Finite Group Databasebeta

← C4×#453 #21469

Subgroups — C4×#453 #21469

Order 256.

789 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 789 conjugacy classes of subgroups — too many to draw. Only its 189 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
4C4×#127C211yesmaximal, characteristic, complemented
5#1887C211yesmaximal, characteristic, complemented
6#1883C212yesmaximal, complemented
7#1859C212yesmaximal, complemented
8#1856C211yesmaximal, characteristic, complemented
9#1792C211yesmaximal, characteristic, complemented
10C4×#104C212yesmaximal, 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×D4—22no—
25C2×C4×D4C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic, complemented
27#453C418yesdirect product factor
28#442—22no—
29#440—22no—
30C4×SD16—22no—
31C4×D8—22no—
32#428C2212yescomplemented
33C2×#104C2212yes—
34C2×#104C2212yes—
35#418C2212yescomplemented
36#417C2212yes—
37C2×#100C2211yescharacteristic
38C2×#100C2211yescharacteristic
39C4×M16C2212yescomplemented
40#390—22no—
41#390C2211yescharacteristic
42#390C2211yescharacteristic, complemented
43#384C2211yescharacteristic, complemented
44C43C2211yescharacteristic, 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×D4—48no—
87C4×D4D412yescomplemented
88C4×D4D412yescomplemented
89#117—24no—
90#117—22no—
91#117—22no—
92#117C2312yes—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C22⋊C4—22no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4D412yes—
105C2×C42—22no—
106C2×C42—21no—
107C2×C42—22no—
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#107—22no—
113#106—22no—
114#104—24no—
115#104C2×C418yescomplemented
116#103—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#100—22no—
121#100C2×C414yescomplemented
122#97—22no—
123C4×C8—22no—
124C24—22no—
125C24—21no—
126C24C2×D412yes—
127#41—28no—
128#41—28no—
129#41—22no—
130#41—22no—
131#41C22×C414yes—
132#41#4114yes—
133C2×Q8C2×D412yes—
134C2×D4—24no—
135C2×D4—28no—
136C2×D4—28no—
137C2×D4—28no—
138C2×D4—22no—
139C2×D4—28no—
140C2×D4—28no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—44no—
147C2×D4—44no—
148C2×D4C22×C412yes—
149C2×D4C22×C412yescomplemented
150C2×D4C2×D412yes—
151C2×D4#4112yes—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—84no—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D412yes—
171C22×C4C2×D411yescharacteristic
172C22×C4C2411yescharacteristic
173SD16—28no—
174D8—28no—
175M16—22no—
176M16—22no—
177M16—22no—
178M16C22×C414yescomplemented
179C2×C8—22no—
180C2×C8—22no—
181C4⋊C4—24no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—44no—
187C4⋊C4C2×D412yes—
188C4⋊C4C2×D412yes—
189C22⋊C4—24no—
190C22⋊C4—24no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—24no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C22⋊C4—42no—
198C42—24no—
199C42—24no—
200C42—22no—
201C42—22no—
202C42—22no—
203C42—21no—
204C42—22no—
205C42—42no—
206C42C22×C412yescomplemented
207C42C2×D412yes—
208C42C2×D412yes—
209C23—28no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—41no—
219C23—41no—
220C23—82no—
221C23—82no—
222C23C4×D414yes—
223C23C22×D411yescharacteristic
224Q8—22no—
225Q8C4×D414yescomplemented
226D4—216no—
227D4—24no—
228D4—24no—
229D4—24no—
230D4—24no—
231D4—22no—
232D4—24no—
233D4—416no—
234D4—42no—
235D4C4×D414yescomplemented
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—28no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—84no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4#12012yes—
264C2×C4#12011yescharacteristic
265C2×C4#12011yescharacteristic
266C2×C4C23×C411yescharacteristic
267C2×C4C22×D412yes—
268C2×C4C2×#4112yes—
269C2×C4C2×#4111yescharacteristic
270C8—24no—
271C8—42no—
272C22—21no—
273C22—21no—
274C22—24no—
275C22—24no—
276C22—44no—
277C22—44no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—82no—
284C22—82no—
285C22—88no—
286C22—81no—
287C22—81no—
288C22C2×C4×D411yescharacteristic
289C22C2×#12011yescharacteristic
290C22#52511yescharacteristic
291C4—24no—
292C4—21no—
293C4—22no—
294C4—24no—
295C4—42no—
296C4—42no—
297C4—42no—
298C4—42no—
299C4—81no—
300C4#45312yesdirect product factor
301C4C2×C4×D412yes—
302C4#52512yes—
303C2—21no—
304C2—21no—
305C2—44no—
306C2—44no—
307C2—82no—
308C2C4×#12011yescharacteristic
309C2C2×#45311yescharacteristic
310C2#301811yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C2×#120
prime radicalC2×#453C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. C4×#127C2semidirect
5. #1887C2semidirect
6. #1883C2semidirect
7. #1859C2semidirect
8. #1856C2semidirect
9. #1792C2semidirect
10. C4×#104C2semidirect
11. C4×#100C2semidirect
26. C2×C4×D4C22semidirect
27. #453C4direct (N × H)
32. #428C22semidirect
35. #418C22semidirect
39. C4×M16C22semidirect
42. #390C22semidirect
43. #384C22semidirect
44. C43C22semidirect
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
87. C4×D4D4semidirect
88. C4×D4D4semidirect
115. #104C2×C4semidirect
121. #100C2×C4semidirect
149. C2×D4C22×C4semidirect
178. M16C22×C4semidirect
206. C42C22×C4semidirect
225. Q8C4×D4semidirect
235. D4C4×D4semidirect
300. C4#453direct (N × H)

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
C4×#1272
#18872
#18832
#18592
#18562
#17922
C4×#1042
C4×#1002