Finite Group Databasebeta

← C22×#16862272 #30680694

Subgroups — C22×#16862272 #30680694

Order 16768.

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×#14523513C211yesmaximal, characteristic, complemented
2C2×#16862272C2124yesmaximal, direct product factor
3C22×#14523501C212yesmaximal, complemented
4C22×#14523498C212yesmaximal, complemented
5C22×#14523497C211yesmaximal, characteristic, complemented
6C23×D4×C131C211yesmaximal, characteristic, complemented
7C2×#14523513—26no—
8C2×#14523513C22112yescomplemented
9C22×Dic262C2211yescharacteristic, complemented
10C22×D524C2211yescharacteristic, complemented
11C22×C22⋊D131—21no—
12C22×C4×D131—21no—
13#16862272C22164yesdirect product factor
14C2×#14523501C22124yescomplemented
15C2×#14523498C22124yescomplemented
16C2×#14523497C22112yescomplemented
17C22×C131⋊C8C2212yescomplemented
18C25×C131—21no—
19C22×D4×C131—21no—
20C22×D4×C131—26no—
21C22×D4×C131C2212yes—
22C22×D4×C131C22112yescomplemented
23C23×C4×C131C2211yescharacteristic
24C23×D131—21no—
25#14523513—224no—
26#14523513C23116yescomplemented
27C2×Dic262—23no—
28C2×Dic262C2316yescomplemented
29C2×D524—23no—
30C2×D524C2316yescomplemented
31C2×C22⋊D131—212no—
32C22×Dic131—21no—
33C2×C4×D131—212no—
34#14523501C23132yescomplemented
35#14523498C23132yescomplemented
36#14523497C23116yescomplemented
37C2×C131⋊C8C23112yescomplemented
38C24×C131—22no—
39C24×C131—212no—
40C24×C131—48no—
41C24×C131D411yescharacteristic
42C2×D4×C131—26no—
43C2×D4×C131—212no—
44C2×D4×C131—224no—
45C2×D4×C131C23112yes—
46C2×D4×C131C23116yescomplemented
47C22×C4×C131D411yescharacteristic
48C22×C4×C131D416yes—
49C22×C4×C131C2311yescharacteristic
50C22×C4×C131C2316yes—
51C22×D131—26no—
52C22×D131—44no—
53Dic262—26no—
54Dic262C2414yescomplemented
55D524—26no—
56D524C2414yescomplemented
57C22⋊D131—216no—
58C2×Dic131—26no—
59C4×D131—216no—
60C131⋊C8C2418yescomplemented
61C23×C131—24no—
62C23×C131—212no—
63C23×C131—216no—
64C23×C131—448no—
65C23×C131—48no—
66C23×C131C2×D416yes—
67C23×C131C2×D411yescharacteristic
68D4×C131—212no—
69D4×C131—216no—
70D4×C131C2418yes—
71C2×C4×C131C2×D416yes—
72C2×C4×C131C2×D413yes—
73C2×C4×C131C2×D4112yes—
74C2×C4×C131C2413yes—
75C2×C4×C131C2414yes—
76C22×C131—212no—
77C22×C131—28no—
78C22×C131—432no—
79C22×C131—424no—
80C22×C131AGL(1,ℤ/8)14yescomplemented
81C22×C131C22×D414yes—
82C22×C131C22×D413yes—
83C2×D131—24no—
84C2×D131—412no—
85C4×C131C22×D414yes—
86C4×C131C22×D413yes—
87C4×C131C2511yescharacteristic
88Dic131—24no—
89C2×C131—24no—
90C2×C131—48no—
91C2×C131C2×AGL(1,ℤ/8)16yescomplemented
92C2×C131C23×D411yescharacteristic
93D131—44no—
94C131C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—1311nomaximal, Sylow
96C22×#41—1311no—
97C23×D4D13111yescharacteristic, complemented
98C2×AGL(1,ℤ/8)—13124no—
99C22×SD16—1312no—
100C22×D8—1312no—
101C22×M16—1311no—
102C25—21no—
103C2×#41—2626no—
104C2×#41—13112no—
105C22×Q8—1311no—
106C22×D4—26no—
107C22×D4—21no—
108C22×D4—2621no—
109C22×D4—1311no—
