Finite Group Databasebeta

← C2×#3151 #60636

Subgroups — C2×#3151 #60636

Order 256.

1975 conjugacy classes of subgroups; only the 583 normal subgroups are shown.

normal subgroups only

This group has 1975 conjugacy classes of subgroups — too many to draw. Only its 583 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×AGL(1,ℤ/8)C214yesmaximal, complemented
2C2×#532C211yesmaximal, characteristic, complemented
3C22×#127C211yesmaximal, characteristic, complemented
4#3151C2116yesmaximal, direct product factor
5C2×#496C214yesmaximal, complemented
6C2×#493C212yesmaximal, complemented
7C2×#492C212yesmaximal, complemented
8C2×C4×M16C211yesmaximal, characteristic, complemented
9C22×#41C2212yes—
10C23×D4—22no—
11C23×D4C2212yes—
12C2×AGL(1,ℤ/8)—232no—
13C2×AGL(1,ℤ/8)C22132yescomplemented
14C22×SD16C2218yescomplemented
15C22×D8C2218yescomplemented
16C22×M16C2212yes—
17#532—24no—
18#532C2218yescomplemented
19C2×C4⋊Q8C2211yescharacteristic, complemented
20C2×#127—22no—
21C2×#127—24no—
22C2×#127C2212yes—
23C2×#127C2212yes—
24C2×#127C2218yescomplemented
25C2×#127C2211yescharacteristic, complemented
26C2×#124C2212yescomplemented
27C2×#121—22no—
28C2×C4×D4—22no—
29C22×C42C2211yescharacteristic
30#496C22132yescomplemented
31#493C22116yescomplemented
32#492C22116yescomplemented
33C4×M16C2218yescomplemented
34C2×#97C2212yescomplemented
35C2×C4×C8C2212yescomplemented
36C25—22no—
37C2×#41—24no—
38C2×#41—216no—
39C2×#41C2318yes—
40C22×Q8C2312yes—
41C22×D4—22no—
42C22×D4—216no—
43C22×D4—22no—
44C22×D4—24no—
45C22×D4—22no—
46C22×D4—24no—
47C22×D4—24no—
48C22×D4—216no—
49C22×D4—416no—
50C22×D4C2312yes—
51C22×D4C2314yes—
52C22×D4C2318yes—
53C23×C4—22no—
54C23×C4D412yes—
55C23×C4C2311yescharacteristic
56AGL(1,ℤ/8)—2128no—
57C2×SD16—232no—
58C2×SD16C23132yescomplemented
59C2×D8—232no—
60C2×D8C23132yescomplemented
61C2×M16D4116yescomplemented
62C2×M16C2318yes—
63C22×C8C2314yes—
64C4⋊Q8—22no—
65C4⋊Q8C2314yescomplemented
66#127—24no—
67#127—24no—
68#127—216no—
69#127—22no—
70#127C2318yes—
71#127C2318yes—
72#127C2314yescomplemented
73#124—24no—
74#124C2318yescomplemented
75#121—216no—
76C4×D4—216no—
77C2×C4⋊C4—22no—
78C2×C22⋊C4—24no—
79C2×C42D412yes—
80C2×C42D414yes—
81C2×C42C2311yescharacteristic
82C2×C42C2311yescharacteristic
83C2×C42C2314yes—
84#97C2318yescomplemented
85C4×C8C2318yescomplemented
86C24—28no—
87C24—22no—
88C24—24no—
89C24—416no—
90C24—48no—
91C24C2×D411yescharacteristic
92#41—232no—
93#41—416no—
94C2×Q8—22no—
95C2×Q8—28no—
96C2×Q8C2414yes—
97C2×D4—28no—
98C2×D4—28no—
99C2×D4—216no—
100C2×D4—232no—
101C2×D4—28no—
102C2×D4—22no—
103C2×D4—28no—
104C2×D4—24no—
105C2×D4—216no—
106C2×D4—216no—
107C2×D4—416no—
108C2×D4—464no—
109C2×D4—48no—
110C2×D4—48no—
111C2×D4—416no—
112C2×D4—48no—
113C2×D4—416no—
114C2×D4C2414yes—
