Finite Group Databasebeta

← D8×D724 #37545397

Subgroups — D8×D724 #37545397

Order 23168.

620 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 141 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×D8×D181C212yesmaximal, complemented
2D4×D724C212yesmaximal, complemented
3#26041656C211yesmaximal, characteristic, complemented
4#26041516C212yesmaximal, complemented
5#26041474C212yesmaximal, complemented
6#26041414C214yesmaximal, complemented
7C8×D724C211yesmaximal, characteristic, complemented
8C4×D8×C181C211yesmaximal, characteristic, complemented
9C22×D724—22no—
10C2×D4×D181C2214yescomplemented
11C2×D1448C2212yescomplemented
12C2×D1448C2212yescomplemented
13C2×#15668434C2214yescomplemented
14D8×D181—28no—
15C2×C8×D181C2212yescomplemented
16#19153813C2212yescomplemented
17#19153791C2214yescomplemented
18#19153790—22no—
19#19153786—24no—
20C4×D724C2211yescharacteristic
21C4⋊#13089678C2211yescharacteristic, complemented
22#19153736C2214yescomplemented
23#19153728C2212yescomplemented
24C2×D8×C181C2211yescharacteristic
25C4×D4×C181C2212yescomplemented
26#107×C181C2211yescharacteristic, complemented
27#102×C181C2212yescomplemented
28C4×C8×C181C2211yescharacteristic, complemented
29C23×D181—24no—
30C2×D724—24no—
31C2×D724—22no—
32C2×D724—22no—
33C2×D724—48no—
34C2×D724D411yescharacteristic
35C2×D724C2314yes—
36C2×C22⋊D181—24no—
37D4×D181—28no—
38D4×D181—44no—
39C2×C4×D181C2312yes—
40D1448—24no—
41D1448—42no—
42#15668434—28no—
43C2×#13089678C2312yes—
44C8×D181—24no—
45C4⋊Dic181D411yescharacteristic
46#15668426—24no—
47#15668426D412yes—
48C2×D4×C181C2312yes—
49C22×C4×C181—22no—
50D8×C181D414yescomplemented
51C2×C8×C181C2311yescharacteristic
52C2×C8×C181C2311yescharacteristic
53C4⋊C4×C181C2312yes—
54C22⋊C4×C181—22no—
55C42×C181C2311yescharacteristic
56C22×D181—24no—
57C22×D181—48no—
58C22×D181—48no—
59C22×D181—84no—
60C22×D181C2×D412yes—
61D724—24no—
62D724—24no—
63D724—48no—
64D724—44no—
65D724—82no—
66D724D814yesdirect product factor
67C22⋊D181—48no—
68C2×Dic181C2×D412yes—
69C4×D181—22no—
70C4×D181—22no—
71#13089678—22no—
72C23×C181—22no—
73D4×C181—22no—
74D4×C181C2×D414yescomplemented
75C2×C4×C181—22no—
76C2×C4×C181—44no—
77C2×C4×C181C2×D411yescharacteristic
78C2×C4×C181C2×D411yescharacteristic
79C2×C4×C181C2411yescharacteristic
80C8×C181—21no—
81C8×C181C2×D412yescomplemented
82C22×C181—24no—
83C22×C181—42no—
84C22×C181—42no—
85C22×C181C22×D411yescharacteristic
86C2×D181—22no—
87C2×D181—22no—
88C2×D181—44no—
89C2×D181—44no—
90C2×D181—44no—
91C2×D181—816no—
92C2×D181C2×D814yescomplemented
93C4×C181—21no—
94C4×C181—42no—
95C4×C181C2×D812yescomplemented
96C4×C181C22×D411yescharacteristic
97C4×C181H(5,𝔽2)11yescharacteristic
98Dic181—22no—
99C2×C181—44no—
100C2×C181D4211yescharacteristic
101C2×C181C22×D811yescharacteristic
102C2×C181#57611yescharacteristic
103D181—24no—
104D181—84no—
105C181D4×D811yescharacteristic, complemented, Sylow
106D4×D8—1811nomaximal, Sylow
107C22×D8—1812no—
108D42—1812no—
109#493—1811no—
110#466—1812no—
111#459—1812no—
112#447—1814no—
113C4×D8D18111yescharacteristic, complemented
114C8×D4—1811no—
115C22×D4—3622no—
116C22×D4—1814no—
117C2×D8—3628no—
118C2×D8—1814no—
