Finite Group Databasebeta

← C22×#11758509 #16860444

Subgroups — C22×#11758509 #16860444

Order 3968.

1720 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 565 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×#92495C211yesmaximal, characteristic, complemented
2C2×#11758509C2124yesmaximal, direct product factor
3C22×#92480C212yesmaximal, complemented
4C22×#92479C212yesmaximal, complemented
5C22×#92474C211yesmaximal, characteristic, complemented
6C23×D4×C31C211yesmaximal, characteristic, complemented
7C25×C31—21no—
8C22×D4×C31—21no—
9C22×D4×C31—26no—
10C22×D4×C31C2212yes—
11C22×D4×C31C22112yescomplemented
12C23×C4×C31C2211yescharacteristic
13C22×C22⋊D31—21no—
14C2×#92495—26no—
15C2×#92495C22112yescomplemented
16C22×D124C2211yescharacteristic, complemented
17C22×C4×D31—21no—
18C22×Dic62C2211yescharacteristic, complemented
19C2×#92480C22124yescomplemented
20#11758509C22164yesdirect product factor
21C2×#92479C22124yescomplemented
22C2×#92474C22112yescomplemented
23C22×C31⋊C8C2212yescomplemented
24C24×C31—22no—
25C24×C31—212no—
26C24×C31—48no—
27C24×C31D411yescharacteristic
28C23×D31—21no—
29C2×D4×C31—26no—
30C2×D4×C31—212no—
31C2×D4×C31—224no—
32C2×D4×C31C23112yes—
33C2×D4×C31C23116yescomplemented
34C22×C4×C31D411yescharacteristic
35C22×C4×C31D416yes—
36C22×C4×C31C2311yescharacteristic
37C22×C4×C31C2316yes—
38C2×C22⋊D31—212no—
39C22×Dic31—21no—
40#92495—224no—
41#92495C23116yescomplemented
42C2×D124—23no—
43C2×D124C2316yescomplemented
44C2×C4×D31—212no—
45C2×Dic62—23no—
46C2×Dic62C2316yescomplemented
47#92480C23132yescomplemented
48#92479C23132yescomplemented
49#92474C23116yescomplemented
50C2×C31⋊C8C23112yescomplemented
51C23×C31—24no—
52C23×C31—212no—
53C23×C31—216no—
54C23×C31—448no—
55C23×C31—48no—
56C23×C31C2×D416yes—
57C23×C31C2×D411yescharacteristic
58C22×D31—26no—
59C22×D31—44no—
60D4×C31—212no—
61D4×C31—216no—
62D4×C31C2418yes—
63C2×C4×C31C2×D413yes—
64C2×C4×C31C2×D416yes—
65C2×C4×C31C2×D4112yes—
66C2×C4×C31C2413yes—
67C2×C4×C31C2414yes—
68C22⋊D31—216no—
69C2×Dic31—26no—
70D124—26no—
71D124C2414yescomplemented
72C4×D31—216no—
73Dic62—26no—
74Dic62C2414yescomplemented
75C31⋊C8C2418yescomplemented
76C22×AGL(1,ℤ/8)—311nomaximal, Sylow
77C22×C31—212no—
78C22×C31—28no—
79C22×C31—432no—
80C22×C31—424no—
81C22×C31AGL(1,ℤ/8)14yescomplemented
82C22×C31C22×D414yes—
83C22×C31C22×D413yes—
84C2×D31—24no—
85C2×D31—412no—
86C4×C31C22×D413yes—
87C4×C31C22×D414yes—
88C4×C31C2511yescharacteristic
89Dic31—24no—
90C22×#41—311no—
91C23×D4D3111yescharacteristic, complemented
92C2×AGL(1,ℤ/8)—3124no—
93C22×SD16—312no—
94C22×D8—312no—
95C22×M16—311no—
96C2×C31—24no—
97C2×C31—48no—
98C2×C31C2×AGL(1,ℤ/8)16yescomplemented
99C2×C31C23×D411yescharacteristic
100D31—44no—
101C25—21no—
102C2×#41—3112no—
103C2×#41—626no—
104C22×Q8—311no—
105C22×D4—26no—
106C22×D4—21no—
107C22×D4—311no—
108C22×D4—621no—
109C22×D4C2×D3112yes—