115C2×D4C2418yes—
116C22×C4—28no—
117C22×C4—22no—
118C22×C4—28no—
119C22×C4—48no—
120C22×C4C2×D418yes—
121C22×C4C2×D412yes—
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D418yes—
125C22×C4C2×D412yes—
126C22×C4C2412yes—
127C22×C4C2411yescharacteristic
128SD16—264no—
129D8—264no—
130M16C2×D4132yescomplemented
131C2×C8C2×D4116yescomplemented
132C2×C8C2418yes—
133C4⋊C4—28no—
134C22⋊C4—216no—
135C42C2×D418yes—
136C42C2×D412yes—
137C42C2×D412yes—
138C42C2412yes—
139C42C2412yes—
140C23—24no—
141C23—24no—
142C23—28no—
143C23—432no—
144C23—44no—
145C23—416no—
146C23—48no—
147C23—416no—
148C23—48no—
149C23—832no—
150C23#12714yes—
151C23C22×D412yes—
152C23C22×D411yescharacteristic
153Q8—28no—
154Q8—44no—
155D4—28no—
156D4—216no—
157D4—432no—
158D4—416no—
159D4—416no—
160D4—48no—
161D4—44no—
162D4—416no—
163D4—432no—
164C2×C4—24no—
165C2×C4—216no—
166C2×C4—24no—
167C2×C4—44no—
168C2×C4—416no—
169C2×C4#12714yes—
170C2×C4#12718yes—
171C2×C4AGL(1,ℤ/8)18yescomplemented
172C2×C4C22×D414yes—
173C2×C4C22×D412yes—
174C2×C4C22×D411yescharacteristic
175C2×C4C22×D414yes—
176C2×C4C22×D418yes—
177C2×C4C22×D414yes—
178C2×C4C2511yescharacteristic
179C8C22×D4116yescomplemented
180C22—28no—
181C22—24no—
182C22—44no—
183C22—48no—
184C22—416no—
185C22—48no—
186C22—832no—
187C22—88no—
188C22—816no—
189C22C2×#12714yes—
190C22C2×#12712yes—
191C22C2×AGL(1,ℤ/8)14yes—
192C22C23×D411yescharacteristic
193C4—24no—
194C4—44no—
195C4C2×#12714yes—
196C4C2×#12712yes—
197C4C2×AGL(1,ℤ/8)18yescomplemented
198C4C23×D412yes—
199C2—24no—
200C2—84no—
201C2—88no—
202C2#315114yesdirect product factor
203C2C22×#12711yescharacteristic
204C2C22×AGL(1,ℤ/8)12yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×AGL(1,ℤ/8)C2semidirect
2. C2×#532C2semidirect
3. C22×#127C2semidirect
4. #3151C2direct (N × H)
5. C2×#496C2semidirect
6. C2×#493C2semidirect
7. C2×#492C2semidirect
8. C2×C4×M16C2semidirect
13. C2×AGL(1,ℤ/8)C22semidirect
14. C22×SD16C22semidirect
15. C22×D8C22semidirect
18. #532C22semidirect
19. C2×C4⋊Q8C22semidirect
24. C2×#127C22semidirect
25. C2×#127C22semidirect
26. C2×#124C22semidirect
30. #496C22semidirect
31. #493C22semidirect
32. #492C22semidirect
33. C4×M16C22semidirect
34. C2×#97C22semidirect
35. C2×C4×C8C22semidirect
58. C2×SD16C23semidirect
60. C2×D8C23semidirect
61. C2×M16D4semidirect
65. C4⋊Q8C23semidirect
72. #127C23semidirect
74. #124C23semidirect
84. #97C23semidirect
85. C4×C8C23semidirect
130. M16C2×D4semidirect
131. C2×C8C2×D4semidirect
171. C2×C4AGL(1,ℤ/8)semidirect
179. C8C22×D4semidirect
197. C4C2×AGL(1,ℤ/8)semidirect
202. C2#3151direct (N × H)

Maximal subgroups

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

subgroupindex
C22×AGL(1,ℤ/8)2
C2×#5322
C22×#1272
#31512
C2×#4962
C2×#4932
C2×#4922
C2×C4×M162