Finite Group Databasebeta

← #24288

Subgroups — #24288

Order 256.

645 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 645 conjugacy classes of subgroups — too many to draw. Only its 133 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#3584C211yesmaximal, characteristic, complemented
2#3067C212yesmaximal, complemented
3#3018C212yesmaximal, complemented
4#2932C211yesmaximal, characteristic, complemented
5#2050C212yesmaximal, complemented
6#2017C212yesmaximal, complemented
7#1975C212yesmaximal, complemented
8#1813C211yesmaximal, characteristic, complemented
9#1791C212yesmaximal, complemented
10#585C2212yes—
11#578—22no—
12#577—24no—
13#576—22no—
14#575—22no—
15#575C2212yes—
16#575C2214yescomplemented
17C2×#135—22no—
18C2×#135C2212yes—
19C2×#131C2211yescharacteristic
20C2×#131C2211yescharacteristic, complemented
21#532C2211yescharacteristic
22#473—22no—
23#472—22no—
24#471—24no—
25#456—22no—
26#455—22no—
27#454—22no—
28#453—22no—
29#435—22no—
30#435—22no—
31C2×#108C2212yescomplemented
32#423C2211yescharacteristic, complemented
33C2×#104C2214yescomplemented
34#413C2212yes—
35#413C2214yescomplemented
36#408C2211yescharacteristic
37#408C2211yescharacteristic, complemented
38#408C2214yescomplemented
39C2×#97C2211yescharacteristic, complemented
40#338C2212yescomplemented
412−1+4—24no—
422−1+4—22no—
43H(5,𝔽2)—24no—
44H(5,𝔽2)—24no—
45H(5,𝔽2)—22no—
46C2×#41—24no—
47C2×#41—44no—
48C2×#41C2311yescharacteristic
49C2×#41C2314yes—
50C2×#41C2311yescharacteristic
51#137—28no—
52#137—24no—
53#137—24no—
54#137—24no—
55AGL(1,ℤ/8)—28no—
56AGL(1,ℤ/8)—24no—
57AGL(1,ℤ/8)—24no—
58AGL(1,ℤ/8)—24no—
59#135—24no—
60#135—24no—
61#135—22no—
62#135—24no—
63#135—22no—
64#135—22no—
65#135—22no—
66#135—44no—
67C2×Q16—22no—
68C2×Q16D412yescomplemented
69C2×SD16—24no—
70C2×SD16D414yescomplemented
71C2×D8—22no—
72C2×D8D412yescomplemented
73#131—22no—
74#131—22no—
75#131—22no—
76#131—22no—
77C2×M16—21no—
78C2×M16D412yes—
79C2×M16C2311yescharacteristic
80C2×M16C2314yes—
81C2×M16C2311yescharacteristic
82C22×C8—21no—
83C22×C8D412yes—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C4⋊Q8D411yescharacteristic
87#127D411yescharacteristic
88#124D411yescharacteristic
89#124D411yescharacteristic
90#121—22no—
91C4×D4—22no—
92C2×C42C2311yescharacteristic
93#108—22no—
94#108—22no—
95C4⋊C8—22no—
96#104—24no—
97#104—24no—
98#101—22no—
99#101—24no—
100#100—22no—
101#100—24no—
102C22⋊C8—22no—
103C22⋊C8—22no—
104C22⋊C8—44no—
105#97—21no—
106#97—21no—
107#41—22no—
108#41—22no—
109#41—28no—
110#41—24no—
111#41—22no—
112#41—24no—
113#41—24no—
114#41—22no—
115#41—24no—
116#41—48no—
117#41—44no—
118#41—44no—
119#41—44no—
120#41—44no—
121C2×Q8—24no—
122C2×Q8—44no—
123C2×Q8C2×D414yes—
124C2×Q8C2×D411yescharacteristic
125C2×Q8C2×D411yescharacteristic
126C2×D4—24no—
127C2×D4—21no—
128C2×D4—24no—
129C2×D4—22no—
130C2×D4—44no—
131C2×D4—48no—
132C2×D4—44no—
133C2×D4C2×D414yes—
134C2×D4C2×D411yescharacteristic
135C2×D4C2×D411yescharacteristic
136C2×D4#4112yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—44no—
141C22×C4C2×D412yes—
142C22×C4C2411yescharacteristic
143Q16—24no—
144Q16—22no—
145Q16—22no—
146Q16—44no—
147SD16—28no—
148SD16—24no—
149SD16—24no—
150SD16—44no—
151SD16—44no—
152D8—24no—
153D8—22no—
154D8—22no—
155D8—44no—
156M16—22no—
157M16—24no—
158M16—22no—
159M16—24no—
160M16—24no—
161M16—41no—
162M16—41no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—22no—
168C2×C8—24no—
169C2×C8—42no—
170C2×C8C2×D412yes—
171C2×C8C2×D411yescharacteristic
172C2×C8C2×D411yescharacteristic
173C2×C8C2×D412yes—
174C2×C8C2×D412yes—
175C4⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C42—21no—
179C42C2×D411yescharacteristic
180C42C2×D411yescharacteristic
181C23—21no—
182C23—24no—
183C23—44no—
184C23#12112yes—
185C23C2×#4111yescharacteristic
186Q8—22no—
187Q8—24no—
188Q8—22no—
189Q8—24no—
190Q8—44no—
191D4—24no—
192D4—24no—
193D4—22no—
194D4—22no—
195D4—24no—
196D4—44no—
197D4—48no—
198D4—44no—
199D4—44no—
200D4—42no—
201D4—82no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—48no—
209C2×C4—44no—
210C2×C4—41no—
211C2×C4—44no—
212C2×C4—41no—
213C2×C4—44no—
214C2×C4—81no—
215C2×C4#12112yes—
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D412yes—
218C2×C4C22×D411yescharacteristic
219C2×C4H(5,𝔽2)12yes—
220C8—22no—
221C8—41no—
222C8—41no—
223C8—44no—
224C8#12011yescharacteristic
225C8#12011yescharacteristic
226C8#12012yes—
227C22—22no—
228C22—41no—
229C22—42no—
230C22—42no—
231C22—44no—
232C22—41no—
233C22—84no—
234C22—164no—
235C22C2×#12111yescharacteristic
236C22#54612yes—
237C4—22no—
238C4—41no—
239C4—41no—
240C4—44no—
241C4—41no—
242C4C2×#12011yescharacteristic
243C4#53511yescharacteristic
244C4D4212yes—
245C2—21no—
246C2—22no—
247C2—42no—
248C2—81no—
249C2—84no—
250C2#240811yescharacteristic
subgroupG/H
centerC4#535
commutator, Frattini, cosocleC22×C4C24
socleC2#2408
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. #3584C2semidirect
2. #3067C2semidirect
3. #3018C2semidirect
4. #2932C2semidirect
5. #2050C2semidirect
6. #2017C2semidirect
7. #1975C2semidirect
8. #1813C2semidirect
9. #1791C2semidirect
16. #575C22semidirect
20. C2×#131C22semidirect
31. C2×#108C22semidirect
32. #423C22semidirect
33. C2×#104C22semidirect
35. #413C22semidirect
37. #408C22semidirect
38. #408C22semidirect
39. C2×#97C22semidirect
40. #338C22semidirect
68. C2×Q16D4semidirect
70. C2×SD16D4semidirect
72. C2×D8D4semidirect

Maximal subgroups

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

subgroupindex
#35842
#30672
#30182
#29322
#20502
#20172
#19752
#18132
#17912