Finite Group Databasebeta

← #29610

Subgroups — #29610

Order 256.

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

normal subgroups only

This group has 797 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
1C2×#535C211yesmaximal, characteristic, complemented
2#3094C211yesmaximal, characteristic, complemented
3#3006C211yesmaximal, characteristic, complemented
4#2433C211yesmaximal, characteristic, complemented
5#2037C214yesmaximal, complemented
6#2002C214yesmaximal, complemented
7#1826C211yesmaximal, characteristic, complemented
8#1789C211yesmaximal, characteristic, complemented
9#1783C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C23×D4C2211yescharacteristic
12C2×AGL(1,ℤ/8)C2212yescomplemented
13#535—22no—
14#535—22no—
15#535—22no—
16#535C2214yes—
17C2×#127—21no—
18C2×#127C2212yescomplemented
19C2×#124—21no—
20C2×#121—21no—
21C2×#121—21no—
22C2×#121C2211yescharacteristic
23C2×#120—22no—
24C2×#120—22no—
25C2×#117C2211yescharacteristic
26#482—22no—
27#481—21no—
28#480—21no—
29#449—22no—
30#448—21no—
31#447—21no—
32#428C2212yescomplemented
33C2×#104C2214yescomplemented
34#418C2212yescomplemented
35#417C2212yescomplemented
36C2×#100C2214yescomplemented
37#407C2211yescharacteristic
38#407C2211yescharacteristic
39#392—22no—
40#390C2212yescomplemented
41C4×C22⋊C4C2211yescharacteristic
42#339C2214yescomplemented
43C25D412yes—
44C2×#41—21no—
45C2×#41—21no—
46C2×#41—44no—
47C2×#41D412yescomplemented
48C2×#41C2312yes—
49C22×Q8D411yescharacteristic, complemented
50C22×D4—21no—
51C22×D4—22no—
52C22×D4—22no—
53C22×D4—21no—
54C22×D4—24no—
55C22×D4—21no—
56C22×D4—21no—
57C22×D4—21no—
58C22×D4—21no—
59C22×D4—44no—
60C22×D4D412yes—
61C22×D4D411yescharacteristic, complemented
62C22×D4C2311yescharacteristic
63C22×D4C2312yes—
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66AGL(1,ℤ/8)—44no—
67C2×SD16—22no—
68C2×D8—22no—
69C2×M16C2312yes—
70C2×M16C2312yes—
71#127—24no—
72#127—24no—
73#127—44no—
74#124—24no—
75#124—24no—
76#121—24no—
77#121—24no—
78#121—24no—
79#121—24no—
80#121—21no—
81#121—21no—
82#121—21no—
83#121—21no—
84#120—28no—
85#120—28no—
86#120—48no—
87#117—21no—
88#117—21no—
89#117D412yescomplemented
90#117C2312yes—
91C2×C4⋊C4D411yescharacteristic, complemented
92C2×C4⋊C4D411yescharacteristic, complemented
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—21no—
100C2×C42—21no—
101C2×C42C2312yes—
102#107—22no—
103#106—22no—
104#104—44no—
105#103—22no—
106#102—22no—
107#102—22no—
108#102—22no—
109#100—44no—
110C22⋊C8—21no—
111C22⋊C8—21no—
112C22⋊C8—21no—
113C22⋊C8—21no—
114#95—21no—
115C24—22no—
116C24—21no—
117C24—22no—
118C24—21no—
119C24—44no—
120C24—44no—
121C24—82no—
122C24C2×D411yescharacteristic
123C24C2×D414yes—
124#41—24no—
125#41—24no—
126#41—42no—
127#41—42no—
128#41—44no—
129#41—44no—
130C2×Q8—21no—
131C2×Q8—42no—
132C2×Q8C2×D412yes—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—28no—
136C2×D4—28no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—21no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—24no—
145C2×D4—24no—
146C2×D4—24no—
147C2×D4—42no—
148C2×D4—44no—
149C2×D4—44no—
150C2×D4—44no—
151C2×D4—44no—
152C2×D4—44no—
153C2×D4—48no—
154C2×D4—44no—
155C2×D4—44no—
156C2×D4—48no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—84no—
160C2×D4—84no—
161C2×D4—84no—
162C2×D4C2×D412yes—
163C22×C4—24no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—44no—
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2411yescharacteristic
178SD16—44no—
179D8—44no—
180M16—24no—
181M16—24no—
182C2×C8—22no—
183C2×C8—22no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4C2×D412yes—
189C4⋊C4C2×D412yes—
190C22⋊C4—28no—
191C22⋊C4—28no—
192C22⋊C4—24no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—48no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C42—22no—
199C42—22no—
200C42—24no—
201C23—22no—
202C23—22no—
203C23—24no—
204C23—22no—
205C23—24no—
206C23—24no—
207C23—41no—
208C23—44no—
209C23—44no—
210C23—42no—
211C23—44no—
212C23—44no—
213C23—42no—
214C23—44no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—44no—
219C23—41no—
220C23—42no—
221C23—82no—
222C23—82no—
223C23—816no—
224C23#12011yescharacteristic
225C23#12011yescharacteristic
226C23C22×D411yescharacteristic
227Q8—22no—
228Q8—22no—
229Q8—42no—
230D4—22no—
231D4—22no—
232D4—44no—
233D4—44no—
234D4—42no—
235D4—44no—
236D4—44no—
237D4—44no—
238D4—42no—
239D4—44no—
240D4—42no—
241D4—88no—
242D4—88no—
243D4—816no—
244D4—82no—
245D4—82no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4#12011yescharacteristic
265C2×C4#12011yescharacteristic
266C2×C4C22×D411yescharacteristic
267C2×C4C22×D411yescharacteristic
268C2×C4H(5,𝔽2)14yes—
269C8—42no—
270C8—42no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—48no—
280C22—42no—
281C22—44no—
282C22—42no—
283C22—44no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—84no—
288C22—88no—
289C22—84no—
290C22—88no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22#45312yes—
295C22C2×#12011yescharacteristic
296C22#53412yes—
297C4—21no—
298C4—21no—
299C4—44no—
300C4—42no—
301C4—42no—
302C4—42no—
303C4—81no—
304C4#53412yes—
305C4#53512yes—
306C2—22no—
307C2—22no—
308C2—41no—
309C2—84no—
310C2—82no—
311C2—161no—
312C2#261311yescharacteristic
313C2C2×#45311yescharacteristic
314C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#535C2semidirect
2. #3094C2semidirect
3. #3006C2semidirect
4. #2433C2semidirect
5. #2037C2semidirect
6. #2002C2semidirect
7. #1826C2semidirect
8. #1789C2semidirect
9. #1783C2semidirect
12. C2×AGL(1,ℤ/8)C22semidirect
18. C2×#127C22semidirect
32. #428C22semidirect
33. C2×#104C22semidirect
34. #418C22semidirect
35. #417C22semidirect
36. C2×#100C22semidirect
40. #390C22semidirect
42. #339C22semidirect
47. C2×#41D4semidirect
49. C22×Q8D4semidirect
61. C22×D4D4semidirect
89. #117D4semidirect
91. C2×C4⋊C4D4semidirect
92. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5352
#30942
#30062
#24332
#20372
#20022
#18262
#17892
#17832