Finite Group Databasebeta

← #23990

Subgroups — #23990

Order 256.

630 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 121 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#3508C211yesmaximal, characteristic, complemented
2#3027C211yesmaximal, characteristic, complemented
3#3021C211yesmaximal, characteristic, complemented
4#2901C211yesmaximal, characteristic, complemented
5#2031C212yesmaximal, complemented
6#2002C212yesmaximal, complemented
7#1914C211yesmaximal, characteristic, complemented
8#1911C212yesmaximal, complemented
9#1910C211yesmaximal, characteristic, complemented
10#1904C212yesmaximal, complemented
11#1827C211yesmaximal, characteristic, complemented
12#585C2211yescharacteristic
13C2×AGL(1,ℤ/8)C2212yes—
14#568C2211yescharacteristic
15#559—22no—
16#550—22no—
17#546—24no—
18#540—22no—
19#535C2211yescharacteristic
20#535C2212yes—
21#535C2211yescharacteristic
22#533C2211yescharacteristic, complemented
23#532—22no—
24C2×#127C2211yescharacteristic, complemented
25#518C2211yescharacteristic, complemented
26#518C2211yescharacteristic, complemented
27C4×#41C2211yescharacteristic
28B(4,𝔽2)—24no—
29#454—22no—
30#453—22no—
31#421—22no—
32#421C2212yes—
33#421C2212yes—
34C2×#104—22no—
35C2×#104C2212yes—
36#418C2212yescomplemented
37#413C2211yescharacteristic
38C2×#100C2212yescomplemented
39#410C2211yescharacteristic
40#410C2212yes—
41#344C2212yescomplemented
42#339C2212yescomplemented
43H(3,ℤ/4)C2212yescomplemented
44H(3,ℤ/4)C2212yescomplemented
452−1+4—22no—
462−1+4—21no—
47H(5,𝔽2)—22no—
48H(5,𝔽2)—22no—
49H(5,𝔽2)—21no—
50C2×#41—24no—
51C2×#41—22no—
52C2×#41—42no—
53C2×#41C2312yes—
54C2×#41C2311yescharacteristic
55C22×D4—22no—
56C22×D4C2311yescharacteristic
57C22×D4C2311yescharacteristic
58AGL(1,ℤ/8)—24no—
59AGL(1,ℤ/8)—42no—
60C2×SD16—22no—
61C2×D8—22no—
62#131—42no—
63C2×M16—22no—
64C2×M16C2312yes—
65C2×M16C2312yes—
66C4⋊Q8—21no—
67#127—22no—
68#127—21no—
69#127—22no—
70#127—21no—
71#127—21no—
72#127D412yescomplemented
73#125—21no—
74#124—24no—
75#124—21no—
76#124—22no—
77#124D412yescomplemented
78#123—24no—
79#123—24no—
80C22⋊Q8—22no—
81#121—24no—
82#121—24no—
83#121—24no—
84#121—22no—
85#121—22no—
86#121—42no—
87#120—22no—
88#120—24no—
89#120D414yes—
90C4×Q8—21no—
91C4×Q8—22no—
92C4×D4—24no—
93C4×D4—22no—
94C4×D4—21no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—22no—
99#117D412yescomplemented
100#117C2312yes—
101#117C2311yescharacteristic
102#117C2311yescharacteristic
103#117C2311yescharacteristic
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C42D412yescomplemented
107C2×C42C2311yescharacteristic
108#104—22no—
109#104—24no—
110#104—22no—
111#104—24no—
112#102—42no—
113#101—21no—
114#100—21no—
115#100—42no—
116#99—22no—
117#99—24no—
118C24—21no—
119C24C2×D412yes—
120#41—22no—
121#41—22no—
122#41—24no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—22no—
127#41—44no—
128#41—42no—
129#41—42no—
130#41—42no—
131#41—42no—
132#41—88no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—42no—
136C2×Q8C2×D411yescharacteristic
137C2×Q8#4112yescomplemented
138C2×D4—22no—
139C2×D4—24no—
140C2×D4—22no—
141C2×D4—24no—
142C2×D4—24no—
143C2×D4—24no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—42no—
152C2×D4—44no—
153C2×D4—42no—
154C2×D4—44no—
155C2×D4—84no—
156C2×D4C2×D412yes—
157C2×D4C2×D412yes—
158C2×D4C2×D411yescharacteristic
159C2×D4#4112yescomplemented
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—42no—
166C22×C4C2×D412yes—
167C22×C4C2411yescharacteristic
168SD16—44no—
169D8—44no—
170M16—22no—
171M16—22no—
172M16—22no—
173M16—22no—
174M16—42no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—42no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—22no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—22no—
194C22⋊C4—44no—
195C22⋊C4C2×D412yes—
196C42—22no—
197C42—21no—
198C42—21no—
199C42—22no—
200C42C2×D411yescharacteristic
201C42C2×D412yes—
202C42C2×D411yescharacteristic
203C42C2×D412yes—
204C23—22no—
205C23—22no—
206C23—24no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—82no—
211C23—82no—
212C23#12012yes—
213C23C22×D411yescharacteristic
214Q8—22no—
215Q8—22no—
216Q8—22no—
217Q8—42no—
218Q8—44no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4—22no—
223D4—24no—
224D4—42no—
225D4—48no—
226D4—44no—
227D4—42no—
228D4—42no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—42no—
234D4—84no—
235D4—84no—
236D4—84no—
237D4—164no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4#12012yes—
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D411yescharacteristic
258C2×C4C2×#4112yes—
259C2×C4H(5,𝔽2)12yes—
260C8—42no—
261C8—42no—
262C22—22no—
263C22—41no—
264C22—42no—
265C22—44no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—82no—
271C22—84no—
272C22—162no—
273C22—162no—
274C22C2×#12011yescharacteristic
275C22#54612yes—
276C4—22no—
277C4—22no—
278C4—42no—
279C4—41no—
280C4—42no—
281C4—42no—
282C4—42no—
283C4—82no—
284C4#52511yescharacteristic
285C4#53511yescharacteristic
286C4#54612yes—
287C2—21no—
288C2—22no—
289C2—42no—
290C2—82no—
291C2—82no—
292C2—82no—
293C2#240411yescharacteristic
subgroupG/H
center, socleC2#2404
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. #3508C2semidirect
2. #3027C2semidirect
3. #3021C2semidirect
4. #2901C2semidirect
5. #2031C2semidirect
6. #2002C2semidirect
7. #1914C2semidirect
8. #1911C2semidirect
9. #1910C2semidirect
10. #1904C2semidirect
11. #1827C2semidirect
22. #533C22semidirect
24. C2×#127C22semidirect
25. #518C22semidirect
26. #518C22semidirect
36. #418C22semidirect
38. C2×#100C22semidirect
41. #344C22semidirect
42. #339C22semidirect
43. H(3,ℤ/4)C22semidirect
44. H(3,ℤ/4)C22semidirect
72. #127D4semidirect
77. #124D4semidirect
99. #117D4semidirect
106. C2×C42D4semidirect
137. C2×Q8#41semidirect
159. C2×D4#41semidirect

Maximal subgroups

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

subgroupindex
#35082
#30272
#30212
#29012
#20312
#20022
#19142
#19112
#19102
#19042
#18272