Finite Group Databasebeta

← C23×#14519825 #30593903

Subgroups — C23×#14519825 #30593903

Order 15616.

59806 conjugacy classes of subgroups; only the 32139 normal subgroups are shown.

normal subgroups only

This group has 59806 conjugacy classes of subgroups — too many to draw. Only its 32139 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×#14519825C21112yesmaximal, direct product factor
2C23×#11758226C214yesmaximal, complemented
3C23×#11758222C216yesmaximal, complemented
4C23×Q8×D61C214yesmaximal, complemented
5C24×Q8×C61C211yesmaximal, characteristic, complemented
6C2×#14519825C2211792yesdirect product factor
7C22×#11758226C221224yescomplemented
8C22×#11758222C221336yescomplemented
9C23×Dic122C2216yes—
10C22×Q8×D61C221224yescomplemented
11C23×D244C2216yes—
12C23×C22⋊D61C2214yes—
13C23×C4×D61C22112yes—
14C23×Q8×C61C2214yes—
15C23×Q8×C61C22156yescomplemented
16C24×C4×C61C2213yes—
17C24×D61C2314yes—
18#14519825C2314096yesdirect product factor
19C2×#11758226C2311792yescomplemented
20C2×#11758222C2312688yescomplemented
21C22×Dic122C231168yes—
22C2×Q8×D61C2311792yescomplemented
23C22×D244C231168yes—
24C22×C22⋊D61C231112yes—
25C23×Dic61C2314yes—
26C22×C4×D61C231336yes—
27C25×C61C2311yescharacteristic
28C22×Q8×C61C231112yes—
29C22×Q8×C61C231448yescomplemented
30C23×C4×C61C2316yes—
31C23×C4×C61C23184yes—
32C24×C61—28no—
33C24×C61C24114yes—
34C24×C61C2411yescharacteristic
35C23×D61—232no—
36C23×D61C24156yes—
37C2×Q8×C61C241448yes—
38C2×Q8×C61C241512yescomplemented
39C22×C4×C61C24184yes—
40C22×C4×C61C241336yes—
41C2×C22⋊D61C241448yes—
42C22×Dic61C24156yes—
43#11758226C2412048yescomplemented
44Q8×D61C2412048yescomplemented
45#11758222C2413072yescomplemented
46C2×D244C241672yes—
47C2×C4×D61C2411344yes—
48C2×Dic122C241672yes—
49C23×C61—256no—
50C23×C612−1+418yescomplemented
51C23×C61C25128yes—
52C23×C61C2517yes—
53C22×D61—2224no—
54C22×D61C251112yes—
55Q8×C61C251256yes—
56C2×C4×C61C251168yes—
57C2×C4×C61C251192yes—
58C22⋊D61C251256yes—
59C2×Dic61C251112yes—
60D244C251384yes—
61C4×D61C251768yes—
62Dic122C251384yes—
63C23×2−1+4—611nomaximal, Sylow
64C22×C61—256no—
65C22×C61C2×2−1+4128yescomplemented
66C22×C61C2618yes—
67C22×C61C2617yes—
68C2×D61—2224no—
69C2×D61C26132yes—
70C4×C61C26148yes—
71Dic61C26132yes—
72C22×2−1+4—61112no—
73C23×#41—614no—
74C23×#41—616no—
75C24×Q8—614no—
76C24×Q8D6111yescharacteristic, complemented
77C2×C61—28no—
78C2×C61C22×2−1+4114yescomplemented
79C2×C61C2711yescharacteristic
80D61—232no—
81C2×2−1+4—611792no—
82C22×#41—61224no—
83C22×#41—61336no—
84C23×Q8—61224no—
85C23×Q8—616no—
86C23×Q8C2×D6114yes—
87C23×Q8C2×D61156yescomplemented
88C23×D4—614no—
89C23×D4—616no—
90C24×C4—6112no—
91C24×C4C2×D6113yes—
92C61C23×2−1+411yescharacteristic, complemented, Sylow
93C25—614no—
94C25C22×D6111yescharacteristic
952−1+4—614096no—
96C2×#41—611792no—
97C2×#41—612688no—
98C22×Q8—611792no—
99C22×Q8—61168no—
100C22×Q8C22×D611112yes—
101C22×Q8C22×D611448yescomplemented
102C22×D4—61112no—
103C22×D4—61168no—
104C23×C4—614no—
105C23×C4—61336no—
106C23×C4C22×D61184yes—
107C23×C4C22×D6116yes—
108C24—28no—
109C24—6156no—
110C24—12232no—
111C24C23×D61114yes—
112C24C23×D6111yescharacteristic
113#41—612048no—
114#41—613072no—
115C2×Q8—612048no—
116C2×Q8—61672no—
117C2×Q8C23×D611448yes—
118C2×Q8C23×D611512yescomplemented
119C2×D4—61448no—
120C2×D4—61672no—
121C22×C4—6156no—
122C22×C4—611344no—
123C22×C4C23×D61184yes—
124C22×C4C23×D611336yes—
125C23—256no—
126C23—61112no—
127C23—122224no—
128C23#1451982518yesdirect product factor
129C23C24×D61128yes—
130C23C24×D6117yes—
131Q8—61384no—
132Q8C24×D611256yes—
133D4—61256no—
134D4—61384no—
135C2×C4—61112no—
136C2×C4—61768no—
137C2×C4C24×D611168yes—
138C2×C4C24×D611192yes—
139C22—256no—
140C22—6132no—
141C22—122224no—
142C22C2×#14519825128yesdirect product factor
143C22C25×D6118yes—
144C22C25×D6117yes—
145C4—6132no—
146C4C25×D61148yes—
147C2—28no—
148C2—12232no—
149C2C22×#14519825114yesdirect product factor
150C2C26×D6111yescharacteristic
subgroupG/H
centerC24C23×D61
commutator, cosocleC2×C61C27
FrattiniC2C26×D61
FittingC24×Q8×C61C2
socleC24×C61C24
hypercenterC24×Q8D61
hypocenterC61C23×2−1+4
LCSG ▷ C2×C61 ▷ C61
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14519825C2direct (N × H)
2. C23×#11758226C2semidirect
3. C23×#11758222C2semidirect
4. C23×Q8×D61C2semidirect
5. C24×Q8×C61C2semidirect
6. C2×#14519825C22direct (N × H)
7. C22×#11758226C22semidirect
8. C22×#11758222C22semidirect
10. C22×Q8×D61C22semidirect
15. C23×Q8×C61C22semidirect
18. #14519825C23direct (N × H)
19. C2×#11758226C23semidirect
20. C2×#11758222C23semidirect
22. C2×Q8×D61C23semidirect
29. C22×Q8×C61C23semidirect
38. C2×Q8×C61C24semidirect
43. #11758226C24semidirect
44. Q8×D61C24semidirect
45. #11758222C24semidirect
50. C23×C612−1+4semidirect
65. C22×C61C2×2−1+4semidirect
76. C24×Q8D61semidirect
78. C2×C61C22×2−1+4semidirect
87. C23×Q8C2×D61semidirect
92. C61C23×2−1+4semidirect
101. C22×Q8C22×D61semidirect
118. C2×Q8C23×D61semidirect
128. C23#14519825direct (N × H)
142. C22C2×#14519825direct (N × H)
149. C2C22×#14519825direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198252
C23×#117582262
C23×#117582222
C23×Q8×D612
C24×Q8×C612
C23×2−1+461