Finite Group Databasebeta

← C4×#455 #21470

Subgroups — C4×#455 #21470

Order 256.

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

normal subgroups only

This group has 721 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
2C2×#455C211yesmaximal, characteristic
3C4×#137C212yesmaximal, complemented
4C4×#124C211yesmaximal, characteristic, complemented
5#1885C211yesmaximal, characteristic, complemented
6#1882C212yesmaximal, complemented
7#1860C212yesmaximal, complemented
8#1856C211yesmaximal, characteristic, complemented
9#1792C211yesmaximal, characteristic
10C4×#104C212yesmaximal, complemented
11C4×#100C211yesmaximal, characteristic, complemented
12C2×H(5,𝔽2)C2211yescharacteristic
13C2×#137C2212yes—
14C2×#124C2211yescharacteristic
15#520—22no—
16#520C2212yes—
17#518—22no—
18#518—22no—
19#518C2211yescharacteristic
20C4×#41—22no—
21C4×#41C2212yes—
22C2×C4×Q8C2211yescharacteristic, complemented
23C2×C4×D4—22no—
24C2×C4×D4—22no—
25C2×C4×D4C2211yescharacteristic
26#455C418yesdirect product factor
27#441—22no—
28#440—22no—
29C4×Q16—22no—
30C4×SD16—22no—
31#428C2212yescomplemented
32C2×#104C2212yes—
33C2×#104C2212yes—
34#419C2212yescomplemented
35#417C2212yes—
36C2×#100C2211yescharacteristic
37C2×#100C2211yescharacteristic
38C4×M16C2212yescomplemented
39#391C2211yescharacteristic, complemented
40#390C2211yescharacteristic
41#388—22no—
42#382C2211yescharacteristic, complemented
43C4×C22⋊C4—22no—
44C43C2211yescharacteristic, complemented
45#343C2212yescomplemented
46#339C2212yescomplemented
47#339C2212yescomplemented
48H(5,𝔽2)—24no—
49H(5,𝔽2)—22no—
50H(5,𝔽2)C2×C414yes—
51C2×#41—22no—
52C2×#41C2312yes—
53C22×Q8C2311yescharacteristic
54C22×D4—22no—
55C22×D4—22no—
56C22×D4C2311yescharacteristic
57C23×C4—22no—
58C23×C4D412yes—
59#137—24no—
60#137C2×C418yescomplemented
61C2×Q16—22no—
62C2×SD16—22no—
63C2×M16C2312yes—
64C2×M16C2312yes—
65#124—22no—
66#124C2×C414yescomplemented
67C4×Q8—22no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8D412yes—
71C4×Q8D412yes—
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×D4D412yes—
85C4×D4D412yes—
86#117—24no—
87#117—22no—
88#117—22no—
89#117C2312yes—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—22no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4D412yes—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108C2×C42C2311yescharacteristic
109#107—22no—
110#106—22no—
111#104—24no—
112#104C2×C418yescomplemented
113#103—22no—
114#103—22no—
115#103—22no—
116#102—22no—
117#100—22no—
118#100C2×C414yescomplemented
119#97—22no—
120C4×C8—22no—
121#95—22no—
122#95—22no—
123C24—22no—
124C24C2×D412yes—
125#41—28no—
126#41—28no—
127#41—22no—
128#41—22no—
129#41C22×C414yes—
130#41#4114yes—
131C2×Q8—21no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8C22×C412yescomplemented
135C2×Q8C2×D412yes—
136C2×D4—24no—
137C2×D4—28no—
138C2×D4—28no—
139C2×D4—28no—
140C2×D4—22no—
141C2×D4—28no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4C22×C412yes—
145C2×D4C2×D412yes—
146C2×D4#4112yes—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—44no—
161C22×C4—84no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D412yes—
164C22×C4C2×D411yescharacteristic
165C22×C4C2411yescharacteristic
166Q16—28no—
167SD16—28no—
168M16—22no—
169M16—22no—
170M16—22no—
171M16C22×C414yescomplemented
172C2×C8—22no—
173C2×C8—22no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4C2×D412yes—
181C4⋊C4C2×D412yes—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—28no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—24no—
189C22⋊C4—44no—
190C42—24no—
191C42—22no—
192C42—22no—
193C42—22no—
194C42—21no—
195C42—22no—
196C42—42no—
197C42—42no—
198C42—41no—
199C42—41no—
200C42C22×C412yescomplemented
201C42C2×D412yes—
202C42C2×D412yes—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—42no—
208C23—42no—
209C23—82no—
210C23—82no—
211C23C4×D414yes—
212C23C22×D411yescharacteristic
213Q8—22no—
214Q8—24no—
215Q8—42no—
216Q8C4×D414yes—
217D4—216no—
218D4—24no—
219D4—24no—
220D4—24no—
221D4—22no—
222D4—24no—
223D4C4×D414yes—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—84no—
251C2×C4—82no—
252C2×C4#12012yes—
253C2×C4#12011yescharacteristic
254C2×C4#12011yescharacteristic
255C2×C4C23×C411yescharacteristic
256C2×C4C22×D412yes—
257C2×C4C2×#4112yes—
258C2×C4C2×#4111yescharacteristic
259C8—24no—
260C8—42no—
261C22—21no—
262C22—21no—
263C22—24no—
264C22—24no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—82no—
271C22—88no—
272C22C2×C4×D411yescharacteristic
273C22C2×#12011yescharacteristic
274C22#52511yescharacteristic
275C4—21no—
276C4—22no—
277C4—24no—
278C4—42no—
279C4—42no—
280C4—42no—
281C4—42no—
282C4—42no—
283C4—42no—
284C4—81no—
285C4#45512yesdirect product factor
286C4C2×C4×D412yes—
287C4#52512yes—
288C2—21no—
289C2—21no—
290C2—44no—
291C2—44no—
292C2C4×#12011yescharacteristic
293C2C2×#45511yescharacteristic
294C2#301811yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C2×#120
prime radicalC2×H(5,𝔽2)C22
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
3. C4×#137C2semidirect
4. C4×#124C2semidirect
5. #1885C2semidirect
6. #1882C2semidirect
7. #1860C2semidirect
8. #1856C2semidirect
10. C4×#104C2semidirect
11. C4×#100C2semidirect
22. C2×C4×Q8C22semidirect
26. #455C4direct (N × H)
31. #428C22semidirect
34. #419C22semidirect
38. C4×M16C22semidirect
39. #391C22semidirect
42. #382C22semidirect
44. C43C22semidirect
45. #343C22semidirect
46. #339C22semidirect
47. #339C22semidirect
60. #137C2×C4semidirect
66. #124C2×C4semidirect
112. #104C2×C4semidirect
118. #100C2×C4semidirect
134. C2×Q8C22×C4semidirect
171. M16C22×C4semidirect
200. C42C22×C4semidirect
285. C4#455direct (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×#4552
C4×#1372
C4×#1242
#18852
#18822
#18602
#18562
#17922
C4×#1042
C4×#1002