Finite Group Databasebeta

← C22×#16862271 #30680693

Subgroups — C22×#16862271 #30680693

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
1C23×D524C211yesmaximal, characteristic, complemented
2C2×#16862271C2124yesmaximal, direct product factor
3C22×#14523500C212yesmaximal, complemented
4C22×#14523498C212yesmaximal, complemented
5C22×#14523497C211yesmaximal, characteristic, complemented
6C22×#41×C131C211yesmaximal, characteristic, complemented
7C24×D131—21no—
8C22×D524—21no—
9C22×D524—26no—
10C22×D524C2212yes—
11C22×D524C22112yescomplemented
12#16862271C22164yesdirect product factor
13C2×#14523500C22124yescomplemented
14C2×#14523498C22124yescomplemented
15C2×#14523497C22112yescomplemented
16C22×C131⋊C8C2212yescomplemented
17C2×#41×C131—26no—
18C2×#41×C131C22112yescomplemented
19C22×Q8×C131C2211yescharacteristic, complemented
20C22×D4×C131—21no—
21C22×D4×C131C2211yescharacteristic, complemented
22C23×C4×C131—21no—
23C23×C4×C131C2211yescharacteristic
24C23×D131—22no—
25C23×D131—212no—
26C23×D131—48no—
27C2×D524—26no—
28C2×D524—212no—
29C2×D524—224no—
30C2×D524C23112yes—
31C2×D524C23116yescomplemented
32#14523500C23132yescomplemented
33#14523498C23132yescomplemented
34#14523497C23116yescomplemented
35C2×C131⋊C8C23112yescomplemented
36C24×C131—21no—
37C24×C131D411yescharacteristic
38#41×C131—224no—
39#41×C131C23116yescomplemented
40C2×Q8×C131—23no—
41C2×Q8×C131C2316yescomplemented
42C2×D4×C131—212no—
43C2×D4×C131—23no—
44C2×D4×C131C2316yescomplemented
45C22×C4×C131—21no—
46C22×C4×C131—212no—
47C22×C4×C131D411yescharacteristic
48C22×C4×C131D416yes—
49C22×C4×C131C2311yescharacteristic
50C22×C4×C131C2316yes—
51C22×D131—212no—
52C22×D131—216no—
53C22×D131—448no—
54C22×D131—48no—
55D524—212no—
56D524—216no—
57D524C2418yes—
58C131⋊C8C2418yescomplemented
59C23×C131—24no—
60C23×C131—26no—
61C23×C131—44no—
62C23×C131C2×D416yes—
63C23×C131C2×D411yescharacteristic
64Q8×C131—26no—
65Q8×C131C2414yescomplemented
66D4×C131—216no—
67D4×C131—26no—
68D4×C131C2414yescomplemented
69C2×C4×C131—26no—
70C2×C4×C131—216no—
71C2×C4×C131C2×D416yes—
72C2×C4×C131C2×D413yes—
73C2×C4×C131C2×D4112yes—
74C2×C4×C131C2413yes—
75C2×C4×C131C2414yes—
76C22×C131—212no—
77C22×C131—24no—
78C22×C131—412no—
79C22×C131AGL(1,ℤ/8)14yescomplemented
80C22×C131C22×D414yes—
81C22×C131C22×D413yes—
82C2×D131—28no—
83C2×D131—432no—
84C2×D131—424no—
85C4×C131—24no—
86C4×C131C22×D414yes—
87C4×C131C22×D413yes—
88C4×C131C2511yescharacteristic
89C2×C131—24no—
90C2×C131—44no—
91C2×C131C2×AGL(1,ℤ/8)16yescomplemented
92C2×C131C23×D411yescharacteristic
93D131—48no—
94C131C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—1311nomaximal, Sylow
96C22×#41D13111yescharacteristic, complemented
97C23×D4—1311no—
98C2×AGL(1,ℤ/8)—13124no—
99C22×SD16—1312no—
100C22×D8—1312no—
101C22×M16—1311no—
102C25—2621no—
103C2×#41—26no—
104C2×#41C2×D131112yescomplemented
105C22×Q8C2×D13111yescharacteristic, complemented
106C22×D4—21no—
107C22×D4—2626no—
108C22×D4—2621no—
109C22×D4—1312no—
110C22×D4—13112no—
111C22×D4C2×D13111yescharacteristic, complemented
112C23×C4—21no—
