Finite Group Databasebeta

← #16860491

Subgroups — #16860491

Order 3968.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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
1#14521885C211yesmaximal, characteristic, complemented
2#14521860C218yesmaximal, complemented
3#14521852C218yesmaximal, complemented
4#14521847C216yesmaximal, complemented
5C2×#11758509C216yesmaximal, complemented
6#14521816C211yesmaximal, characteristic, complemented
7C2×C31×H(5,𝔽2)C211yesmaximal, characteristic, complemented
8C31×H(5,𝔽2)—24no—
9C31×H(5,𝔽2)C2218yescomplemented
10C2×#41×C31—21no—
11C2×#41×C31C2211yescharacteristic
12C2×#41×C31C2213yescomplemented
13C22×D4×C31—23no—
14C22×D4×C31C2213yescomplemented
15#11758567C2214yescomplemented
16#11758566C2214yescomplemented
17#41×D31—24no—
18#11758564—21no—
19#11758561—23no—
20C2×#92495C2216yescomplemented
21#11758528C22112yescomplemented
22#11758527C22124yescomplemented
23#11758526C22112yescomplemented
24#11758525C2218yescomplemented
25#11758519C2214yescomplemented
26C2×#92480C22112yescomplemented
27#11758509C22112yescomplemented
28#11758509C22124yescomplemented
29C2×#92479C22112yescomplemented
30C2×#92474C2216yescomplemented
31C24×C31—23no—
32#41×C31—28no—
33#41×C31—26no—
34#41×C31D414yes—
35#41×C31C23112yes—
36#41×C31C2314yes—
37C2×Q8×C31D411yescharacteristic
38C2×Q8×C31C2311yescharacteristic
39C2×D4×C31—23no—
40C2×D4×C31—224no—
41C2×D4×C31—23no—
42C2×D4×C31—26no—
43C2×D4×C31D413yes—
44C2×D4×C31C23112yes—
45C2×D4×C31C2313yes—
46C22×C4×C31—23no—
47C22×C4×C31C2313yes—
48C2×C22⋊D31—26no—
49#92499—24no—
50Q8×D31—24no—
51#92497—212no—
52D4×D31—212no—
53#92495—26no—
54#92495—26no—
55#92495C2314yes—
56#92495C23112yescomplemented
57C2×D124C2316yescomplemented
58C2×C4×D31—26no—
59C2×Dic62C2316yescomplemented
60#92482C2318yescomplemented
61#92481C2318yescomplemented
62#92480C23124yescomplemented
63#92479C23124yescomplemented
64#92474C2314yes—
65#92474C23112yescomplemented
66C2×C31⋊C8C23112yescomplemented
67C23×C31—23no—
68C23×C31—212no—
69C23×C31—86no—
70C23×C31C2×D413yes—
71C22×D31—26no—
72Q8×C31—21no—
73Q8×C31C2×D414yes—
74Q8×C31C2412yes—
75D4×C31—212no—
76D4×C31—212no—
77D4×C31—23no—
78D4×C31C2×D4112yes—
79D4×C31C2416yes—
80C2×C4×C31—21no—
81C2×C4×C31—212no—
82C2×C4×C31C2×D416yes—
83C2×C4×C31C2×D413yes—
84C2×C4×C31C2416yes—
85C2×C4×C31C2411yescharacteristic
86C22⋊D31—24no—
87C22⋊D31—212no—
88C2×Dic31—26no—
89D124—26no—
90D124C2414yes—
91C4×D31—24no—
92C4×D31—212no—
93Dic62—26no—
94Dic62C2414yes—
95C31⋊C8C2418yes—
96#3585—311nomaximal, Sylow
97C22×C31—26no—
98C22×C31—43no—
99C22×C31—46no—
100C22×C31—812no—
101C22×C31C22×D416yes—
102C22×C31C22×D411yescharacteristic
103C2×D31—24no—
104C2×D31—86no—
105C4×C31—22no—
106C4×C31C22×D416yes—
107C4×C31C22×D411yescharacteristic
108C4×C31C2511yescharacteristic
109Dic31—24no—
110#585—311no—
111C2×H(5,𝔽2)D3111yescharacteristic, complemented
112#577—318no—
113#576—318no—
114#575—316no—
115C2×AGL(1,ℤ/8)—316no—
116#568—311no—
117C2×C31—26no—
118C2×C31—21no—
119C2×C31—46no—
120C2×C31C23×D411yescharacteristic
121D31—44no—
1222−1+4—314no—
1232−1+4—621no—
124H(5,𝔽2)—24no—
125H(5,𝔽2)—314no—
126H(5,𝔽2)—623no—
127H(5,𝔽2)C2×D3118yescomplemented
128C2×#41—21no—
129C2×#41—316no—
