Finite Group Databasebeta

← #30496

Subgroups — #30496

Order 256.

522 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 113 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#3151C211yesmaximal, characteristic, complemented
2#3068C212yesmaximal, complemented
3#3021C212yesmaximal, complemented
4#3018C211yesmaximal, characteristic, complemented
5#2048C212yesmaximal, complemented
6#2008C212yesmaximal, complemented
7#2002C212yesmaximal, complemented
8#1932C212yesmaximal, complemented
9#1884C211yesmaximal, characteristic, complemented
10#585C2211yescharacteristic
11#575C2212yes—
12C2×AGL(1,ℤ/8)C2212yes—
13C2×AGL(1,ℤ/8)C2212yes—
14C2×AGL(1,ℤ/8)C2212yescomplemented
15#568C2211yescharacteristic, complemented
16#535C2212yes—
17#532C2211yescharacteristic
18C2×#127C2211yescharacteristic, complemented
19#496—21no—
20#496—21no—
21#493—21no—
22#492—21no—
23#472—24no—
24#471—24no—
25#456—21no—
26#455—21no—
27#454—21no—
28#454—24no—
29#453—21no—
30#453—24no—
31#430C2212yescomplemented
32#421C2212yes—
33#421C2212yes—
34C2×#104C2212yes—
35#418C2212yescomplemented
36#413C2211yescharacteristic
37#413C2212yes—
38C2×#100C2212yescomplemented
39#410C2211yescharacteristic, complemented
40C4×M16C2211yescharacteristic, complemented
41#344C2212yescomplemented
42#339C2212yescomplemented
432−1+4—22no—
442−1+4—21no—
45H(5,𝔽2)—22no—
46H(5,𝔽2)—21no—
47H(5,𝔽2)—41no—
48C2×#41—41no—
49C2×#41—42no—
50C2×#41C2311yescharacteristic
51C2×#41C2311yescharacteristic
52C2×#41C2312yes—
53C22×D4—41no—
54C22×D4C2312yes—
55#137—22no—
56#137—22no—
57AGL(1,ℤ/8)—22no—
58AGL(1,ℤ/8)—22no—
59AGL(1,ℤ/8)—24no—
60AGL(1,ℤ/8)—24no—
61AGL(1,ℤ/8)—24no—
62AGL(1,ℤ/8)—42no—
63AGL(1,ℤ/8)—42no—
64AGL(1,ℤ/8)—42no—
65#135—42no—
66C2×SD16—22no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×D8—22no—
70C2×D8—22no—
71C2×D8—22no—
72#131—24no—
73C2×M16—41no—
74C2×M16C2312yes—
75C2×M16C2312yes—
76C2×M16C2312yes—
77C4⋊Q8D411yescharacteristic, complemented
78#127—22no—
79#127—21no—
80#127—21no—
81#127—41no—
82#127D411yescharacteristic, complemented
83#124—22no—
84#124D411yescharacteristic, complemented
85#124D411yescharacteristic, complemented
86#121—42no—
87#121—41no—
88#120—42no—
89C4×D4—41no—
90#117C2312yes—
91C2×C42C2311yescharacteristic
92#108—24no—
93#104—24no—
94#104—24no—
95#104—22no—
96#104—22no—
97#102—42no—
98#101—21no—
99#101D414yescomplemented
100#100—21no—
101#100—24no—
102#100D414yescomplemented
103#99—41no—
104#97—21no—
105C4×C8—21no—
106C24—22no—
107#41—22no—
108#41—22no—
109#41—22no—
110#41—22no—
111#41—22no—
112#41—22no—
113#41—22no—
114#41—22no—
115#41—42no—
116#41—44no—
117#41—42no—
118#41—42no—
119#41—42no—
120#41—42no—
121C2×Q8—41no—
122C2×Q8—42no—
123C2×Q8C2×D411yescharacteristic
124C2×Q8C2×D412yes—
125C2×Q8C2×D411yescharacteristic
126C2×D4—21no—
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—41no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—41no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—44no—
142C2×D4—44no—
143C2×D4—84no—
144C2×D4C2×D411yescharacteristic
145C2×D4C2×D412yes—
146C2×D4C2×D411yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4C2411yescharacteristic
153Q16—42no—
154SD16—44no—
155SD16—44no—
156SD16—44no—
157SD16—42no—
158SD16—42no—
159D8—44no—
160D8—44no—
161D8—44no—
162D8—42no—
163M16—22no—
164M16—22no—
165M16—22no—
166M16—22no—
167M16—42no—
168M16C2×D414yes—
169M16C2×D414yes—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—42no—
174C4⋊C4—22no—
175C4⋊C4—41no—
176C22⋊C4—22no—
177C22⋊C4—41no—
178C22⋊C4—41no—
179C22⋊C4—44no—
180C42—21no—
181C42—22no—
182C42C2×D411yescharacteristic
183C42C2×D411yescharacteristic
184C23—21no—
185C23—21no—
186C23—22no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—84no—
192C23H(5,𝔽2)11yescharacteristic
193Q8—22no—
194Q8—22no—
195Q8—22no—
196Q8—22no—
197Q8—42no—
198D4—22no—
199D4—22no—
200D4—22no—
201D4—22no—
202D4—42no—
203D4—44no—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—42no—
208D4—42no—
209D4—42no—
210D4—42no—
211D4—42no—
212D4—82no—
213D4—84no—
214D4—84no—
215D4—164no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4C22×D411yescharacteristic
232C2×C4C22×D412yes—
233C2×C4H(5,𝔽2)11yescharacteristic
234C2×C4H(5,𝔽2)12yes—
235C8—24no—
236C8—24no—
237C8—42no—
238C22—41no—
239C22—41no—
240C22—42no—
241C22—41no—
242C22—81no—
243C22—81no—
244C22—82no—
245C22—84no—
246C22—84no—
247C22—82no—
248C22—162no—
249C22—162no—
250C22#53411yescharacteristic
251C22#53512yes—
252C4—22no—
253C4—41no—
254C4—41no—
255C4—42no—
256C4—41no—
257C4—82no—
258C4D4211yescharacteristic
259C4D4212yes—
260C4#56011yescharacteristic
261C2—21no—
262C2—22no—
263C2—81no—
264C2—81no—
265C2—82no—
266C2—162no—
267C2#266911yescharacteristic
subgroupG/H
center, socleC2#2669
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3151C2semidirect
2. #3068C2semidirect
3. #3021C2semidirect
4. #3018C2semidirect
5. #2048C2semidirect
6. #2008C2semidirect
7. #2002C2semidirect
8. #1932C2semidirect
9. #1884C2semidirect
14. C2×AGL(1,ℤ/8)C22semidirect
15. #568C22semidirect
18. C2×#127C22semidirect
31. #430C22semidirect
35. #418C22semidirect
38. C2×#100C22semidirect
39. #410C22semidirect
40. C4×M16C22semidirect
41. #344C22semidirect
42. #339C22semidirect
77. C4⋊Q8D4semidirect
82. #127D4semidirect
84. #124D4semidirect
85. #124D4semidirect
99. #101D4semidirect
102. #100D4semidirect

Maximal subgroups

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

subgroupindex
#31512
#30682
#30212
#30182
#20482
#20082
#20022
#19322
#18842