Finite Group Databasebeta

← C2×C4×D3×#3861 #72245830

Subgroups — C2×C4×D3×#3861 #72245830

Order 7056.

1160 conjugacy classes of subgroups; only the 385 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 385 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×Dic3×#3861C211yesmaximal, characteristic, complemented
2C2×C3×C4×#3861C211yesmaximal, characteristic, complemented
3C4×D3×#3861C214yesmaximal, direct product factor
4C2×C4×D3×C72C311yesmaximal, characteristic, complemented
5C22×C4×#3861—31nomaximal
6C22×D3×#3861C211yesmaximal, characteristic
7C22×C3×#3861C2211yescharacteristic
8C2×D3×#3861C414yesdirect product factor
9C2×D3×#3861C2212yes—
10C3×C4×#3861C2212yescomplemented
11Dic3×#3861C2212yescomplemented
12C22×D3×C72C2×C311yescharacteristic
13C23×#3861—31no—
14C2×C3×C4×C72C2×C311yescharacteristic, complemented
15C2×C72×Dic3C2×C311yescharacteristic, complemented
16C4×D3×C72C2×C314yescomplemented
17C2×C4×#3861—21no—
18C2×C4×#3861—31no—
19C2×C4×#3861—34no—
20C2×C4×#3861D311yescharacteristic, direct product factor
21C2×C4×D3×ΓL(1,𝔽8)—78nomaximal
22C2×C3×#3861C2×C412yescomplemented
23C2×C3×#3861C2311yescharacteristic
24D3×#3861C2×C414yesdirect product factor
25C22×C4×C72—31noHall
26C22×C3×C72C22×C311yescharacteristic
27C2×D3×C72C3×C414yescomplemented
28C2×D3×C72C22×C312yes—
29C22×#3861—21no—
30C22×#3861—34no—
31C22×#3861—32no—
32C22×#3861C2×D311yescharacteristic
33C3×C4×C72C22×C312yescomplemented
34C72×Dic3C22×C312yescomplemented
35C4×#3861—22no—
36C4×#3861—32no—
37C4×#3861C2×D312yesdirect product factor
38C22×D3×ΓL(1,𝔽8)—78no—
39C2×C3×C4×ΓL(1,𝔽8)—78no—
40C2×Dic3×ΓL(1,𝔽8)—78no—
41C4×D3×ΓL(1,𝔽8)—732no—
42C3×#3861C22×C411yescharacteristic, complemented, Hall
43C23×C72—31no—
44C2×C4×C72—31no—
45C2×C4×C72—34no—
46C2×C4×C72C3×D311yescharacteristic, complemented
47C2×C4×D3×C7ΓL(1,𝔽8)18yescomplemented
48C22×C4×ΓL(1,𝔽8)—218no—
49C2×C3×C72C2×C3×C412yescomplemented
50C2×C3×C72C23×C311yescharacteristic
51D3×C72C2×C3×C414yescomplemented
52C2×#3861—22no—
53C2×#3861—21no—
54C2×#3861—34no—
55C2×#3861C4×D312yesdirect product factor
56C2×#3861C22×D311yescharacteristic
57C22×C3×ΓL(1,𝔽8)—78no—
58C2×D3×ΓL(1,𝔽8)—732no—
59C2×D3×ΓL(1,𝔽8)—716no—
60C3×C4×ΓL(1,𝔽8)—716no—
61Dic3×ΓL(1,𝔽8)—716no—
62C22×C72—34no—
63C22×C72—32no—
64C22×C72C2×C3×D311yescharacteristic
65C4×C72—32no—
66C4×C72C2×C3×D312yescomplemented
67C22×D3×C7C2×ΓL(1,𝔽8)18yes—
68C23×ΓL(1,𝔽8)—218no—
69C2×C3×C4×C7C2×ΓL(1,𝔽8)18yescomplemented
70C2×C7×Dic3C2×ΓL(1,𝔽8)18yescomplemented
71C4×D3×C7C2×ΓL(1,𝔽8)132yescomplemented
72C2×C4×ΓL(1,𝔽8)—78no—
