Finite Group Databasebeta

← #30497

Subgroups — #30497

Order 256.

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

normal subgroups only

This group has 490 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#3152C211yesmaximal, characteristic, complemented
2#3069C212yesmaximal, complemented
3#3022C212yesmaximal, complemented
4#3018C211yesmaximal, characteristic
5#2049C212yesmaximal, complemented
6#2009C212yesmaximal, complemented
7#2005C212yesmaximal, complemented
8#1932C212yesmaximal, complemented
9#1884C211yesmaximal, characteristic, complemented
10#585C2211yescharacteristic
11#575C2212yes—
12C2×#137C2212yes—
13C2×#137C2212yes—
14C2×#137C2212yescomplemented
15#568C2211yescharacteristic, complemented
16#536C2212yes—
17#532C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic, complemented
19#497—21no—
20#497—21no—
21#494—21no—
22#492—21no—
23#473—24no—
24#471—24no—
25#456—21no—
26#456—24no—
27#455—21no—
28#455—24no—
29#454—21no—
30#453—21no—
31#430C2212yescomplemented
32#421C2212yes—
33#421C2212yes—
34C2×#104C2212yes—
35#419C2212yescomplemented
36#413C2211yescharacteristic
37#413C2212yes—
38C2×#101C2212yescomplemented
39#410C2211yescharacteristic, complemented
40C4×M16C2211yescharacteristic, complemented
41#344C2212yescomplemented
42#339C2212yescomplemented
432−1+4—21no—
442−1+4—22no—
45H(5,𝔽2)—21no—
46H(5,𝔽2)—22no—
47H(5,𝔽2)—41no—
48C2×#41—41no—
49C2×#41—42no—
50C2×#41C2311yescharacteristic
51C2×#41C2311yescharacteristic
52C2×#41C2312yes—
53C22×Q8C2312yes—
54#137—22no—
55#137—22no—
56#137—24no—
57#137—24no—
58#137—24no—
59#137—42no—
60#137—42no—
61#137—42no—
62AGL(1,ℤ/8)—22no—
63AGL(1,ℤ/8)—22no—
64#135—42no—
65C2×Q16—22no—
66C2×Q16—22no—
67C2×Q16—22no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71#131—24no—
72C2×M16—41no—
73C2×M16C2312yes—
74C2×M16C2312yes—
75C2×M16C2312yes—
76C4⋊Q8—22no—
77C4⋊Q8—21no—
78C4⋊Q8—21no—
79C4⋊Q8—41no—
80C4⋊Q8D411yescharacteristic
81#127D411yescharacteristic
82#124—22no—
83#124D411yescharacteristic
84#124D411yescharacteristic
85#123—42no—
86C22⋊Q8—42no—
87#121—41no—
88C4×D4—41no—
89#117C2312yes—
90C2×C4⋊C4—41no—
91C2×C42C2311yescharacteristic
92#108—24no—
93#104—24no—
94#104—24no—
95#104—22no—
96#104—22no—
97#103—42no—
98#101—21no—
99#101—24no—
100#101D414yescomplemented
101#100—21no—
102#100D414yescomplemented
103#99—41no—
104#97—21no—
105C4×C8—21no—
106#41—22no—
107#41—22no—
108#41—22no—
109#41—22no—
110#41—22no—
111#41—22no—
112#41—22no—
113#41—22no—
114#41—42no—
115#41—44no—
116#41—42no—
117#41—42no—
118#41—42no—
119#41—42no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—42no—
127C2×Q8—41no—
128C2×Q8C2×D411yescharacteristic
129C2×Q8C2×D412yes—
130C2×Q8C2×D411yescharacteristic
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—22no—
134C2×D4—41no—
135C2×D4—42no—
136C2×D4—41no—
137C2×D4—42no—
138C2×D4—44no—
139C2×D4C2×D411yescharacteristic
140C2×D4C2×D412yes—
141C2×D4C2×D411yescharacteristic
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—42no—
148C22×C4C2411yescharacteristic
149Q16—44no—
150Q16—44no—
151Q16—44no—
152Q16—42no—
153SD16—44no—
154SD16—44no—
155SD16—44no—
156SD16—42no—
157SD16—42no—
158D8—42no—
159M16—22no—
160M16—22no—
161M16—22no—
162M16—22no—
163M16—42no—
164M16C2×D414yes—
165M16C2×D414yes—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—42no—
170C4⋊C4—22no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C22⋊C4—22no—
176C22⋊C4—41no—
177C22⋊C4—41no—
178C22⋊C4—44no—
179C42—21no—
180C42—22no—
181C42C2×D411yescharacteristic
182C42C2×D411yescharacteristic
183C23—21no—
184C23—21no—
185C23—22no—
186C23—42no—
187C23H(5,𝔽2)11yescharacteristic
188Q8—22no—
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—42no—
194Q8—42no—
195Q8—42no—
196Q8—42no—
197D4—22no—
198D4—22no—
199D4—22no—
200D4—22no—
201D4—42no—
202D4—44no—
203D4—42no—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—82no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4C22×D411yescharacteristic
227C2×C4C22×D412yes—
228C2×C4H(5,𝔽2)11yescharacteristic
229C2×C4H(5,𝔽2)12yes—
230C8—24no—
231C8—24no—
232C8—42no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22—81no—
239C22—82no—
240C22—162no—
241C22#53411yescharacteristic
242C22#53512yes—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—41no—
248C4—82no—
249C4—82no—
250C4D4211yescharacteristic
251C4D4212yes—
252C4#56011yescharacteristic
253C2—21no—
254C2—22no—
255C2—81no—
256C2—81no—
257C2—82no—
258C2#266911yescharacteristic
subgroupG/H
center, socleC2#2669
commutator, Frattini, cosocleC22×C4C24
prime radical#3018C2
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. #3152C2semidirect
2. #3069C2semidirect
3. #3022C2semidirect
5. #2049C2semidirect
6. #2009C2semidirect
7. #2005C2semidirect
8. #1932C2semidirect
9. #1884C2semidirect
14. C2×#137C22semidirect
15. #568C22semidirect
18. C2×C4⋊Q8C22semidirect
31. #430C22semidirect
35. #419C22semidirect
38. C2×#101C22semidirect
39. #410C22semidirect
40. C4×M16C22semidirect
41. #344C22semidirect
42. #339C22semidirect
100. #101D4semidirect
102. #100D4semidirect

Maximal subgroups

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

subgroupindex
#31522
#30692
#30222
#30182
#20492
#20092
#20052
#19322
#18842