Finite Group Databasebeta

← C22×#453 #60378

Subgroups — C22×#453 #60378

Order 256.

2835 conjugacy classes of subgroups; only the 567 normal subgroups are shown.

normal subgroups only

This group has 2835 conjugacy classes of subgroups — too many to draw. Only its 567 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
1C22×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C22×AGL(1,ℤ/8)C212yesmaximal, complemented
3C22×#127C211yesmaximal, characteristic, complemented
4C2×#453C2124yesmaximal, direct product factor
5C22×#104C212yesmaximal, complemented
6C22×#100C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)—212no—
8C2×H(5,𝔽2)—26no—
9C2×H(5,𝔽2)C22112yescomplemented
10C22×#41—22no—
11C22×#41C2212yes—
12C23×D4—22no—
13C23×D4—22no—
14C23×D4—22no—
15C23×D4C2211yescharacteristic
16C23×D4C2211yescharacteristic, complemented
17C2×AGL(1,ℤ/8)—212no—
18C2×AGL(1,ℤ/8)C22124yescomplemented
19C22×SD16—22no—
20C22×D8—22no—
21C22×M16C2212yescomplemented
22C2×#127—26no—
23C2×#127C22112yescomplemented
24C22×C42C2211yescharacteristic, complemented
25#453C22164yesdirect product factor
26C2×#104C22124yescomplemented
27C2×#100C22112yescomplemented
28C25—22no—
29C25—21no—
30C25D412yes—
31H(5,𝔽2)—248no—
32H(5,𝔽2)—248no—
33H(5,𝔽2)—224no—
34H(5,𝔽2)C23116yescomplemented
35C2×#41—224no—
36C2×#41—224no—
37C2×#41—26no—
38C2×#41—26no—
39C2×#41D4112yescomplemented
40C2×#41C23112yes—
41C22×Q8D412yescomplemented
42C22×D4—24no—
43C22×D4—224no—
44C22×D4—224no—
45C22×D4—22no—
46C22×D4—224no—
47C22×D4—224no—
48C22×D4—224no—
49C22×D4—22no—
50C22×D4—26no—
51C22×D4—22no—
52C22×D4—26no—
53C22×D4—23no—
54C22×D4—416no—
55C22×D4D416yes—
56C22×D4D412yescomplemented
57C22×D4C2316yes—
58C22×D4C2316yescomplemented
59C23×C4—22no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4C2311yescharacteristic
63AGL(1,ℤ/8)—248no—
64AGL(1,ℤ/8)C23132yescomplemented
65C2×SD16—224no—
66C2×D8—224no—
67C2×M16—26no—
68C2×M16C23112yescomplemented
69C22×C8—22no—
70#127—224no—
71#127C23116yescomplemented
72C2×C42—23no—
73C2×C42C2316yescomplemented
74#104C23132yescomplemented
75#100C23116yescomplemented
76C24—224no—
77C24—22no—
78C24—22no—
79C24—26no—
80C24—26no—
81C24—48no—
82C24—48no—
83C24—43no—
84C24—41no—
85C24—88no—
86C24C2×D411yescharacteristic
87C24C2×D4112yes—
88#41—296no—
89#41—232no—
90#41—224no—
91#41—212no—
92#41—212no—
93#41C2×D4124yescomplemented
94#41C2418yes—
95C2×Q8—26no—
96C2×Q8C2×D4112yescomplemented
97C2×D4—248no—
98C2×D4—296no—
99C2×D4—296no—
100C2×D4—212no—
101C2×D4—212no—
102C2×D4—232no—
103C2×D4—232no—
104C2×D4—232no—
105C2×D4—212no—
106C2×D4—26no—
107C2×D4—212no—
108C2×D4—212no—
109C2×D4—212no—
110C2×D4—212no—
111C2×D4—212no—
112C2×D4—26no—
113C2×D4—496no—
114C2×D4—46no—