73C2×C4×ΓL(1,𝔽8)—148no—
74C2×C4×ΓL(1,𝔽8)—2132no—
75C2×C4×ΓL(1,𝔽8)—218no—
76C3×C72C22×C3×C411yescharacteristic, complemented
77#3861—21no—
78#3861C2×C4×D311yescharacteristic, direct product factor
79C2×C3×C4×D3—491noHall
80C2×C3×ΓL(1,𝔽8)—716no—
81C2×C3×ΓL(1,𝔽8)—78no—
82D3×ΓL(1,𝔽8)—732no—
83C22×C4×C7—38no—
84C2×C72—34no—
85C2×C72C3×C4×D312yescomplemented
86C2×C72C22×C3×D311yescharacteristic
87C22×C3×C7C22×ΓL(1,𝔽8)18yes—
88C2×D3×C7C4×ΓL(1,𝔽8)132yescomplemented
89C2×D3×C7C22×ΓL(1,𝔽8)116yes—
90C22×ΓL(1,𝔽8)—78no—
91C22×ΓL(1,𝔽8)—148no—
92C22×ΓL(1,𝔽8)—2116no—
93C22×ΓL(1,𝔽8)—2132no—
94C3×C4×C7C22×ΓL(1,𝔽8)116yescomplemented
95C7×Dic3C22×ΓL(1,𝔽8)116yescomplemented
96C4×ΓL(1,𝔽8)—716no—
97C4×ΓL(1,𝔽8)—1416no—
98C4×ΓL(1,𝔽8)—2116no—
99C22×C3×D3—491no—
100C2×C32×C4—491no—
101C2×C3×Dic3—491no—
102C3×C4×D3—494no—
103C3×ΓL(1,𝔽8)—78no—
104C23×C7—38no—
105C2×C4×C7—38no—
106C2×C4×C7—332no—
107C2×C4×C7D3×ΓL(1,𝔽8)18yescomplemented
108C72C2×C3×C4×D311yescharacteristic, complemented, Sylow
109C22×C3×C4—1471no—
110C2×C4×D3#386111yescharacteristic, direct product factor
111C2×C3×C7C2×C4×ΓL(1,𝔽8)116yescomplemented
112C2×C3×C7C23×ΓL(1,𝔽8)18yes—
113D3×C7C2×C4×ΓL(1,𝔽8)132yescomplemented
114C2×ΓL(1,𝔽8)—78no—
115C2×ΓL(1,𝔽8)—716no—
116C2×ΓL(1,𝔽8)—148no—
117C2×ΓL(1,𝔽8)—1416no—
118C2×ΓL(1,𝔽8)—2132no—
119C22×C32—491no—
120C2×C3×D3—494no—
121C2×C3×D3—492no—
122C32×C4—492no—
123C3×Dic3—492no—
124C22×C7—316no—
125C22×C7—332no—
126C22×C7C2×D3×ΓL(1,𝔽8)18yes—
127C4×C7—316no—
128C4×C7C2×D3×ΓL(1,𝔽8)116yescomplemented
129C23×C3—1471no—
130C22×D3C2×#386111yescharacteristic
131C2×C3×C4—491no—
132C2×C3×C4—981no—
133C2×C3×C4—1474no—
134C2×C3×C4—1471no—
135C2×C3×C4C2×#386111yescharacteristic, complemented
136C2×Dic3C2×#386111yescharacteristic, complemented
137C4×D3C2×#386114yesdirect product factor
138C3×C7C22×C4×ΓL(1,𝔽8)18yescomplemented
139ΓL(1,𝔽8)—78no—
140ΓL(1,𝔽8)—148no—
141C2×C32—492no—
142C2×C32—491no—
143C3×D3—494no—
144C22×C4—31noSylow
145C2×C7—332no—
146C2×C7C4×D3×ΓL(1,𝔽8)116yescomplemented
147C2×C7C22×D3×ΓL(1,𝔽8)18yes—
148C22×C3—491no—
149C22×C3—981no—
150C22×C3—1472no—
151C22×C3—1474no—
152C22×C3C22×#386111yescharacteristic
153C2×D3C4×#386114yesdirect product factor
154C2×D3C22×#386112yes—
155C3×C4—492no—
156C3×C4—982no—
157C3×C4—1472no—
158C3×C4C22×#386112yescomplemented
159Dic3C22×#386112yescomplemented
160C32—491noSylow
161C23—31no—
162C2×C4—31no—
163C2×C4—34no—
164C2×C4D3×#386111yescharacteristic, direct product factor
165C7C2×C4×D3×ΓL(1,𝔽8)18yescomplemented
166C2×C3—492no—
167C2×C3—491no—
168C2×C3—982no—
169C2×C3—981no—
170C2×C3—1474no—
