Finite Group Databasebeta

← C2×#3199 #60743

Subgroups — C2×#3199 #60743

Order 256.

1253 conjugacy classes of subgroups; only the 447 normal subgroups are shown.

normal subgroups only

This group has 1253 conjugacy classes of subgroups — too many to draw. Only its 447 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×#535C212yesmaximal, complemented
2#3199C2116yesmaximal, direct product factor
3C2×#482C214yesmaximal, complemented
4C2×#469C212yesmaximal, complemented
5C2×#466C212yesmaximal, complemented
6C2×#447C214yesmaximal, complemented
7C2×#408C211yesmaximal, characteristic, complemented
8C22×#41C2211yescharacteristic
9C23×D4C2212yescomplemented
10C22×D8C2212yes—
11C22×D8C2212yes—
12C22×M16C2211yescharacteristic, complemented
13C23×C8C2211yescharacteristic, complemented
14#535—28no—
15#535C22116yescomplemented
16C2×#127—22no—
17C2×#124—22no—
18C2×#121C2214yes—
19C2×#121C2214yescomplemented
20C2×#120—22no—
21C2×#120—22no—
22C2×#117C2212yescomplemented
23#482C22132yescomplemented
24#469C22116yescomplemented
25#466C22116yescomplemented
26#447C22132yescomplemented
27C2×#107C2212yescomplemented
28C2×#106C2212yescomplemented
29C2×#102C2214yescomplemented
30C2×#102C2214yescomplemented
31#408C2218yescomplemented
32C2×C22⋊C8C2214yescomplemented
33C25—22no—
34C2×#41—42no—
35C2×#41—42no—
36C2×#41D414yes—
37C2×#41C2314yes—
38C22×Q8D411yescharacteristic
39C22×D4—24no—
40C22×D4—24no—
41C22×D4—22no—
42C22×D4—24no—
43C22×D4—44no—
44C22×D4—44no—
45C22×D4D411yescharacteristic
46C22×D4D412yes—
47C22×D4C2314yes—
48C22×D4C2318yescomplemented
49C23×C4C2312yes—
50C23×C4C2311yescharacteristic
51C2×D8—44no—
52C2×D8—44no—
53C2×D8C2318yes—
54C2×D8C2318yes—
55C2×M16—42no—
56C2×M16C2314yescomplemented
57C22×C8—42no—
58C22×C8C2312yes—
59C22×C8C2312yes—
60C22×C8C2314yescomplemented
61#127—216no—
62#124—216no—
63#121—28no—
64#121—28no—
65#121C23116yes—
66#121C23116yescomplemented
67#120—216no—
68#120—216no—
69#117—24no—
70#117C2318yescomplemented
71C2×C4⋊C4C2314yes—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C42—22no—
76#107C2318yescomplemented
77#106C2318yescomplemented
78#102C23116yescomplemented
79#102C23116yescomplemented
80C22⋊C8C23116yescomplemented
81C24—24no—
82C24—28no—
83C24—48no—
84C24—88no—
85C24C2×D412yes—
86C24C2×D411yescharacteristic
87#41—48no—
88#41—44no—
89#41—44no—
90C2×Q8—42no—
91C2×Q8C2×D414yes—
92C2×D4—216no—
93C2×D4—216no—
94C2×D4—24no—
95C2×D4—28no—
96C2×D4—28no—
97C2×D4—28no—
98C2×D4—48no—
99C2×D4—48no—
100C2×D4—44no—
101C2×D4—42no—
102C2×D4—48no—
103C2×D4—416no—
104C2×D4—88no—
105C2×D4—88no—
106C2×D4C2×D414yes—
107C2×D4C2×D418yes—
108C2×D4C2418yes—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4C2×D414yes—
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2414yes—
119C22×C4C2412yes—
120C22×C4C2411yescharacteristic
121D8—48no—
122D8—48no—
123M16—44no—
124C2×C8—24no—
125C2×C8—24no—
126C2×C8—44no—
127C2×C8C2414yes—
128C2×C8C2414yes—
129C4⋊C4—24no—
130C4⋊C4C2418yes—
131C22⋊C4—28no—
132C22⋊C4—216no—
133C22⋊C4—216no—
134C42—28no—
135C23—24no—
136C23—22no—
137C23—24no—
138C23—28no—
139C23—28no—
140C23—416no—
141C23—48no—
142C23—48no—
143C23—48no—
144C23—816no—
145C23—832no—
146C23C22×D414yes—
147C23C22×D412yes—
148C23C22×D411yescharacteristic
149Q8—44no—
150D4—28no—
151D4—28no—
152D4—48no—
153D4—44no—
154D4—416no—
155D4—48no—
156D4—816no—
157D4—816no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—28no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—28no—
167C2×C4—48no—
168C2×C4C22×D414yes—
169C2×C4C22×D412yes—
170C2×C4C22×D411yescharacteristic
171C2×C4H(5,𝔽2)12yes—
172C2×C4H(5,𝔽2)12yes—
173C2×C4C2511yescharacteristic
174C8—24no—
175C8—24no—
176C22—24no—
177C22—22no—
178C22—24no—
179C22—28no—
180C22—44no—
181C22—48no—
182C22—48no—
183C22—48no—
184C22—48no—
185C22—832no—
186C22—816no—
187C22#53412yes—
188C22#57614yes—
189C22C23×D411yescharacteristic
190C4—24no—
191C4—22no—
192C4—48no—
193C4#53412yes—
194C4C2×H(5,𝔽2)11yescharacteristic
195C4C2×H(5,𝔽2)11yescharacteristic
196C2—42no—
197C2—44no—
198C2—88no—
199C2#319914yesdirect product factor
200C2C2×#53411yescharacteristic
201C2C2×#57612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#535C2semidirect
2. #3199C2direct (N × H)
3. C2×#482C2semidirect
4. C2×#469C2semidirect
5. C2×#466C2semidirect
6. C2×#447C2semidirect
7. C2×#408C2semidirect
9. C23×D4C22semidirect
12. C22×M16C22semidirect
13. C23×C8C22semidirect
15. #535C22semidirect
19. C2×#121C22semidirect
22. C2×#117C22semidirect
23. #482C22semidirect
24. #469C22semidirect
25. #466C22semidirect
26. #447C22semidirect
27. C2×#107C22semidirect
28. C2×#106C22semidirect
29. C2×#102C22semidirect
30. C2×#102C22semidirect
31. #408C22semidirect
32. C2×C22⋊C8C22semidirect
48. C22×D4C23semidirect
56. C2×M16C23semidirect
60. C22×C8C23semidirect
66. #121C23semidirect
70. #117C23semidirect
76. #107C23semidirect
77. #106C23semidirect
78. #102C23semidirect
79. #102C23semidirect
80. C22⋊C8C23semidirect
199. C2#3199direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5352
#31992
C2×#4822
C2×#4692
C2×#4662
C2×#4472
C2×#4082