130C2×#41—624no—
131C2×#41C2×D3111yescharacteristic
132C2×#41C2×D3113yescomplemented
133C22×D4—23no—
134C22×D4C2×D3113yescomplemented
135#137—3112no—
136#137—314no—
137AGL(1,ℤ/8)—3112no—
138AGL(1,ℤ/8)—3112no—
139AGL(1,ℤ/8)—3124no—
140#135—3124no—
141C2×SD16—3112no—
142C2×D8—3112no—
143#131—318no—
144C2×M16—316no—
145C31#358511yescharacteristic, complemented, Sylow
146C24—23no—
147#41—28no—
148#41—26no—
149#41—314no—
150#41—3112no—
151#41—624no—
152#41—6212no—
153#41—626no—
154#41—626no—
155#41C22⋊D3114yes—
156#41C22×D31112yes—
157#41C22×D3114yes—
158C2×Q8—316no—
159C2×Q8—624no—
160C2×Q8C22⋊D3111yescharacteristic
161C2×Q8C22×D3111yescharacteristic
162C2×D4—23no—
163C2×D4—224no—
164C2×D4—23no—
165C2×D4—26no—
166C2×D4—316no—
167C2×D4—626no—
168C2×D4—6212no—
169C2×D4C22⋊D3113yes—
170C2×D4C22×D31112yes—
171C2×D4C22×D3113yes—
172C22×C4—23no—
173C22×C4—626no—
174C22×C4C22×D3113yes—
175Q16—318no—
176SD16—318no—
177SD16—3124no—
178D8—3124no—
179M16—314no—
180M16—3112no—
181C2×C8—3112no—
182C23—23no—
183C23—212no—
184C23—86no—
185C23—626no—
186C23C2×C22⋊D3113yes—
187Q8—21no—
188Q8—314no—
189Q8—626no—
190Q8C2×C22⋊D3114yes—
191Q8C23×D3112yes—
192D4—212no—
193D4—212no—
194D4—23no—
195D4—314no—
196D4—624no—
197D4—6212no—
198D4—626no—
199D4C2×C22⋊D31112yes—
200D4C23×D3116yes—
201C2×C4—21no—
202C2×C4—212no—
203C2×C4—626no—
204C2×C4—624no—
205C2×C4—6212no—
206C2×C4C2×C22⋊D3113yes—
207C2×C4C2×C22⋊D3116yes—
208C2×C4C23×D3116yes—
209C2×C4C23×D3111yescharacteristic
210C8—318no—
211C22—26no—
212C22—43no—
213C22—46no—
214C22—812no—
215C22—624no—
216C22—2486no—
217C22C22×C22⋊D3111yescharacteristic
218C22C22×C22⋊D3116yes—
219C4—22no—
220C4—624no—
221C4C22×C22⋊D3116yes—
222C4C22×C22⋊D3111yescharacteristic
223C4C24×D3111yescharacteristic
224C2—21no—
225C2—26no—
226C2—46no—
227C2—1244no—
228C2C23×C22⋊D3111yescharacteristic
subgroupG/H
centerC2C23×C22⋊D31
commutator, cosocleC4×C31C25
FrattiniC4C24×D31
FittingC2×C31×H(5,𝔽2)C2
socleC2×C31C23×D4
hypercenterC2×H(5,𝔽2)D31
hypocenterC31#3585
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #14521885C2semidirect
2. #14521860C2semidirect
3. #14521852C2semidirect
4. #14521847C2semidirect
5. C2×#11758509C2semidirect
6. #14521816C2semidirect
7. C2×C31×H(5,𝔽2)C2semidirect
9. C31×H(5,𝔽2)C22semidirect
12. C2×#41×C31C22semidirect
14. C22×D4×C31C22semidirect
15. #11758567C22semidirect
16. #11758566C22semidirect
20. C2×#92495C22semidirect
21. #11758528C22semidirect
22. #11758527C22semidirect
23. #11758526C22semidirect
24. #11758525C22semidirect
25. #11758519C22semidirect
26. C2×#92480C22semidirect
27. #11758509C22semidirect
28. #11758509C22semidirect
29. C2×#92479C22semidirect
30. C2×#92474C22semidirect
56. #92495C23semidirect
57. C2×D124C23semidirect
59. C2×Dic62C23semidirect
60. #92482C23semidirect
61. #92481C23semidirect
62. #92480C23semidirect
63. #92479C23semidirect
65. #92474C23semidirect
66. C2×C31⋊C8C23semidirect
111. C2×H(5,𝔽2)D31semidirect
127. H(5,𝔽2)C2×D31semidirect
132. C2×#41C2×D31semidirect
134. C22×D4C2×D31semidirect
145. C31#3585semidirect

Maximal subgroups

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

subgroupindex
#145218852
#145218602
#145218522
#145218472
C2×#117585092
#145218162
C2×C31×H(5,𝔽2)2
#358531