Finite Group Databasebeta

← #21473

Subgroups — #21473

Order 256.

743 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 743 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C4×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C2×#453C211yesmaximal, characteristic
3C4×AGL(1,ℤ/8)C212yesmaximal, complemented
4#2328C211yesmaximal, characteristic, complemented
5#1887C211yesmaximal, characteristic, complemented
6#1883C212yesmaximal, complemented
7#1860C212yesmaximal, complemented
8#1856C211yesmaximal, characteristic, complemented
9#1792C211yesmaximal, characteristic, complemented
10#1759C212yesmaximal, complemented
11C4×#100C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)C2211yescharacteristic
13C2×AGL(1,ℤ/8)C2212yes—
14C2×#127C2211yescharacteristic
15#520—22no—
16#520C2212yes—
17#518—22no—
18#518—22no—
19#518C2211yescharacteristic
20C4×#41—22no—
21C4×#41C2212yes—
22C2×C4×D4—22no—
23C2×C4×D4—22no—
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic, complemented
26#453C418yescomplemented
27#442—22no—
28#440—22no—
29C4×SD16—22no—
30C4×D8—22no—
31#428C2212yescomplemented
32C2×#104C2212yes—
33C2×#104C2212yes—
34#418C2212yescomplemented
35#417C2212yes—
36C2×#100C2211yescharacteristic
37C2×#100C2211yescharacteristic
38C4×M16C2212yescomplemented
39#390C2211yescharacteristic
40#390C2211yescharacteristic, complemented
41#386—22no—
42#386—22no—
43#382C2211yescharacteristic, complemented
44#376C2211yescharacteristic, complemented
45#343C2212yescomplemented
46#339C2212yescomplemented
47#339C2212yescomplemented
48H(5,𝔽2)—24no—
49H(5,𝔽2)—22no—
50H(5,𝔽2)C2×C414yescomplemented
51C2×#41—22no—
52C2×#41C2312yes—
53C22×D4—22no—
54C22×D4—22no—
55C22×D4—22no—
56C22×D4C2311yescharacteristic
57C22×D4C2311yescharacteristic
58C23×C4—22no—
59C23×C4—21no—
60C23×C4D412yes—
61AGL(1,ℤ/8)—24no—
62AGL(1,ℤ/8)C2×C418yescomplemented
63C2×SD16—22no—
64C2×D8—22no—
65C2×M16C2312yes—
66C2×M16C2312yes—
67#127—22no—
68#127C2×C414yescomplemented
69C4×Q8—22no—
70C4×Q8D412yescomplemented
71C4×Q8D412yescomplemented
72C4×D4—24no—
73C4×D4—24no—
74C4×D4—24no—
75C4×D4—24no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—24no—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4D412yescomplemented
87C4×D4D412yescomplemented
88#117—24no—
89#117—22no—
90#117—22no—
91#117C2312yes—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—22no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4D412yes—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#107—22no—
109#106—22no—
110#104—24no—
111#104C2×C418yescomplemented
112#103—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116#100—22no—
117#100C2×C414yescomplemented
118#97—22no—
119C4×C8—22no—
120#95—21no—
121#95—21no—
122#95—22no—
123C24—22no—
124C24—21no—
125C24C2×D412yes—
126#41—28no—
127#41—28no—
128#41—22no—
129#41—22no—
130#41C22×C414yes—
131#41#4114yes—
132C2×Q8C2×D412yes—
133C2×D4—24no—
134C2×D4—28no—
135C2×D4—28no—
136C2×D4—28no—
137C2×D4—22no—
138C2×D4—28no—
139C2×D4—28no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—21no—
145C2×D4—82no—
146C2×D4—82no—
147C2×D4C22×C412yes—
148C2×D4C22×C412yescomplemented
149C2×D4C2×D412yes—
150C2×D4#4112yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—82no—
165C22×C4—84no—
166C22×C4—82no—
167C22×C4C2×D412yes—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171SD16—28no—
172D8—28no—
173M16—22no—
174M16—22no—
175M16—22no—
176M16C22×C414yescomplemented
177C2×C8—22no—
178C2×C8—22no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D412yes—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—24no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—82no—
195C22⋊C4—82no—
196C42—24no—
197C42—22no—
198C42—22no—
199C42—22no—
200C42—21no—
201C42—22no—
202C42C22×C412yescomplemented
203C42C2×D412yes—
204C42C2×D412yes—
205C23—28no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—41no—
215C23—41no—
216C23—82no—
217C23—82no—
218C23C4×D414yes—
219C23C22×D411yescharacteristic
220Q8—22no—
221Q8C4×D414yes—
222D4—216no—
223D4—24no—
224D4—24no—
225D4—24no—
226D4—24no—
227D4—22no—
228D4—24no—
229D4—42no—
230D4—88no—
231D4C4×D414yes—
232C2×C4—28no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—84no—
255C2×C4—82no—
256C2×C4—84no—
257C2×C4—82no—
258C2×C4#12011yescharacteristic
259C2×C4#12012yes—
260C2×C4#12011yescharacteristic
261C2×C4C23×C411yescharacteristic
262C2×C4C22×D412yes—
263C2×C4C2×#4112yes—
264C2×C4C2×#4111yescharacteristic
265C8—24no—
266C8—42no—
267C22—21no—
268C22—21no—
269C22—24no—
270C22—24no—
271C22—44no—
272C22—44no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—82no—
279C22—82no—
280C22—88no—
281C22—81no—
282C22—81no—
283C22C2×C4×D411yescharacteristic
284C22C2×#12011yescharacteristic
285C22#52511yescharacteristic
286C4—21no—
287C4—21no—
288C4—22no—
289C4—24no—
290C4—42no—
291C4—42no—
292C4—42no—
293C4—42no—
294C4—81no—
295C4—81no—
296C4C2×C4×D412yes—
297C4#52512yes—
298C2—21no—
299C2—21no—
300C2—44no—
301C2—44no—
302C2—82no—
303C2C4×#12011yescharacteristic
304C2#301911yescharacteristic
305C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#453C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×H(5,𝔽2)C2semidirect
3. C4×AGL(1,ℤ/8)C2semidirect
4. #2328C2semidirect
5. #1887C2semidirect
6. #1883C2semidirect
7. #1860C2semidirect
8. #1856C2semidirect
9. #1792C2semidirect
10. #1759C2semidirect
11. C4×#100C2semidirect
25. C2×C4×D4C22semidirect
26. #453C4semidirect
31. #428C22semidirect
34. #418C22semidirect
38. C4×M16C22semidirect
40. #390C22semidirect
43. #382C22semidirect
44. #376C22semidirect
45. #343C22semidirect
46. #339C22semidirect
47. #339C22semidirect
50. H(5,𝔽2)C2×C4semidirect
62. AGL(1,ℤ/8)C2×C4semidirect
68. #127C2×C4semidirect
70. C4×Q8D4semidirect
71. C4×Q8D4semidirect
86. C4×D4D4semidirect
87. C4×D4D4semidirect
111. #104C2×C4semidirect
117. #100C2×C4semidirect
148. C2×D4C22×C4semidirect
176. M16C22×C4semidirect
202. C42C22×C4semidirect

Maximal subgroups

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

subgroupindex
C4×H(5,𝔽2)2
C2×#4532
C4×AGL(1,ℤ/8)2
#23282
#18872
#18832
#18602
#18562
#17922
#17592
C4×#1002