119C2×D8—1812no—
120C2×D8—1812no—
121C2×D8C2×D18111yescharacteristic
122C22×C8—1812no—
123#127—1812no—
124#121—3622no—
125#121—1814no—
126#120—3624no—
127C4×D4—1811no—
128C4×D4C2×D18112yescomplemented
129#107C2×D18111yescharacteristic, complemented
130C4⋊C8—1811no—
131#102—1814no—
132#102C2×D18112yescomplemented
133C22⋊C8—1812no—
134C4×C8C2×D18111yescharacteristic, complemented
135C24—3624no—
136C2×D4—3624no—
137C2×D4—3628no—
138C2×D4—3624no—
139C2×D4—3622no—
140C2×D4—3622no—
141C2×D4—1811no—
142C2×D4—1814no—
143C2×D4—7248no—
144C2×D4—7244no—
145C2×D4C22×D18112yes—
146C22×C4—22no—
147C22×C4—1812no—
148D8—3628no—
149D8—3624no—
150D8—7242no—
151D8D72414yesdirect product factor
152C2×C8—3624no—
153C2×C8—1812no—
154C2×C8C22×D18111yescharacteristic
155C2×C8C22×D18111yescharacteristic
156C4⋊C4—1811no—
157C4⋊C4C22×D18112yes—
158C22⋊C4—22no—
159C22⋊C4—3624no—
160C22⋊C4—1812no—
161C42C22×D18111yescharacteristic
162C23—22no—
163C23—3624no—
164C23—14484no—
165C23—1812no—
166C23—7248no—
167C23—7248no—
168D4—22no—
169D4—3624no—
170D4—3624no—
171D4—14482no—
172D4—1814no—
173D4—7248no—
174D4—7248no—
175D4—7244no—
176D4C2×D72414yescomplemented
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—3622no—
180C2×C4—3622no—
181C2×C4—1812no—
182C2×C4D4×D18111yescharacteristic
183C2×C4D4×D18111yescharacteristic
184C2×C4C23×D18111yescharacteristic
185C8—21no—
186C8—3622no—
187C8C2×D72412yescomplemented
188C22—24no—
189C22—42no—
190C22—42no—
191C22—3622no—
192C22—3622no—
193C22—144816no—
194C22—1814no—
195C22—7244no—
196C22—7244no—
197C22—7244no—
198C22C2×D4×D18111yescharacteristic
199C4—21no—
200C4—42no—
201C4—3622no—
202C4D8×D18112yescomplemented
203C4C22×D72411yescharacteristic
204C4#1915386211yescharacteristic
205C2—44no—
206C2—3624no—
207C2—14484no—
208C2D4×D72411yescharacteristic
209C2C2×D8×D18111yescharacteristic
210C2#2604208111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC4×D8×C181C2
socleC22×C181C22×D4
hypercenterC4×D8D181
hypocenterC181D4×D8
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×D8×D181C2semidirect
2. D4×D724C2semidirect
3. #26041656C2semidirect
4. #26041516C2semidirect
5. #26041474C2semidirect
6. #26041414C2semidirect
7. C8×D724C2semidirect
8. C4×D8×C181C2semidirect
10. C2×D4×D181C22semidirect
11. C2×D1448C22semidirect
12. C2×D1448C22semidirect
13. C2×#15668434C22semidirect
15. C2×C8×D181C22semidirect
16. #19153813C22semidirect
17. #19153791C22semidirect
21. C4⋊#13089678C22semidirect
22. #19153736C22semidirect
23. #19153728C22semidirect
25. C4×D4×C181C22semidirect
26. #107×C181C22semidirect
27. #102×C181C22semidirect
28. C4×C8×C181C22semidirect
50. D8×C181D4semidirect
66. D724D8direct (N × H)
74. D4×C181C2×D4semidirect
81. C8×C181C2×D4semidirect
92. C2×D181C2×D8semidirect
95. C4×C181C2×D8semidirect
105. C181D4×D8semidirect
113. C4×D8D181semidirect
128. C4×D4C2×D181semidirect
129. #107C2×D181semidirect
132. #102C2×D181semidirect
134. C4×C8C2×D181semidirect
151. D8D724direct (N × H)
176. D4C2×D724semidirect
187. C8C2×D724semidirect
202. C4D8×D181semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1812
D4×D7242
#260416562
#260415162
#260414742
#260414142
C8×D7242
C4×D8×C1812
D4×D8181