Finite Group Databasebeta

← #23993

Subgroups — #23993

Order 256.

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

normal subgroups only

This group has 576 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#3510C211yesmaximal, characteristic, complemented
2#3028C211yesmaximal, characteristic, complemented
3#3021C211yesmaximal, characteristic
4#2901C211yesmaximal, characteristic, complemented
5#2031C212yesmaximal, complemented
6#2003C212yesmaximal, complemented
7#1915C211yesmaximal, characteristic, complemented
8#1912C212yesmaximal, 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#556—22no—
17#551—24no—
18#542—22no—
19#536C2211yescharacteristic
20#536C2212yes—
21#535C2211yescharacteristic
22#533C2211yescharacteristic, complemented
23#529—22no—
24C2×#124C2211yescharacteristic, complemented
25#519C2211yescharacteristic, complemented
26#518C2211yescharacteristic, complemented
27C4×#41C2211yescharacteristic
28#458—24no—
29#454—22no—
30#453—22no—
31#421—22no—
32#421C2212yes—
33#421C2212yes—
34C2×#104—22no—
35C2×#104C2212yes—
36#418C2212yescomplemented
37#413C2211yescharacteristic
38C2×#101C2212yescomplemented
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—22no—
51C2×#41—42no—
52C2×#41C2312yes—
53C2×#41C2311yescharacteristic
54C22×Q8C2311yescharacteristic
55C22×D4C2311yescharacteristic
56AGL(1,ℤ/8)—24no—
57AGL(1,ℤ/8)—42no—
58C2×SD16—22no—
59C2×D8—22no—
60#131—42no—
61C2×M16—22no—
62C2×M16C2312yes—
63C2×M16C2312yes—
64C4⋊Q8—21no—
65C4⋊Q8—22no—
66C4⋊Q8—21no—
67#127D412yes—
68#126—24no—
69#126—24no—
70#125—21no—
71#124—24no—
72#124—22no—
73#124—21no—
74#124—22no—
75#124—21no—
76#124—21no—
77#124D412yes—
78#123—24no—
79#123—24no—
80#123—24no—
81#123D414yes—
82C22⋊Q8—24no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C22⋊Q8—42no—
86#121—22no—
87#120—22no—
88C4×Q8—21no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×D4—24no—
93C4×D4—22no—
94C4×D4—21no—
95C4×D4—22no—
96C4×D4—22no—
97#117—24no—
98#117—22no—
99#117D412yes—
100#117C2312yes—
101#117C2311yescharacteristic
102#117C2311yescharacteristic
103#117C2311yescharacteristic
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C42D412yes—
107C2×C42C2311yescharacteristic
108#104—22no—
109#104—24no—
110#104—22no—
111#104—24no—
112#102—42no—
113#101—21no—
114#101—42no—
115#100—21no—
116#99—22no—
117#99—24no—
118C24—21no—
119#41—22no—
120#41—22no—
121#41—24no—
122#41—22no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—44no—
127#41—42no—
128#41—42no—
129#41—42no—
130#41—42no—
131C2×Q8—22no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8—42no—
138C2×Q8C2×D411yescharacteristic
139C2×Q8#4112yes—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—24no—
143C2×D4—21no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—42no—
147C2×D4—44no—
148C2×D4—42no—
149C2×D4C2×D412yes—
150C2×D4C2×D412yes—
151C2×D4C2×D411yescharacteristic
152C2×D4#4112yes—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—42no—
158C22×C4C2×D412yes—
159C22×C4C2×D412yes—
160C22×C4C2411yescharacteristic
161SD16—44no—
162D8—44no—
163M16—22no—
164M16—22no—
165M16—22no—
166M16—22no—
167M16—42no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—42no—
171C4⋊C4—22no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—24no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—44no—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—22no—
186C22⋊C4—24no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—24no—
190C22⋊C4—22no—
191C22⋊C4—44no—
192C22⋊C4C2×D412yes—
193C42—24no—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—22no—
198C42C2×D411yescharacteristic
199C42C2×D412yes—
200C42C2×D411yescharacteristic
201C42C2×D412yes—
202C23—22no—
203C23—22no—
204C23—42no—
205C23—42no—
206C23—82no—
207C23#12012yes—
208C23C22×D411yescharacteristic
209Q8—22no—
210Q8—22no—
211Q8—22no—
212Q8—42no—
213Q8—41no—
214Q8—41no—
215Q8—42no—
216Q8—82no—
217D4—24no—
218D4—22no—
219D4—22no—
220D4—22no—
221D4—24no—
222D4—42no—
223D4—44no—
224D4—42no—
225D4—42no—
226D4—41no—
227D4—41no—
228D4—84no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4—84no—
246C2×C4—81no—
247C2×C4#12012yes—
248C2×C4C22×D411yescharacteristic
249C2×C4C22×D411yescharacteristic
250C2×C4C2×#4112yes—
251C2×C4H(5,𝔽2)12yes—
252C8—42no—
253C8—42no—
254C22—22no—
255C22—41no—
256C22—42no—
257C22—44no—
258C22—42no—
259C22—42no—
260C22—82no—
261C22—162no—
262C22—162no—
263C22C2×#12011yescharacteristic
264C22#54612yes—
265C4—22no—
266C4—42no—
267C4—41no—
268C4—42no—
269C4—41no—
270C4—42no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4#52511yescharacteristic
275C4#53511yescharacteristic
276C4#54612yes—
277C2—21no—
278C2—22no—
279C2—42no—
280C2—82no—
281C2—82no—
282C2#240411yescharacteristic
subgroupG/H
center, socleC2#2404
commutator, Frattini, cosocleC22×C4C24
prime radical#3021C2
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. #3510C2semidirect
2. #3028C2semidirect
4. #2901C2semidirect
5. #2031C2semidirect
6. #2003C2semidirect
7. #1915C2semidirect
8. #1912C2semidirect
9. #1910C2semidirect
10. #1904C2semidirect
11. #1827C2semidirect
22. #533C22semidirect
24. C2×#124C22semidirect
25. #519C22semidirect
26. #518C22semidirect
36. #418C22semidirect
38. C2×#101C22semidirect
41. #344C22semidirect
42. #339C22semidirect
43. H(3,ℤ/4)C22semidirect
44. 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
#35102
#30282
#30212
#29012
#20312
#20032
#19152
#19122
#19102
#19042
#18272