110C22×D4C2×D13112yes—
111C22×D4C2×D131112yescomplemented
112C23×C4—2621no—
113C23×C4C2×D13111yescharacteristic
114AGL(1,ℤ/8)—13164no—
115C2×SD16—13124no—
116C2×D8—13124no—
117C2×M16—13112no—
118C22×C8—1312no—
119C24—212no—
120C24—22no—
121C24—48no—
122C24—2621no—
123C24C22⋊D13111yescharacteristic
124#41—26224no—
125#41—13116no—
126C2×Q8—2623no—
127C2×Q8—1316no—
128C2×D4—26no—
129C2×D4—212no—
130C2×D4—224no—
131C2×D4—26212no—
132C2×D4—2623no—
133C2×D4—1316no—
134C2×D4C22×D131112yes—
135C2×D4C22×D131116yescomplemented
136C22×C4—2621no—
137C22×C4—26212no—
138C22×C4C22⋊D13116yes—
139C22×C4C22⋊D13111yescharacteristic
140C22×C4C22×D13116yes—
141C22×C4C22×D13111yescharacteristic
142SD16—13132no—
143D8—13132no—
144M16—13116no—
145C2×C8—13112no—
146C23—24no—
147C23—212no—
148C23—216no—
149C23—448no—
150C23—48no—
151C23—2626no—
152C23—5244no—
153C23C2×C22⋊D13116yes—
154C23C2×C22⋊D13111yescharacteristic
155Q8—2626no—
156Q8—1314no—
157D4—212no—
158D4—216no—
159D4—26216no—
160D4—2626no—
161D4—1314no—
162D4C23×D13118yes—
163C2×C4—2626no—
164C2×C4—26216no—
165C2×C4C2×C22⋊D13116yes—
166C2×C4C2×C22⋊D13113yes—
167C2×C4C2×C22⋊D131112yes—
168C2×C4C23×D13113yes—
169C2×C4C23×D13114yes—
170C8—1318no—
171C22—212no—
172C22—28no—
173C22—432no—
174C22—424no—
175C22—2624no—
176C22—52412no—
177C22#1686227214yesdirect product factor
178C22C22×C22⋊D13114yes—
179C22C22×C22⋊D13113yes—
180C4—2624no—
181C4C22×C22⋊D13114yes—
182C4C22×C22⋊D13113yes—
183C4C24×D13111yescharacteristic
184C2—24no—
185C2—48no—
186C2—5244no—
187C2C2×#1686227216yesdirect product factor
188C2C23×C22⋊D13111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D131
commutator, cosocleC4×C131C25
FrattiniC4C24×D131
FittingC23×D4×C131C2
socleC23×C131C2×D4
hypercenterC23×D4D131
hypocenterC131C22×AGL(1,ℤ/8)
LCSG ▷ C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14523513C2semidirect
2. C2×#16862272C2direct (N × H)
3. C22×#14523501C2semidirect
4. C22×#14523498C2semidirect
5. C22×#14523497C2semidirect
6. C23×D4×C131C2semidirect
8. C2×#14523513C22semidirect
9. C22×Dic262C22semidirect
10. C22×D524C22semidirect
13. #16862272C22direct (N × H)
14. C2×#14523501C22semidirect
15. C2×#14523498C22semidirect
16. C2×#14523497C22semidirect
17. C22×C131⋊C8C22semidirect
22. C22×D4×C131C22semidirect
26. #14523513C23semidirect
28. C2×Dic262C23semidirect
30. C2×D524C23semidirect
34. #14523501C23semidirect
35. #14523498C23semidirect
36. #14523497C23semidirect
37. C2×C131⋊C8C23semidirect
46. C2×D4×C131C23semidirect
54. Dic262C24semidirect
56. D524C24semidirect
60. C131⋊C8C24semidirect
80. C22×C131AGL(1,ℤ/8)semidirect
91. C2×C131C2×AGL(1,ℤ/8)semidirect
94. C131C22×AGL(1,ℤ/8)semidirect
97. C23×D4D131semidirect
111. C22×D4C2×D131semidirect
135. C2×D4C22×D131semidirect
177. C22#16862272direct (N × H)
187. C2C2×#16862272direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145235132
C2×#168622722
C22×#145235012
C22×#145234982
C22×#145234972
C23×D4×C1312
C22×AGL(1,ℤ/8)131