Finite Group Databasebeta

← C22×#22575478 #46825617

Subgroups — C22×#22575478 #46825617

Order 34432.

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×D1076C211yesmaximal, characteristic, complemented
2C2×#22575478C2124yesmaximal, direct product factor
3C22×#16864389C212yesmaximal, complemented
4C22×#16864387C212yesmaximal, complemented
5C22×#16864386C211yesmaximal, characteristic, complemented
6C22×#41×C269C211yesmaximal, characteristic, complemented
7C24×D269—21no—
8C22×D1076—21no—
9C22×D1076—26no—
10C22×D1076C2212yes—
11C22×D1076C22112yescomplemented
12#22575478C22164yesdirect product factor
13C2×#16864389C22124yescomplemented
14C2×#16864387C22124yescomplemented
15C2×#16864386C22112yescomplemented
16C22×#14524130C2212yescomplemented
17C2×#41×C269—26no—
18C2×#41×C269C22112yescomplemented
19C22×Q8×C269C2211yescharacteristic, complemented
20C22×D4×C269—21no—
21C22×D4×C269C2211yescharacteristic, complemented
22C23×C4×C269—21no—
23C23×C4×C269C2211yescharacteristic
24C23×D269—22no—
25C23×D269—212no—
26C23×D269—48no—
27C2×D1076—26no—
28C2×D1076—212no—
29C2×D1076—224no—
30C2×D1076C23112yes—
31C2×D1076C23116yescomplemented
32#16864389C23132yescomplemented
33#16864387C23132yescomplemented
34#16864386C23116yescomplemented
35C2×#14524130C23112yescomplemented
36C24×C269—21no—
37C24×C269D411yescharacteristic
38#41×C269—224no—
39#41×C269C23116yescomplemented
40C2×Q8×C269—23no—
41C2×Q8×C269C2316yescomplemented
42C2×D4×C269—212no—
43C2×D4×C269—23no—
44C2×D4×C269C2316yescomplemented
45C22×C4×C269—21no—
46C22×C4×C269—212no—
47C22×C4×C269D411yescharacteristic
48C22×C4×C269D416yes—
49C22×C4×C269C2311yescharacteristic
50C22×C4×C269C2316yes—
51C22×D269—212no—
52C22×D269—216no—
53C22×D269—448no—
54C22×D269—48no—
55D1076—212no—
56D1076—216no—
57D1076C2418yes—
58#14524130C2418yescomplemented
59C23×C269—24no—
60C23×C269—26no—
61C23×C269—44no—
62C23×C269C2×D416yes—
63C23×C269C2×D411yescharacteristic
64Q8×C269—26no—
65Q8×C269C2414yescomplemented
66D4×C269—216no—
67D4×C269—26no—
68D4×C269C2414yescomplemented
69C2×C4×C269—26no—
70C2×C4×C269—216no—
71C2×C4×C269C2×D416yes—
72C2×C4×C269C2×D413yes—
73C2×C4×C269C2×D4112yes—
74C2×C4×C269C2413yes—
75C2×C4×C269C2414yes—
76C22×C269—212no—
77C22×C269—24no—
78C22×C269—412no—
79C22×C269AGL(1,ℤ/8)14yescomplemented
80C22×C269C22×D414yes—
81C22×C269C22×D413yes—
82C2×D269—28no—
83C2×D269—432no—
84C2×D269—424no—
85C4×C269—24no—
86C4×C269C22×D414yes—
87C4×C269C22×D413yes—
88C4×C269C2511yescharacteristic
89C2×C269—24no—
90C2×C269—44no—
91C2×C269C2×AGL(1,ℤ/8)16yescomplemented
92C2×C269C23×D411yescharacteristic
93D269—48no—
94C269C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—2691nomaximal, Sylow
96C22×#41D26911yescharacteristic, complemented
97C23×D4—2691no—
98C2×AGL(1,ℤ/8)—26924no—
99C22×SD16—2692no—
100C22×D8—2692no—
101C22×M16—2691no—
102C25—5381no—
103C2×#41—26no—
104C2×#41C2×D269112yescomplemented
105C22×Q8C2×D26911yescharacteristic, complemented
106C22×D4—21no—
107C22×D4—2692no—
108C22×D4—26912no—
109C22×D4—5386no—
110C22×D4—5381no—
111C22×D4C2×D26911yescharacteristic, complemented
112C23×C4—21no—