171C2×C3C2×C4×#386112yescomplemented
172C2×C3C23×#386111yescharacteristic
173D3C2×C4×#386114yesdirect product factor
174C22—34no—
175C22—32no—
176C22C2×D3×#386111yescharacteristic
177C4—32no—
178C4C2×D3×#386112yesdirect product factor
179C3—491no—
180C3—981no—
181C3C22×C4×#386111yescharacteristic, complemented
182C2—34no—
183C2C22×D3×#386111yescharacteristic
184C2C4×D3×#386112yesdirect product factor
subgroupG/H
center, hypercenterC2×C4D3×#3861
commutator, hypocenterC3×C72C22×C3×C4
FrattiniC2C22×D3×#3861
FittingC2×C3×C4×C72C2×C3
socleC22×C3×C72C22×C3
prime radicalC22×D3×#3861C2
cosocleC2×C3×C72C23×C3
LCSG ▷ C3×C72
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×#3861C2semidirect
2. C2×C3×C4×#3861C2semidirect
3. C4×D3×#3861C2direct (N × H)
4. C2×C4×D3×C72C3semidirect
8. C2×D3×#3861C4direct (N × H)
10. C3×C4×#3861C22semidirect
11. Dic3×#3861C22semidirect
14. C2×C3×C4×C72C2×C3semidirect
15. C2×C72×Dic3C2×C3semidirect
16. C4×D3×C72C2×C3semidirect
20. C2×C4×#3861D3direct (N × H)
22. C2×C3×#3861C2×C4semidirect
24. D3×#3861C2×C4direct (N × H)
27. C2×D3×C72C3×C4semidirect
33. C3×C4×C72C22×C3semidirect
34. C72×Dic3C22×C3semidirect
37. C4×#3861C2×D3direct (N × H)
42. C3×#3861C22×C4semidirect
46. C2×C4×C72C3×D3semidirect
47. C2×C4×D3×C7ΓL(1,𝔽8)semidirect
49. C2×C3×C72C2×C3×C4semidirect
51. D3×C72C2×C3×C4semidirect
55. C2×#3861C4×D3direct (N × H)
66. C4×C72C2×C3×D3semidirect
69. C2×C3×C4×C7C2×ΓL(1,𝔽8)semidirect
70. C2×C7×Dic3C2×ΓL(1,𝔽8)semidirect
71. C4×D3×C7C2×ΓL(1,𝔽8)semidirect
76. C3×C72C22×C3×C4semidirect
78. #3861C2×C4×D3direct (N × H)
85. C2×C72C3×C4×D3semidirect
88. C2×D3×C7C4×ΓL(1,𝔽8)semidirect
94. C3×C4×C7C22×ΓL(1,𝔽8)semidirect
95. C7×Dic3C22×ΓL(1,𝔽8)semidirect
107. C2×C4×C7D3×ΓL(1,𝔽8)semidirect
108. C72C2×C3×C4×D3semidirect
110. C2×C4×D3#3861direct (N × H)
111. C2×C3×C7C2×C4×ΓL(1,𝔽8)semidirect
113. D3×C7C2×C4×ΓL(1,𝔽8)semidirect
128. C4×C7C2×D3×ΓL(1,𝔽8)semidirect
135. C2×C3×C4C2×#3861semidirect
136. C2×Dic3C2×#3861semidirect
137. C4×D3C2×#3861direct (N × H)
138. C3×C7C22×C4×ΓL(1,𝔽8)semidirect
146. C2×C7C4×D3×ΓL(1,𝔽8)semidirect
153. C2×D3C4×#3861direct (N × H)
158. C3×C4C22×#3861semidirect
159. Dic3C22×#3861semidirect
164. C2×C4D3×#3861direct (N × H)
165. C7C2×C4×D3×ΓL(1,𝔽8)semidirect
171. C2×C3C2×C4×#3861semidirect
173. D3C2×C4×#3861direct (N × H)
178. C4C2×D3×#3861direct (N × H)
181. C3C22×C4×#3861semidirect
184. C2C4×D3×#3861direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Dic3×#38612
C2×C3×C4×#38612
C4×D3×#38612
C2×C4×D3×C723
C22×C4×#38613
C22×D3×#38612
C2×C4×D3×ΓL(1,𝔽8)7