Finite Group Databasebeta

← #23992

Subgroups — #23992

Order 256.

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

normal subgroups only

This group has 564 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#3509C211yesmaximal, characteristic, complemented
2#3028C211yesmaximal, characteristic
3#3022C211yesmaximal, characteristic
4#2901C211yesmaximal, characteristic
5#2031C212yesmaximal, complemented
6#2005C212yesmaximal, complemented
7#1915C211yesmaximal, characteristic, complemented
8#1912C212yesmaximal, complemented
9#1909C211yesmaximal, characteristic, complemented
10#1905C212yesmaximal, complemented
11#1827C211yesmaximal, characteristic, complemented
12#585C2211yescharacteristic
13C2×#137C2212yes—
14#568C2211yescharacteristic
15#557—22no—
16#556—22no—
17#552—24no—
18#541—22no—
19#536C2211yescharacteristic
20#536C2211yescharacteristic
21#536C2212yes—
22#533—22no—
23#533C2211yescharacteristic, complemented
24C2×C4⋊Q8C2211yescharacteristic, complemented
25#519C2211yescharacteristic, complemented
26#519C2211yescharacteristic, complemented
27C4×#41C2211yescharacteristic
28#458—24no—
29#456—22no—
30#455—22no—
31#421—22no—
32#421C2212yes—
33#421C2212yes—
34C2×#104—22no—
35C2×#104C2212yes—
36#419C2212yescomplemented
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×Q8C2311yescharacteristic
56#137—24no—
57#137—42no—
58C2×Q16—22no—
59C2×SD16—22no—
60#131—42no—
61C2×M16—22no—
62C2×M16C2312yes—
63C2×M16C2312yes—
64C4⋊Q8—22no—
65C4⋊Q8—21no—
66C4⋊Q8—22no—
67C4⋊Q8—21no—
68C4⋊Q8—21no—
69C4⋊Q8—21no—
70C4⋊Q8D412yes—
71#126—24no—
72#126—24no—
73#125—24no—
74#125—24no—
75#125—24no—
76#125—21no—
77#124—21no—
78#124—22no—
79#124D412yes—
80#123—22no—
81#123—24no—
82#123D414yes—
83C22⋊Q8—24no—
84C22⋊Q8—22no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—42no—
88C4×Q8—24no—
89C4×Q8—21no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×D4—22no—
96C4×D4—21no—
97#117—24no—
98#117—22no—
99#117—22no—
100#117D412yes—
101#117C2312yes—
102#117C2311yescharacteristic
103#117C2311yescharacteristic
104#117C2311yescharacteristic
105C2×C4⋊C4—22no—
106C2×C42D412yes—
107C2×C42C2311yescharacteristic
108#104—22no—
109#104—24no—
110#104—22no—
111#104—24no—
112#103—42no—
113#101—21no—
114#101—42no—
115#100—21no—
116#99—22no—
117#99—24no—
118#41—22no—
119#41—22no—
120#41—24no—
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125#41—44no—
126#41—42no—
127#41—42no—
128#41—42no—
129#41—42no—
130C2×Q8—22no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8—42no—
140C2×Q8C2×D412yes—
141C2×Q8C2×D411yescharacteristic
142C2×Q8#4112yes—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—42no—
146C2×D4—44no—
147C2×D4—42no—
148C2×D4C2×D412yes—
149C2×D4C2×D411yescharacteristic
150C2×D4#4112yes—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—42no—
156C22×C4C2×D412yes—
157C22×C4C2×D412yes—
158C22×C4C2411yescharacteristic
159Q16—44no—
160SD16—44no—
161M16—22no—
162M16—22no—
163M16—22no—
164M16—22no—
165M16—42no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—42no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—22no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—44no—
184C22⋊C4—24no—
185C22⋊C4—24no—
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—22no—
190C22⋊C4—44no—
191C22⋊C4C2×D412yes—
192C42—24no—
193C42—24no—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—22no—
198C42C2×D411yescharacteristic
199C42C2×D412yes—
200C42C2×D411yescharacteristic
201C42C2×D412yes—
202C23—22no—
203C23—42no—
204C23#12012yes—
205C23C22×D411yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—24no—
209Q8—22no—
210Q8—44no—
211Q8—42no—
212Q8—41no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216Q8—42no—
217D4—24no—
218D4—22no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—44no—
223D4—42no—
224D4—42no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4#12012yes—
245C2×C4C22×D411yescharacteristic
246C2×C4C22×D411yescharacteristic
247C2×C4C2×#4112yes—
248C2×C4H(5,𝔽2)12yes—
249C8—42no—
250C8—42no—
251C22—22no—
252C22—41no—
253C22—42no—
254C22—42no—
255C22—82no—
256C22—162no—
257C22C2×#12011yescharacteristic
258C22#54612yes—
259C4—22no—
260C4—22no—
261C4—42no—
262C4—41no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—42no—
267C4—42no—
268C4—82no—
269C4#52511yescharacteristic
270C4#53511yescharacteristic
271C4#54612yes—
272C2—21no—
273C2—22no—
274C2—42no—
275C2—82no—
276C2#240411yescharacteristic
subgroupG/H
center, socleC2#2404
commutator, Frattini, cosocleC22×C4C24
prime radical#585C22
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. #3509C2semidirect
5. #2031C2semidirect
6. #2005C2semidirect
7. #1915C2semidirect
8. #1912C2semidirect
9. #1909C2semidirect
10. #1905C2semidirect
11. #1827C2semidirect
23. #533C22semidirect
24. C2×C4⋊Q8C22semidirect
25. #519C22semidirect
26. #519C22semidirect
36. #419C22semidirect
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
#35092
#30282
#30222
#29012
#20312
#20052
#19152
#19122
#19092
#19052
#18272