113C23×C4C2×D26911yescharacteristic
114AGL(1,ℤ/8)—26964no—
115C2×SD16—26924no—
116C2×D8—26924no—
117C2×M16—26912no—
118C22×C8—2692no—
119C24—21no—
120C24—53812no—
121C24—5382no—
122C24—10768no—
123C24C22⋊D26911yescharacteristic
124#41—224no—
125#41C22×D269116yescomplemented
126C2×Q8—23no—
127C2×Q8C22×D26916yescomplemented
128C2×D4—212no—
129C2×D4—23no—
130C2×D4—26912no—
131C2×D4—26916no—
132C2×D4—53812no—
133C2×D4—5386no—
134C2×D4—53824no—
135C2×D4C22×D26916yescomplemented
136C22×C4—21no—
137C22×C4—212no—
138C22×C4C22⋊D26916yes—
139C22×C4C22⋊D26911yescharacteristic
140C22×C4C22×D26916yes—
141C22×C4C22×D26911yescharacteristic
142SD16—26932no—
143D8—26932no—
144M16—26916no—
145C2×C8—26912no—
146C23—24no—
147C23—26no—
148C23—44no—
149C23—53816no—
150C23—53812no—
151C23—107648no—
152C23—10768no—
153C23C2×C22⋊D26916yes—
154C23C2×C22⋊D26911yescharacteristic
155Q8—26no—
156Q8C23×D26914yescomplemented
157D4—216no—
158D4—26no—
159D4—2698no—
160D4—53816no—
161D4—53812no—
162D4C23×D26914yescomplemented
163C2×C4—26no—
164C2×C4—216no—
165C2×C4C2×C22⋊D26916yes—
166C2×C4C2×C22⋊D26913yes—
167C2×C4C2×C22⋊D269112yes—
168C2×C4C23×D26913yes—
169C2×C4C23×D26914yes—
170C8—2698no—
171C22—212no—
172C22—24no—
173C22—412no—
174C22—5388no—
175C22—107632no—
176C22—107624no—
177C22#2257547814yesdirect product factor
178C22C22×C22⋊D26914yes—
179C22C22×C22⋊D26913yes—
180C4—24no—
181C4C22×C22⋊D26914yes—
182C4C22×C22⋊D26913yes—
183C4C24×D26911yescharacteristic
184C2—24no—
185C2—44no—
186C2—10768no—
187C2C2×#2257547816yesdirect product factor
188C2C23×C22⋊D26911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D269
commutator, cosocleC4×C269C25
FrattiniC4C24×D269
FittingC22×#41×C269C2
socleC23×C269C2×D4
hypercenterC22×#41D269
hypocenterC269C22×AGL(1,ℤ/8)
LCSG ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C23×D1076C2semidirect
2. C2×#22575478C2direct (N × H)
3. C22×#16864389C2semidirect
4. C22×#16864387C2semidirect
5. C22×#16864386C2semidirect
6. C22×#41×C269C2semidirect
11. C22×D1076C22semidirect
12. #22575478C22direct (N × H)
13. C2×#16864389C22semidirect
14. C2×#16864387C22semidirect
15. C2×#16864386C22semidirect
16. C22×#14524130C22semidirect
18. C2×#41×C269C22semidirect
19. C22×Q8×C269C22semidirect
21. C22×D4×C269C22semidirect
31. C2×D1076C23semidirect
32. #16864389C23semidirect
33. #16864387C23semidirect
34. #16864386C23semidirect
35. C2×#14524130C23semidirect
39. #41×C269C23semidirect
41. C2×Q8×C269C23semidirect
44. C2×D4×C269C23semidirect
58. #14524130C24semidirect
65. Q8×C269C24semidirect
68. D4×C269C24semidirect
79. C22×C269AGL(1,ℤ/8)semidirect
91. C2×C269C2×AGL(1,ℤ/8)semidirect
94. C269C22×AGL(1,ℤ/8)semidirect
96. C22×#41D269semidirect
104. C2×#41C2×D269semidirect
105. C22×Q8C2×D269semidirect
111. C22×D4C2×D269semidirect
125. #41C22×D269semidirect
127. C2×Q8C22×D269semidirect
135. C2×D4C22×D269semidirect
156. Q8C23×D269semidirect
162. D4C23×D269semidirect
177. C22#22575478direct (N × H)
187. C2C2×#22575478direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D10762
C2×#225754782
C22×#168643892
C22×#168643872
C22×#168643862
C22×#41×C2692
C22×AGL(1,ℤ/8)269