115C2×D4—46no—
116C2×D4C2×D4112yes—
117C2×D4C2×D4112yescomplemented
118C2×D4C2414yes—
119C2×D4C2414yescomplemented
120C22×C4—224no—
121C22×C4—28no—
122C22×C4—22no—
123C22×C4—26no—
124C22×C4—212no—
125C22×C4—212no—
126C22×C4C2×D413yes—
127C22×C4C2×D416yes—
128C22×C4C2×D412yes—
129C22×C4C2413yes—
130SD16—232no—
131D8—232no—
132M16—212no—
133M16C2418yescomplemented
134C2×C8—212no—
135C42—26no—
136C42C2414yescomplemented
137C23—24no—
138C23—232no—
139C23—212no—
140C23—212no—
141C23—212no—
142C23—24no—
143C23—424no—
144C23—424no—
145C23—48no—
146C23—48no—
147C23—46no—
148C23—46no—
149C23—812no—
150C23—812no—
151C23—848no—
152C23—84no—
153C23#12013yes—
154C23#12011yescharacteristic
155C23C22×D413yes—
156C23C22×D418yes—
157Q8—212no—
158Q8C22×D418yescomplemented
159D4—264no—
160D4—224no—
161D4—28no—
162D4—224no—
163D4—212no—
164D4—28no—
165D4—28no—
166D4—464no—
167D4—412no—
168D4C22×D418yescomplemented
169C2×C4—232no—
170C2×C4—224no—
171C2×C4—212no—
172C2×C4—24no—
173C2×C4—224no—
174C2×C4—28no—
175C2×C4#12016yes—
176C2×C4#12016yes—
177C2×C4C22×D413yes—
178C2×C4C22×D416yes—
179C2×C4C22×D416yes—
180C2×C4C2511yescharacteristic
181C8—28no—
182C22—26no—
183C22—26no—
184C22—28no—
185C22—28no—
186C22—48no—
187C22—48no—
188C22—424no—
189C22—424no—
190C22—44no—
191C22—812no—
192C22—812no—
193C22—832no—
194C22—812no—
195C22#45314yesdirect product factor
196C22C2×#12013yes—
197C22C2×#12013yes—
198C22C23×D411yescharacteristic
199C4—28no—
200C4—28no—
201C4C2×#12016yes—
202C4C23×D412yes—
203C2—21no—
204C2—23no—
205C2—48no—
206C2—48no—
207C2—84no—
208C2C2×#45316yesdirect product factor
209C2C22×#12011yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×H(5,𝔽2)C2semidirect
2. C22×AGL(1,ℤ/8)C2semidirect
3. C22×#127C2semidirect
4. C2×#453C2direct (N × H)
5. C22×#104C2semidirect
6. C22×#100C2semidirect
9. C2×H(5,𝔽2)C22semidirect
16. C23×D4C22semidirect
18. C2×AGL(1,ℤ/8)C22semidirect
21. C22×M16C22semidirect
23. C2×#127C22semidirect
24. C22×C42C22semidirect
25. #453C22direct (N × H)
26. C2×#104C22semidirect
27. C2×#100C22semidirect
34. H(5,𝔽2)C23semidirect
39. C2×#41D4semidirect
41. C22×Q8D4semidirect
56. C22×D4D4semidirect
58. C22×D4C23semidirect
64. AGL(1,ℤ/8)C23semidirect
68. C2×M16C23semidirect
71. #127C23semidirect
73. C2×C42C23semidirect
74. #104C23semidirect
75. #100C23semidirect
93. #41C2×D4semidirect
96. C2×Q8C2×D4semidirect
117. C2×D4C2×D4semidirect
119. C2×D4C24semidirect
133. M16C24semidirect
136. C42C24semidirect
158. Q8C22×D4semidirect
168. D4C22×D4semidirect
195. C22#453direct (N × H)
208. C2C2×#453direct (N × H)

Maximal subgroups

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

subgroupindex
C22×H(5,𝔽2)2
C22×AGL(1,ℤ/8)2
C22×#1272
C2×#4532
C22×#1042
C22×#1002