Finite Group Databasebeta

← #31280

Subgroups — #31280

Order 256.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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#3135C211yesmaximal, characteristic, complemented
2#3018C211yesmaximal, characteristic
3#2080C212yesmaximal, complemented
4#2053C212yesmaximal, complemented
5#2031C212yesmaximal, complemented
6#1938C212yesmaximal, complemented
7#1884C211yesmaximal, characteristic, complemented
8#1881C212yesmaximal, complemented
9#1794C212yesmaximal, complemented
10#585C2211yescharacteristic
11#575C2212yes—
12#568C2211yescharacteristic, complemented
13#533C2211yescharacteristic, complemented
14#532C2211yescharacteristic
15#491—21no—
16#490—21no—
17#489—21no—
18#489—21no—
19#456—21no—
20#455—21no—
21#454—21no—
22#453—21no—
23#424C2212yescomplemented
24#421C2212yes—
25#421C2212yes—
26C2×#104C2212yes—
27#416C2212yes—
28#416C2212yes—
29#416C2212yescomplemented
30#413C2211yescharacteristic
31#410C2212yes—
32#410C2211yescharacteristic, complemented
33#408C2212yes—
34C2×#97C2211yescharacteristic, complemented
35#344C2212yescomplemented
36#341C2212yescomplemented
37#339C2212yescomplemented
38H(3,ℤ/4)C2212yescomplemented
392−1+4—21no—
402−1+4—22no—
41H(5,𝔽2)—21no—
42H(5,𝔽2)—22no—
43H(5,𝔽2)—41no—
44C2×#41—42no—
45C2×#41—41no—
46C2×#41C2311yescharacteristic
47C2×#41C2311yescharacteristic
48C2×#41C2312yes—
49#137—22no—
50#137—22no—
51AGL(1,ℤ/8)—22no—
52AGL(1,ℤ/8)—22no—
53#135—42no—
54#131—42no—
55C2×M16—41no—
56C2×M16C2312yes—
57C2×M16C2312yes—
58C22×C8C2312yes—
59C4⋊Q8—21no—
60C4⋊Q8D411yescharacteristic
61#127D411yescharacteristic
62#125—21no—
63#124D411yescharacteristic
64#124D411yescharacteristic
65C22⋊Q8—41no—
66#121—41no—
67C4×Q8—41no—
68C4×D4—41no—
69#117C2312yes—
70#117C2312yes—
71C2×C42C2311yescharacteristic
72C4⋊C8—42no—
73#104—22no—
74#104—22no—
75#104—42no—
76#104—42no—
77#103—22no—
78#103—22no—
79#103—22no—
80#102—22no—
81#102—22no—
82#102—22no—
83#101—21no—
84#100—21no—
85#99—42no—
86#99—41no—
87C22⋊C8—42no—
88#97—21no—
89#97—21no—
90#41—22no—
91#41—22no—
92#41—22no—
93#41—22no—
94#41—22no—
95#41—22no—
96#41—22no—
97#41—22no—
98#41—42no—
99#41—44no—
100#41—42no—
101#41—42no—
102#41—42no—
103#41—42no—
104C2×Q8—41no—
105C2×Q8—42no—
106C2×Q8—41no—
107C2×Q8C2×D411yescharacteristic
108C2×Q8C2×D411yescharacteristic
109C2×Q8#4112yes—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—41no—
114C2×D4—42no—
115C2×D4—41no—
116C2×D4—42no—
117C2×D4—44no—
118C2×D4C2×D411yescharacteristic
119C2×D4C2×D411yescharacteristic
120C2×D4#4112yes—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—42no—
126C22×C4C2411yescharacteristic
127Q16—42no—
128SD16—42no—
129SD16—42no—
130D8—42no—
131M16—22no—
132M16—22no—
133M16—22no—
134M16—22no—
135M16—42no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—42no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—41no—
145C4⋊C4—42no—
146C4⋊C4—41no—
147C22⋊C4—22no—
148C22⋊C4—22no—
149C22⋊C4—41no—
150C22⋊C4—41no—
151C42—21no—
152C42—22no—
153C42—22no—
154C42C2×D411yescharacteristic
155C42C2×D411yescharacteristic
156C23—21no—
157C23—21no—
158C23—22no—
159C23—42no—
160C23H(5,𝔽2)11yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—22no—
164Q8—22no—
165Q8—42no—
166Q8—82no—
167D4—22no—
168D4—22no—
169D4—22no—
170D4—22no—
171D4—42no—
172D4—44no—
173D4—42no—
174D4—42no—
175D4—42no—
176D4—42no—
177D4—82no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4C22×D411yescharacteristic
195C2×C4C2×#4112yes—
196C2×C4H(5,𝔽2)12yes—
197C2×C4H(5,𝔽2)11yescharacteristic
198C8—42no—
199C8—42no—
200C8—42no—
201C22—41no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22—81no—
207C22—82no—
208C22—162no—
209C22#53411yescharacteristic
210C22#53812yes—
211C4—41no—
212C4—41no—
213C4—41no—
214C4—42no—
215C4—41no—
216C4—82no—
217C4—82no—
218C4#54612yes—
219C4#55111yescharacteristic
220C4H(3,𝔽4)11yescharacteristic
221C2—21no—
222C2—22no—
223C2—81no—
224C2—81no—
225C2—82no—
226C2#268411yescharacteristic
subgroupG/H
centerC4H(3,𝔽4)
commutator, Frattini, cosocleC22×C4C24
socleC2#2684
prime radical#3018C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3135C2semidirect
3. #2080C2semidirect
4. #2053C2semidirect
5. #2031C2semidirect
6. #1938C2semidirect
7. #1884C2semidirect
8. #1881C2semidirect
9. #1794C2semidirect
12. #568C22semidirect
13. #533C22semidirect
23. #424C22semidirect
29. #416C22semidirect
32. #410C22semidirect
34. C2×#97C22semidirect
35. #344C22semidirect
36. #341C22semidirect
37. #339C22semidirect
38. H(3,ℤ/4)C22semidirect

Maximal subgroups

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

subgroupindex
#31352
#30182
#20802
#20532
#20312
#19382
#18842
#18812
#17942