110C22×D4C2×D31112yescomplemented
111C23×C4—621no—
112C23×C4C2×D3111yescharacteristic
113AGL(1,ℤ/8)—3164no—
114C2×SD16—3124no—
115C2×D8—3124no—
116C2×M16—3112no—
117C22×C8—312no—
118C31C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
119C24—212no—
120C24—22no—
121C24—48no—
122C24—621no—
123C24C22⋊D3111yescharacteristic
124#41—3116no—
125#41—6224no—
126C2×Q8—316no—
127C2×Q8—623no—
128C2×D4—26no—
129C2×D4—212no—
130C2×D4—224no—
131C2×D4—316no—
132C2×D4—6212no—
133C2×D4—623no—
134C2×D4C22×D31112yes—
135C2×D4C22×D31116yescomplemented
136C22×C4—621no—
137C22×C4—6212no—
138C22×C4C22⋊D3116yes—
139C22×C4C22⋊D3111yescharacteristic
140C22×C4C22×D3116yes—
141C22×C4C22×D3111yescharacteristic
142SD16—3132no—
143D8—3132no—
144M16—3116no—
145C2×C8—3112no—
146C23—24no—
147C23—212no—
148C23—216no—
149C23—448no—
150C23—48no—
151C23—626no—
152C23—1244no—
153C23C2×C22⋊D3116yes—
154C23C2×C22⋊D3111yescharacteristic
155Q8—314no—
156Q8—626no—
157D4—212no—
158D4—216no—
159D4—314no—
160D4—6216no—
161D4—626no—
162D4C23×D3118yes—
163C2×C4—626no—
164C2×C4—6216no—
165C2×C4C2×C22⋊D3116yes—
166C2×C4C2×C22⋊D3113yes—
167C2×C4C2×C22⋊D31112yes—
168C2×C4C23×D3113yes—
169C2×C4C23×D3114yes—
170C8—318no—
171C22—212no—
172C22—28no—
173C22—432no—
174C22—424no—
175C22—624no—
176C22—12412no—
177C22#1175850914yesdirect product factor
178C22C22×C22⋊D3114yes—
179C22C22×C22⋊D3113yes—
180C4—624no—
181C4C22×C22⋊D3114yes—
182C4C22×C22⋊D3113yes—
183C4C24×D3111yescharacteristic
184C2—24no—
185C2—48no—
186C2—1244no—
187C2C2×#1175850916yesdirect product factor
188C2C23×C22⋊D3111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D31
commutator, cosocleC4×C31C25
FrattiniC4C24×D31
FittingC23×D4×C31C2
socleC23×C31C2×D4
hypercenterC23×D4D31
hypocenterC31C22×AGL(1,ℤ/8)
LCSG ▷ C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#92495C2semidirect
2. C2×#11758509C2direct (N × H)
3. C22×#92480C2semidirect
4. C22×#92479C2semidirect
5. C22×#92474C2semidirect
6. C23×D4×C31C2semidirect
11. C22×D4×C31C22semidirect
15. C2×#92495C22semidirect
16. C22×D124C22semidirect
18. C22×Dic62C22semidirect
19. C2×#92480C22semidirect
20. #11758509C22direct (N × H)
21. C2×#92479C22semidirect
22. C2×#92474C22semidirect
23. C22×C31⋊C8C22semidirect
33. C2×D4×C31C23semidirect
41. #92495C23semidirect
43. C2×D124C23semidirect
46. C2×Dic62C23semidirect
47. #92480C23semidirect
48. #92479C23semidirect
49. #92474C23semidirect
50. C2×C31⋊C8C23semidirect
71. D124C24semidirect
74. Dic62C24semidirect
75. C31⋊C8C24semidirect
81. C22×C31AGL(1,ℤ/8)semidirect
91. C23×D4D31semidirect
98. C2×C31C2×AGL(1,ℤ/8)semidirect
110. C22×D4C2×D31semidirect
118. C31C22×AGL(1,ℤ/8)semidirect
135. C2×D4C22×D31semidirect
177. C22#11758509direct (N × H)
187. C2C2×#11758509direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#924952
C2×#117585092
C22×#924802
C22×#924792
C22×#924742
C23×D4×C312
C22×AGL(1,ℤ/8)31