Finite Group Databasebeta

← D4×#127 #36119

Subgroups — D4×#127 #36119

Order 256.

2865 conjugacy classes of subgroups; only the 635 normal subgroups are shown.

normal subgroups only

This group has 2865 conjugacy classes of subgroups — too many to draw. Only its 635 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×D42C216yesmaximal, complemented
2C22×#127C212yesmaximal, complemented
3#2867C211yesmaximal, characteristic, complemented
4#2856C212yesmaximal, complemented
5#2541C216yesmaximal, complemented
6#2433C2112yesmaximal, complemented
7C4×#127C211yesmaximal, characteristic, complemented
8C42×D4C211yesmaximal, characteristic, complemented
9C23×D4C2214yes—
10C23×D4C22112yescomplemented
11D42—248no—
12C2×#127—216no—
13C2×#127C2211yescharacteristic
14C2×#127C2212yescomplemented
15C2×#127C2216yescomplemented
16C2×#127C22112yescomplemented
17C2×#127C2216yescomplemented
18C2×#121C22112yescomplemented
19C2×#121C22112yescomplemented
20C2×#120C22124yescomplemented
21C2×C4×D4C2213yes—
22C2×C4×D4C2216yescomplemented
23C22×C42C2212yescomplemented
24#392C22124yescomplemented
25#390C2216yescomplemented
26#390C22112yescomplemented
27#384C2211yescharacteristic, complemented
28C4×C4⋊C4C2213yescomplemented
29C4×C22⋊C4C2216yescomplemented
30C43C2211yescharacteristic, complemented
31C25C2318yes—
32C22×D4—248no—
33C22×D4—248no—
34C22×D4—248no—
35C22×D4—424no—
36C22×D4C2318yes—
37C22×D4C2314yes—
38C22×D4C23112yes—
39C22×D4C23124yes—
40C22×D4C2316yes—
41C22×D4C23112yes—
42C22×D4C2311yescharacteristic
43C23×C4C2314yes—
44C23×C4C2316yes—
45#127—248no—
46#127—28no—
47#127—224no—
48#127—412no—
49#127D418yesdirect product factor
50#121—248no—
51#121—248no—
52#120—296no—
53C4×D4—224no—
54C4×D4D4124yescomplemented
55C2×C4⋊C4C2316yes—
56C2×C4⋊C4C2313yes—
57C2×C22⋊C4C23112yes—
58C2×C22⋊C4C23124yes—
59C2×C22⋊C4C2316yes—
60C2×C42—28no—
61C2×C42C2313yes—
62C2×C42C2316yes—
63C2×C42C2313yes—
64C2×C42C2311yescharacteristic
65#95C2316yes—
66C24—216no—
67C24—248no—
68C24—424no—
69C24C2418yes—
70C24C2414yes—
71C24C2412yes—
72C2×D4—248no—
73C2×D4—248no—
74C2×D4—212no—
75C2×D4—248no—
76C2×D4—224no—
77C2×D4—224no—
78C2×D4—224no—
79C2×D4—224no—
80C2×D4—212no—
81C2×D4—224no—
82C2×D4—4192no—
83C2×D4—424no—
84C2×D4—424no—
85C2×D4—464no—
86C2×D4C2×D4112yes—
87C2×D4C2×D4124yescomplemented
88C22×C4—212no—
89C22×C4—212no—
90C22×C4—224no—
91C22×C4C2×D4124yescomplemented
92C22×C4C2414yes—
93C22×C4C2413yes—
94C22×C4C2416yes—
95C22×C4C2411yescharacteristic
96C22×C4C2413yes—
97C4⋊C4—212no—
98C4⋊C4C2×D4112yescomplemented
99C22⋊C4—224no—
100C22⋊C4—248no—
101C22⋊C4C2×D4124yescomplemented
102C42—212no—
103C42—26no—
104C42C2×D414yescomplemented
105C42C2×D4112yescomplemented
106C23—22no—
107C23—26no—
108C23—212no—
109C23—212no—
110C23—224no—
111C23—28no—
112C23—224no—
113C23—496no—
114C23—432no—
115C23—848no—
116C23C22×D418yes—
117C23C22×D4112yes—
118C23C2511yescharacteristic
119D4—248no—
120D4—496no—
121D4—432no—
122D4—416no—
123D4—448no—
124D4—824no—
125D4#127116yesdirect product factor
126C2×C4—26no—
127C2×C4—26no—
128C2×C4—26no—
129C2×C4—248no—
130C2×C4—212no—
131C2×C4—416no—
132C2×C4C22×D416yes—
133C2×C4C22×D4112yes—
134C2×C4C22×D416yes—
135C2×C4C22×D416yes—
136C2×C4C22×D416yes—
137C2×C4H(5,𝔽2)16yes—
138C22—212no—
139C22—212no—
140C22—224no—
141C22—44no—
142C22—412no—
143C22—48no—
144C22—424no—
145C22—424no—
146C22—864no—
147C22C2×#12718yescomplemented
148C22C23×D413yes—
149C22C23×D411yescharacteristic
150C22C2×H(5,𝔽2)13yes—
151C4—212no—
152C4—26no—
153C4—44no—
154C4C2×#12714yescomplemented
155C4D42112yescomplemented
156C2—28no—
157C2—48no—
158C2—88no—
159C2C22×#12711yescharacteristic
160C2C2×D4213yes—
161C2#349813yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×D42C2semidirect
2. C22×#127C2semidirect
3. #2867C2semidirect
4. #2856C2semidirect
5. #2541C2semidirect
6. #2433C2semidirect
7. C4×#127C2semidirect
8. C42×D4C2semidirect
10. C23×D4C22semidirect
14. C2×#127C22semidirect
15. C2×#127C22semidirect
16. C2×#127C22semidirect
17. C2×#127C22semidirect
18. C2×#121C22semidirect
19. C2×#121C22semidirect
20. C2×#120C22semidirect
22. C2×C4×D4C22semidirect
23. C22×C42C22semidirect
24. #392C22semidirect
25. #390C22semidirect
26. #390C22semidirect
27. #384C22semidirect
28. C4×C4⋊C4C22semidirect
29. C4×C22⋊C4C22semidirect
30. C43C22semidirect
49. #127D4direct (N × H)
54. C4×D4D4semidirect
87. C2×D4C2×D4semidirect
91. C22×C4C2×D4semidirect
98. C4⋊C4C2×D4semidirect
101. C22⋊C4C2×D4semidirect
104. C42C2×D4semidirect
105. C42C2×D4semidirect
125. D4#127direct (N × H)
147. C22C2×#127semidirect
154. C4C2×#127semidirect
155. C4D42semidirect

Maximal subgroups

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

subgroupindex
C2×D422
C22×#1272
#28672
#28562
#25412
#24332
C4×#1272
C42×D42