect
9. #1958C2semidirect
10. C2×2−1+4C22semidirect
27. C2×#108C22semidirect
29. C2×#104C22semidirect
31. #417C22semidirect
32. C2×#103C22semidirect
34. C2×#101C22semidirect
37. #408C22semidirect
38. C2×C4×C8C22semidirect
39. #382C22semidirect
40. #343C22semidirect
75. #124D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4542
C2×#4552
#30132
#30122
#20522
#20512
#20402
#19742
#19582