Finite Group Databasebeta

← #30551

Subgroups — #30551

Order 256.

472 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 107 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#3136C212yesmaximal, complemented
2#3008C212yesmaximal, complemented
3#2051C212yesmaximal, complemented
4#2027C211yesmaximal, characteristic, complemented
5#2011C211yesmaximal, characteristic, complemented
6#2002C212yesmaximal, complemented
7#1978C212yesmaximal, complemented
8#1929C212yesmaximal, complemented
9#1792C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)C2211yescharacteristic
11C2×AGL(1,ℤ/8)C2212yes—
12C22×D8C2212yes—
13#544C2211yescharacteristic, complemented
14#535C2212yes—
15C2×#127C2211yescharacteristic, complemented
16#490—22no—
17#489—22no—
18#488—22no—
19#486—22no—
20#449—22no—
21#449—22no—
22#447—22no—
23#447—22no—
24C2×#108C2212yescomplemented
25C2×#104C2212yes—
26#418C2212yes—
27#418C2212yescomplemented
28#417C2212yes—
29#416C2212yes—
30C2×#102C2212yescomplemented
31C2×#100C2211yescharacteristic
32C2×#100C2212yescomplemented
33#408C2212yes—
34#405C2212yescomplemented
35#390C2211yescharacteristic
36#343C2211yescharacteristic, complemented
37#339C2212yescomplemented
38#336C2211yescharacteristic, complemented
39H(5,𝔽2)—22no—
40H(5,𝔽2)—42no—
41H(5,𝔽2)—41no—
42C2×#41—22no—
43C2×#41C2312yes—
44C22×D4—22no—
45C22×D4—41no—
46C22×D4—41no—
47C22×D4C2312yes—
48C22×D4C2311yescharacteristic
49AGL(1,ℤ/8)—42no—
50AGL(1,ℤ/8)—42no—
51C2×SD16—22no—
52C2×D8—22no—
53C2×D8—24no—
54C2×D8—22no—
55C2×D8—22no—
56C2×D8—42no—
57C2×M16C2312yes—
58C2×M16C2312yes—
59C22×C8C2312yes—
60C4⋊Q8—22no—
61#127—22no—
62#127—42no—
63#125—22no—
64#124—22no—
65C22⋊Q8—41no—
66#121—22no—
67#121—22no—
68#120—42no—
69#117C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—41no—
72C2×C42C2311yescharacteristic
73#108—42no—
74#104—42no—
75#103—22no—
76#103D412yescomplemented
77#103D412yescomplemented
78#102—22no—
79#102—22no—
80#102—22no—
81#102—22no—
82#102—22no—
83#102—22no—
84#102—22no—
85#102D412yescomplemented
86#102D412yescomplemented
87#100—41no—
88#100—42no—
89C22⋊C8—22no—
90C22⋊C8—22no—
91#97—22no—
92C4×C8—22no—
93C24—21no—
94C24—22no—
95C24—41no—
96#41—22no—
97#41—22no—
98#41—22no—
99#41—22no—
100#41—44no—
101#41—44no—
102C2×Q8—41no—
103C2×Q8C2×D412yes—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4—44no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—44no—
121C2×D4—44no—
122C2×D4—82no—
123C2×D4—82no—
124C2×D4C2×D412yes—
125C2×D4#4112yes—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4C2411yescharacteristic
131SD16—44no—
132D8—44no—
133D8—44no—
134D8—44no—
135M16—22no—
136M16—22no—
137M16—22no—
138M16—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8C2×D412yes—
144C2×C8C2×D412yes—
145C4⋊C4—22no—
146C4⋊C4—41no—
147C4⋊C4—42no—
148C4⋊C4C2×D412yes—
149C4⋊C4C2×D412yes—
150C22⋊C4—22no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—82no—
154C42—22no—
155C42—22no—
156C23—22no—
157C23—22no—
158C23—44no—
159C23—41no—
160C23—41no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23—81no—
165C23—82no—
166C23—82no—
167C23—81no—
168C23—161no—
169C23—161no—
170C23H(5,𝔽2)11yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—161no—
174D4—22no—
175D4—24no—
176D4—22no—
177D4—48no—
178D4—42no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—42no—
183D4—42no—
184D4—42no—
185D4—88no—
186D4—162no—
187D4—162no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—44no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—81no—
199C2×C4C22×D412yes—
200C2×C4C2×#4111yescharacteristic
201C2×C4H(5,𝔽2)11yescharacteristic
202C2×C4H(5,𝔽2)12yes—
203C8—22no—
204C8—22no—
205C8—42no—
206C8—42no—
207C22—21no—
208C22—21no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—82no—
214C22—84no—
215C22—81no—
216C22—81no—
217C22—82no—
218C22—82no—
219C22—82no—
220C22—161no—
221C22—162no—
222C22—164no—
223C22#53811yescharacteristic
224C22D4211yescharacteristic
225C22H(3,𝔽4)11yescharacteristic
226C4—42no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—81no—
231C4#53512yes—
232C4#54612yes—
233C2—21no—
234C2—21no—
235C2—82no—
236C2—82no—
237C2—162no—
238C2#267011yescharacteristic
239C2#328811yescharacteristic
240C2#329211yescharacteristic
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. #3136C2semidirect
2. #3008C2semidirect
3. #2051C2semidirect
4. #2027C2semidirect
5. #2011C2semidirect
6. #2002C2semidirect
7. #1978C2semidirect
8. #1929C2semidirect
9. #1792C2semidirect
13. #544C22semidirect
15. C2×#127C22semidirect
24. C2×#108C22semidirect
27. #418C22semidirect
30. C2×#102C22semidirect
32. C2×#100C22semidirect
34. #405C22semidirect
36. #343C22semidirect
37. #339C22semidirect
38. #336C22semidirect
76. #103D4semidirect
77. #103D4semidirect
85. #102D4semidirect
86. #102D4semidirect

Maximal subgroups

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

subgroupindex
#31362
#30082
#20512
#20272
#20112
#20022
#19782
#19292
#17922