113C23×C4C2×D13111yescharacteristic
114AGL(1,ℤ/8)—13164no—
115C2×SD16—13124no—
116C2×D8—13124no—
117C2×M16—13112no—
118C22×C8—1312no—
119C24—21no—
120C24—26212no—
121C24—2622no—
122C24—5248no—
123C24C22⋊D13111yescharacteristic
124#41—224no—
125#41C22×D131116yescomplemented
126C2×Q8—23no—
127C2×Q8C22×D13116yescomplemented
128C2×D4—212no—
129C2×D4—23no—
130C2×D4—26212no—
131C2×D4—2626no—
132C2×D4—26224no—
133C2×D4—13112no—
134C2×D4—13116no—
135C2×D4C22×D13116yescomplemented
136C22×C4—21no—
137C22×C4—212no—
138C22×C4C22⋊D13116yes—
139C22×C4C22⋊D13111yescharacteristic
140C22×C4C22×D13116yes—
141C22×C4C22×D13111yescharacteristic
142SD16—13132no—
143D8—13132no—
144M16—13116no—
145C2×C8—13112no—
146C23—24no—
147C23—26no—
148C23—44no—
149C23—26216no—
150C23—26212no—
151C23—52448no—
152C23—5248no—
153C23C2×C22⋊D13116yes—
154C23C2×C22⋊D13111yescharacteristic
155Q8—26no—
156Q8C23×D13114yescomplemented
157D4—216no—
158D4—26no—
159D4—26216no—
160D4—26212no—
161D4—1318no—
162D4C23×D13114yescomplemented
163C2×C4—26no—
164C2×C4—216no—
165C2×C4C2×C22⋊D13116yes—
166C2×C4C2×C22⋊D13113yes—
167C2×C4C2×C22⋊D131112yes—
168C2×C4C23×D13113yes—
169C2×C4C23×D13114yes—
170C8—1318no—
171C22—212no—
172C22—24no—
173C22—412no—
174C22—2628no—
175C22—52432no—
176C22—52424no—
177C22#1686227114yesdirect product factor
178C22C22×C22⋊D13114yes—
179C22C22×C22⋊D13113yes—
180C4—24no—
181C4C22×C22⋊D13114yes—
182C4C22×C22⋊D13113yes—
183C4C24×D13111yescharacteristic
184C2—24no—
185C2—44no—
186C2—5248no—
187C2C2×#1686227116yesdirect product factor
188C2C23×C22⋊D13111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D131
commutator, cosocleC4×C131C25
FrattiniC4C24×D131
FittingC22×#41×C131C2
socleC23×C131C2×D4
hypercenterC22×#41D131
hypocenterC131C22×AGL(1,ℤ/8)
LCSG ▷ C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C23×D524C2semidirect
2. C2×#16862271C2direct (N × H)
3. C22×#14523500C2semidirect
4. C22×#14523498C2semidirect
5. C22×#14523497C2semidirect
6. C22×#41×C131C2semidirect
11. C22×D524C22semidirect
12. #16862271C22direct (N × H)
13. C2×#14523500C22semidirect
14. C2×#14523498C22semidirect
15. C2×#14523497C22semidirect
16. C22×C131⋊C8C22semidirect
18. C2×#41×C131C22semidirect
19. C22×Q8×C131C22semidirect
21. C22×D4×C131C22semidirect
31. C2×D524C23semidirect
32. #14523500C23semidirect
33. #14523498C23semidirect
34. #14523497C23semidirect
35. C2×C131⋊C8C23semidirect
39. #41×C131C23semidirect
41. C2×Q8×C131C23semidirect
44. C2×D4×C131C23semidirect
58. C131⋊C8C24semidirect
65. Q8×C131C24semidirect
68. D4×C131C24semidirect
79. C22×C131AGL(1,ℤ/8)semidirect
91. C2×C131C2×AGL(1,ℤ/8)semidirect
94. C131C22×AGL(1,ℤ/8)semidirect
96. C22×#41D131semidirect
104. C2×#41C2×D131semidirect
105. C22×Q8C2×D131semidirect
111. C22×D4C2×D131semidirect
125. #41C22×D131semidirect
127. C2×Q8C22×D131semidirect
135. C2×D4C22×D131semidirect
156. Q8C23×D131semidirect
162. D4C23×D131semidirect
177. C22#16862271direct (N × H)
187. C2C2×#16862271direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D5242
C2×#168622712
C22×#145235002
C22×#145234982
C22×#145234972
C22×#41×C1312
C22×AGL(1,ℤ/8)131